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45°-45°-90° triangles can be used to evaluate trigonometric functions for multiples of π/4.

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How to Find the Length of the Hypotenuse

Last Updated: September 1, 2024 References

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been viewed 1,539,050 times.

All right triangles have one right (90-degree) angle, and the hypotenuse is the side that is opposite or the right angle, or the longest side of the right triangle. [1] X Research source The hypotenuse is the longest side of the triangle, and it’s also very easy to find using a couple of different methods. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem when you know the length of the other two sides of the triangle. It will then teach you to recognize the hypotenuse of some special right triangles that often appear on tests. It will finally teach you to find the length of the hypotenuse using the Law of Sines when you only know the length of one side and the measure of one additional angle.

Using the Pythagorean Theorem

• Right angles are often notated in textbooks and on tests with a small square in the corner of the angle. This special mark means "90 degrees."

• If your triangle has sides of 3 and 4, and you have assigned letters to those sides such that a = 3 and b = 4, then you should write your equation out as: 3 2 + 4 2 = c 2 .

• If a = 3, a 2 = 3 x 3, or 9. If b = 4, then b 2 = 4 x 4, or 16.
• When you plug those values into your equation, it should now look like this: 9 + 16 = c 2 .

• In our example, 9 + 16 = 25 , so you should write down 25 = c 2 .

• In our example, c 2 = 25 . The square root of 25 is 5 ( 5 x 5 = 25 , so Sqrt(25) = 5 ). That means c = 5 , the length of our hypotenuse!

Joseph Meyer

Use this visual trick to understand the Pythagorean Theorem. Imagine a right triangle with squares constructed on each leg and the hypotenuse. by rearranging the smaller squares within the larger square, the areas of the smaller squares (a² and b²) will add up visually to the area of the larger square (c²).

Finding the Hypotenuse of Special Right Triangles

• The first Pythagorean triple is 3-4-5 (3 2 + 4 2 = 5 2 , 9 + 16 = 25). When you see a right triangle with legs of length 3 and 4, you can instantly be certain that the hypotenuse will be 5 without having to do any calculations.
• The ratio of a Pythagorean triple holds true even when the sides are multiplied by another number. For example a right triangle with legs of length 6 and 8 will have a hypotenuse of 10 (6 2 + 8 2 = 10 2 , 36 + 64 = 100). The same holds true for 9-12-15 , and even 1.5-2-2.5 . Try the math and see for yourself!
• The second Pythagorean triple that commonly appears on tests is 5-12-13 (5 2 + 12 2 = 13 2 , 25 + 144 = 169). Also be on the lookout for multiples like 10-24-26 and 2.5-6-6.5 .

• To calculate the hypotenuse of this triangle based on the length of one of the legs, simply multiply the leg length by Sqrt(2).
• Knowing this ratio comes in especially handy when your test or homework question gives you the side lengths in terms of variables instead of integers.

• If you are given the length of the shortest leg (opposite the 30-degree angle,) simply multiply the leg length by 2 to find the length of the hypotenuse. For instance, if the length of the shortest leg is 4 , you know that the hypotenuse length must be 8 .
• If you are given the length of the longer leg (opposite the 60-degree angle,) multiply that length by 2/Sqrt(3) to find the length of the hypotenuse. For instance, if the length of the longer leg is 4 , you know that the hypotenuse length must be 4.62 .

Finding the Hypotenuse Using the Law of Sines

• To find the sine of an 80 degree angle, you will either need to key in sin 80 followed by the equal sign or enter key, or 80 sin . (The answer is -0.9939.)
• You can also type in "sine calculator" into a web search, and find a number of easy-to-use calculators that will remove any guesswork. [13] X Research source

• The Law of Sines can actually be used to solve any triangle, but only a right triangle will have a hypotenuse.

• For example, if you know that A = 40 degrees , then B = 180 – (90 + 40) . Simplify this to B = 180 – 130 , and you can quickly determine that B = 50 degrees .

• To continue our example, let's say that the length of side a = 10. Angle C = 90 degrees, angle A = 40 degrees, and angle B = 50 degrees.

• Using our example, we find that sin 40 = 0.64278761. To find the value of c, we simply divide the length of a by this number, and learn that 10 / 0.64278761 = 15.6 , the length of our hypotenuse!

Finding the Hypotenuse Using the Area

Practice Problems and Answers

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• ↑ https://www.mathsisfun.com/definitions/hypotenuse.html
• ↑ https://www.mathsisfun.com/definitions/pythagoras-theorem.html
• ↑ https://www.mathsisfun.com/pythagoras.html
• ↑ https://www.palmbeachstate.edu/prepmathlw/Documents/the_pythagorean_theorem.pdf
• ↑ https://www.omnicalculator.com/math/pythagorean-theorem
• ↑ https://www.learnalberta.ca/content/memg/division03/pythagorean%20theorem/index.html
• ↑ https://www.mathsisfun.com/pythagorean_triples.html
• ↑ https://www.ck12.org/book/ck-12-precalculus-concepts/section/4.3/
• ↑ https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/05%3A_Trigonometric_Functions/5.02%3A_Unit_Circle_-_Sine_and_Cosine_Functions
• ↑ https://www.rapidtables.com/calc/math/Sin_Calculator.html
• ↑ https://www.mathsisfun.com/algebra/trig-sine-law.html
• ↑ https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/11%3A_Applications_of_Trigonometry/11.02%3A_The_Law_of_Sines
• ↑ https://mathbitsnotebook.com/Geometry/TrigApps/TAUsingLawSines.html
• ↑ https://www.cut-the-knot.org/pythagoras/cosine2.shtml

About This Article

If you need to find the length of the hypotenuse of a right triangle, you can use the Pythagorean theorem if you know the length of the other two sides. Square the length of the 2 sides, called a and b, then add them together. Take the square root of the result to get the hypotenuse. If you want to learn how to find the hypotenuse using trigonometric functions, keep reading the article! Did this summary help you? Yes No

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Angle Properties, Postulates, and Theorems

In order to study geometry in a logical way, it will be important to understand key mathematical properties and to know how to apply useful postulates and theorems. A postulate is a proposition that has not been proven true, but is considered to be true on the basis for mathematical reasoning. Theorems , on the other hand, are statements that have been proven to be true with the use of other theorems or statements. While some postulates and theorems have been introduced in the previous sections, others are new to our study of geometry. We will apply these properties, postulates, and theorems to help drive our mathematical proofs in a very logical, reason-based way.

Before we begin, we must introduce the concept of congruency. Angles are congruent if their measures, in degrees, are equal. Note : “congruent” does not mean “equal.” While they seem quite similar, congruent angles do not have to point in the same direction. The only way to get equal angles is by piling two angles of equal measure on top of each other.

We will utilize the following properties to help us reason through several geometric proofs.

Reflexive Property

A quantity is equal to itself.

Symmetric Property

If A = B , then B = A .

Transitive Property

If A = B and B = C , then A = C .

Addition Property of Equality

If A = B , then A + C = B + C .

Angle Postulates

Angle addition postulate.

If a point lies on the interior of an angle, that angle is the sum of two smaller angles with legs that go through the given point.

Consider the figure below in which point T lies on the interior of ?QRS . By this postulate, we have that ?QRS = ?QRT + ?TRS . We have actually applied this postulate when we practiced finding the complements and supplements of angles in the previous section.

Corresponding Angles Postulate

If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.

Converse also true : If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.

The figure above yields four pairs of corresponding angles.

Parallel Postulate

Given a line and a point not on that line, there exists a unique line through the point parallel to the given line.

The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry .

There are an infinite number of lines that pass through point E , but only the red line runs parallel to line CD . Any other line through E will eventually intersect line CD .

Angle Theorems

Alternate exterior angles theorem.

If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate exterior angles are congruent, then the lines are parallel.

The alternate exterior angles have the same degree measures because the lines are parallel to each other.

Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel.

The alternate interior angles have the same degree measures because the lines are parallel to each other.

Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Right Angles Theorem

All right angles are congruent.

Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

Converse also true : If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

The sum of the degree measures of the same-side interior angles is 180°.

Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.

The vertical angles have equal degree measures. There are two pairs of vertical angles.

(1) Given: m?DGH = 131

Find: m?GHK

First, we must rely on the information we are given to begin our proof. In this exercise, we note that the measure of ?DGH is 131° .

From the illustration provided, we also see that lines DJ and EK are parallel to each other. Therefore, we can utilize some of the angle theorems above in order to find the measure of ?GHK .

We realize that there exists a relationship between ?DGH and ?EHI : they are corresponding angles. Thus, we can utilize the Corresponding Angles Postulate to determine that ?DGH??EHI .

Directly opposite from ?EHI is ?GHK . Since they are vertical angles, we can use the Vertical Angles Theorem , to see that ?EHI??GHK .

Now, by transitivity , we have that ?DGH??GHK .

Congruent angles have equal degree measures, so the measure of ?DGH is equal to the measure of ?GHK .

Finally, we use substitution to conclude that the measure of ?GHK is 131° . This argument is organized in two-column proof form below.

(2) Given: m?1 = m?3

Prove: m?PTR = m?STQ

We begin our proof with the fact that the measures of ?1 and ?3 are equal.

In our second step, we use the Reflexive Property to show that ?2 is equal to itself.

Though trivial, the previous step was necessary because it set us up to use the Addition Property of Equality by showing that adding the measure of ?2 to two equal angles preserves equality.

Then, by the Angle Addition Postulate we see that ?PTR is the sum of ?1 and ?2 , whereas ?STQ is the sum of ?3 and ?2 .

Ultimately, through substitution , it is clear that the measures of ?PTR and ?STQ are equal. The two-column proof for this exercise is shown below.

(3) Given: m?DCJ = 71 , m?GFJ = 46

Prove: m?AJH = 117

We are given the measure of ?DCJ and ?GFJ to begin the exercise. Also, notice that the three lines that run horizontally in the illustration are parallel to each other. The diagram also shows us that the final steps of our proof may require us to add up the two angles that compose ?AJH .

We find that there exists a relationship between ?DCJ and ?AJI : they are alternate interior angles. Thus, we can use the Alternate Interior Angles Theorem to claim that they are congruent to each other.

By the definition of congruence , their angles have the same measures, so they are equal.

Now, we substitute the measure of ?DCJ with 71 since we were given that quantity. This tells us that ?AJI is also 71° .

Since ?GFJ and ?HJI are also alternate interior angles, we claim congruence between them by the Alternate Interior Angles Theorem .

The definition of congruent angles once again proves that the angles have equal measures. Since we knew the measure of ?GFJ , we just substitute to show that 46 is the degree measure of ?HJI .

As predicted above, we can use the Angle Addition Postulate to get the sum of ?AJI and ?HJI since they compose ?AJH . Ultimately, we see that the sum of these two angles gives us 117° . The two-column proof for this exercise is shown below.

(4) Given: m?1 = 4x + 9 , m?2 = 7(x + 4)

In this exercise, we are not given specific degree measures for the angles shown. Rather, we must use some algebra to help us determine the measure of ?3 . As always, we begin with the information given in the problem. In this case, we are given equations for the measures of ?1 and ?2 . Also, we note that there exists two pairs of parallel lines in the diagram.

By the Same-Side Interior Angles Theorem , we know that that sum of ?1 and ?2 is 180 because they are supplementary.

After substituting these angles by the measures given to us and simplifying, we have 11x + 37 = 180 . In order to solve for x , we first subtract both sides of the equation by 37 , and then divide both sides by 11 .

Once we have determined that the value of x is 13 , we plug it back in to the equation for the measure of ?2 with the intention of eventually using the Corresponding Angles Postulate . Plugging 13 in for x gives us a measure of 119 for ?2 .

Finally, we conclude that ?3 must have this degree measure as well since ?2 and ?3 are congruent . The two-column proof that shows this argument is shown below.

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HA Theorem (Hypotenuse Angle)

Hypotenuse angle (ha) theorem (proof & examples).

Geometry may seem like no laughing matter, but this lesson has more than one HA moment. That's because this is all about the Hypotenuse Angle Theorem, or HA Theorem, which allows you to prove congruence of two right triangles using only their hypotenuses and acute angles.

What are right triangles?

Before we start roaring with all the laughs this lesson brings (!), let's make sure we have a firm understanding of  right triangles . A right triangle is called that because it has one right angle ( 90° ), which means the other two angles must be acute ( less than 90° ).

This right angle limits the possible measurements of the other two angles. Together they must add up to  90°  because all interior angles of  any  triangle -- right, scalene, obtuse -- must add to  180° . Subtracting the one right angle from  180° leaves only  90° to be shared by the two remaining angles, making both of them acute angles.

Can you see what would happen if we knew something about one of those two acute angles? We would know something about the remaining angle. Let's take a look at how that plays a role in the HA Theorem.

Now that we have right triangles right in our heads, let's look at the HA Theorem.

The HA Theorem states; If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another triangle, then the two triangles are congruent.

Congruence does not mean just somewhat alike; it means the two triangles will be identical; every side and every angle, equal between the triangles. That's a tall order, and we are claiming to get it just by knowing one side (the hypotenuse) and one angle.

Remember, though, that we already know a second angle. We know the right angle that forms a square corner.

Here are two right triangles,  △ZAP  and  △HOT:

They are both facing with their hypotenuses to the right, which means their right angles are to the left -- HA!  (A small touch of triangle humor.)

Proving the HA Theorem

Notice  ∠A  and  ∠O  are right angles, indicated by the little square  □  tucked into the interior angles. We are told that the hypotenuses,  ZP  and  HT , are congruent, which is why they have the little matching hash marks. We are also told that acute  ∠Z  and  ∠H  are congruent, shown by their own hash marks.

If we knew only that much geometry, we would be stumped. We could say the six parts (three sides and three angles) have only three parts congruent, and they are not all touching.

Look carefully --  ∠A  and  ∠Z  are consecutive angles in our left right triangle … uh … our right triangle on the left  (HA humor again) . Those two angles do not include a known side between them. We have no idea if  ZA  is congruent to  HO .

Check out the remaining angles.  ∠P  and  ∠T . What do we know about them? We know they are congruent. Why?

They must be congruent because of what we said earlier. Given two of the angles, the third angle is found by subtracting the two given angles from  180° . We do not even need numbers for  ∠Z  and  ∠H ; they are congruent, so  ∠P  and  ∠T  are congruent.

So what, you say? If we know that all three angles are congruent, and we know that included sides between angles are congruent, then we have the ASA Postulate !

Recall that ASA tells us that triangles are congruent if any two angles and their included side are equal in the triangles.

Building off that handy right angle, we worked out two included angles, on either side of the hypotenuse. Now we have all these congruences:

∠A ≅ ∠O (two right angles, which we used to deduce ∠P ≅ ∠T )

∠Z ≅ ∠H (a given)

Hypotenuse ZP ≅ hypotenuse HT (a given)

∠P ≅ ∠T (deduced from ∠Z ≅ ∠H and ∠A ≅ ∠O )

The last three congruences are the ASA Postulate at work. HA! We did some amazing detective work there.

Do you need to go through all that every time you want to show two right triangles are congruent? No. You can use the HA Theorem! HA!  (We told you this would have more than one HA moment.)

Instead of going through the lengthy process of finding the third angle congruent, hauling out the ASA Postulate, and declaring the two right triangles congruent, you can easily apply the HA Theorem.

HA theorem practice proof

You cannot show off the HA Theorem with something as simple as two twin right triangles, charming as  △ZAP  and  △HOT  are. What about something trickier, like two right triangles seeming to slide past each other, like these:

These two right triangles were constructed from line  OA , intersected by line  FB , crossing at  Point G .

Right  △FOG  shares a vertex,  Point G , with  △BAG . We see that  ∠O  and  ∠A  are right angles, and the little hash marks tell us hypotenuses  FG  and  BG  are congruent. What else are we told? Nothing!

Are you ready to have a HA moment? We know sides  OG  and  AG  form a straight line, because they are segments of line  OA . We know that both right triangles share  Point G , creating two interior angles ( ∠FGO  and  ∠BGA ).

Those interior angles are vertical angles of two crossing lines! HA! Vertical angles are congruent. Now we have another set of congruences. Let's make a list:

∠FGO ≅ ∠BGA

Hypotenuse FG ≅ Hypotenuse GB

With just the hypotenuse and one acute angle, we now release the power of the HA Theorem and state that:

Lesson summary

Though it may not have been a barrel of laughs, by exploring the HA Theorem you are now able to recall and state the  Hypotenuse Angle (HA) Theorem , demonstrate the HA Theorem's connection to the Angle Side Angle Theorem, and mathematically prove the HA Theorem.

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The Pythagorean Theorem

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One of the best known mathematical formulas is Pythagorean Theorem, which provides us with the relationship between the sides in a right triangle. A right triangle consists of two legs and a hypotenuse. The two legs meet at a 90° angle and the hypotenuse is the longest side of the right triangle and is the side opposite the right angle.

The Pythagorean Theorem tells us that the relationship in every right triangle is:

$$a^{2}+b^{2}=c^{2}$$

$$C^{2}=6^{2}+4^{2}$$

$$C^{2}=36+16$$

$$C^{2}=52$$

$$C=\sqrt{52}$$

$$C\approx 7.2$$

There are a couple of special types of right triangles, like the 45°-45° right triangles and the 30°-60° right triangle.

Because of their angles it is easier to find the hypotenuse or the legs in these right triangles than in all other right triangles.

In a 45°-45° right triangle we only need to multiply one leg by √2 to get the length of the hypotenuse.

We multiply the length of the leg which is 7 inches by √2 to get the length of the hypotenuse.

$$7\cdot \sqrt{2}\approx 9.9$$

In a 30°-60° right triangle we can find the length of the leg that is opposite the 30° angle by using this formula:

$$a=\frac{1}{2}\cdot c$$

To find a, we use the formula above.

$$a=\frac{1}{2}\cdot 14$$

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Find the sides of this right triangle

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Right Triangle Calculator

Please provide values for any three of the six fields below. At least one of those values must be a side length.

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About the Right Triangle Calculator

This right triangle calculator lets you calculate the length of the hypotenuse or a leg or the area of a right triangle. For each case, you may choose from different combinations of values to input.

Also, the calculator will give you not just the answer, but also a step by step solution. So you can use it as a great tool to learn about right triangles.

Usage Guide

I. valid inputs.

The calculator needs exactly two inputs, at least one of which must be a side (a leg or the hypotenuse).

All inputs can be in any of the three number formats listed below.

Note — The input for each of the two angles must be between 0 \hspace{0.2em} 0 \hspace{0.2em} 0 and 90 \hspace{0.2em} 90 \hspace{0.2em} 90 .

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what a right triangle is a few of the basic concepts related to it.

Right Triangles

A right triangle (or right-angled triangle) is a triangle in which one of the three internal angles is a right angle ( 90 ° \hspace{0.2em} 90 \degree \hspace{0.2em} 90° ).

The longest side in a right triangle (known as the hypotenuse) is the side opposite the right angle.

Pythagorean Theorem

Now one feature of right triangles that makes them so useful and important is that they obey the pythagorean theorem.

And according to the Pythagorean theorem , the square of the hypotenuse is equal to the sum of the squares of the other two sides (called legs). So, for the triangle above –

The two legs of a right triangle measure 5  cm \hspace{0.2em} 5 \text{ cm} \hspace{0.2em} 5  cm and 12  cm \hspace{0.2em} 12 \text{ cm} \hspace{0.2em} 12  cm in length. Find the length of the hypotenuse.

If x \hspace{0.2em} x \hspace{0.2em} x denotes the length of the hypotenuse, according the pythagorean theorem —

Taking the square root on both sides, we have

So the hypotenuse has a length of 13  cm \hspace{0.2em} 13 \text{ cm} \hspace{0.2em} 13  cm .

Trigonometric Ratios

Another concept that makes right triangles great for the study of triangles is that of trigonometric ratios. Here's a brief explanation.

We'll start with the figure below.

Now, when we talk about trigonometric ratios, those ratios are with respect to a reference angle. And that reference angle can be any of the two acute angles in a right triangle.

Also, as the figure shows, we have special names for the two legs. The one opposite to the angle is termed "opposite" and the one adjacent to it is called "adjacent".

Trigonometric ratios are ratios between the side lengths of a right triangle. And the value of a trigonometric ratio depends on the reference angle alone.

Here's a table listing the six trigonometric ratios.

As mentioned earlier, values for trigonometric ratios depend only on the reference angle ( θ ) \hspace{0.2em} (\theta) \hspace{0.2em} ( θ ) . This is crucial, as you'll see in the second example below.

In △ A B C \hspace{0.2em} \triangle ABC \hspace{0.2em} △ A BC , ∠ C = 90 ° \hspace{0.2em} \angle C = 90 \degree \hspace{0.2em} ∠ C = 90° , ∠ B = 40 ° \hspace{0.2em} \angle B = 40 \degree \hspace{0.2em} ∠ B = 40° , and A C = 5  in \hspace{0.2em} AC = 5 \text{ in} \hspace{0.2em} A C = 5  in . Find the lengths of A B \hspace{0.2em} AB \hspace{0.2em} A B and B C \hspace{0.2em} BC \hspace{0.2em} BC .

Let's start with a rough labeled sketch of the triangle.

Next, we know

So, for the given triangle

Now, because the value of a trigonometric ratio depends only on the angle, sin ⁡ 40 ° \hspace{0.2em} \sin 40 \degree \hspace{0.2em} sin 40° would be a constant. As a calculator would tell you, sin ⁡ 40 ° ≈ 0.59 \hspace{0.2em} \sin 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.59 \hspace{0.2em} sin 40° ≈ 0.59 .

Substituting the value of sin ⁡ 40 ° \hspace{0.2em} \sin 40 \degree \hspace{0.2em} sin 40° into the equation above, we have

Substituting tan ⁡ 40 ° ≈ 0.73 \hspace{0.2em} \tan 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.73 \hspace{0.2em} tan 40° ≈ 0.73 ,

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Right Triangle Calculator

Table of contents

The right triangle calculator will help you find the lengths of the sides of a right-angled triangle . This triangle solver will also teach you how to find the area of a right triangle as well as give plenty of information about the practical uses of a right triangle.

What is a right triangle (or right-angled triangle)?

First things first, let's explain what a right triangle is. The definition is very simple and might even seem obvious for those who already know it: a right-angled triangle is a triangle where one and only one of the angles is exactly 90° . The other two angles will clearly be smaller than the right angle because the sum of all angles in a triangle is always 180°.

In a right-angled triangle, we define the sides in a special way. The side opposing the right angle is always the biggest in the triangle and receives the name of "hypotenuse". The other two sides are called catheti. The relationship between the hypotenuse and each cathetus is straightforward, as we will see when we talk about Pythagoras' theorem .

Hypotenuse calculator

If all you want to calculate is the hypotenuse of a right triangle, this page and its right triangle calculator will work just fine. However, we would also recommend using the dedicated tool we have developed at Omni Calculators: the hypotenuse calculator . The hypotenuse is opposite the right angle and can be solved by using the Pythagorean theorem. In a right triangle with cathetus a and b and with hypotenuse c , Pythagoras' theorem states that: a² + b² = c² .

To solve for c , take the square root of both sides to get c = √(b²+a²) . We can consider this extension of the Pythagorean theorem as a "hypotenuse formula". A Pythagorean theorem calculator is also an excellent tool for calculating the hypotenuse.

Let's now solve a practical example of what it would take to calculate the hypotenuse of a right triangle without using any calculators available at Omni:

• Obtain the values of a and b .
• Square a and b .
• Sum up both values: a² + b² .
• Take the square root of the result.
• The square root will yield positive and negative results. Since we are dealing with length, disregard the negative one.
• The resulting value is the value of the hypotenuse c .

Now let's see what the process would be using one of Omni's calculators , for example, the right triangle calculator on this web page:

• Insert the value of a and b into the calculator; and
• Obtain the value of c immediately;
• As a bonus, you will get the value of the area for such a triangle.

How to find the area of a right triangle

We have already seen that calculating the area of a right angle triangle is very easy with the right triangle calculator. At Omni Calculators, we have a calculator specifically designed for that purpose as well: area of a right triangle calculator . Let's now see a bit more in-depth how to calculate areas of right triangles.

The method for finding the area of a right triangle is quite simple. All that you need are the lengths of the base and the height . In a right triangle, the base and the height are the two sides that form the right angle. Since multiplying these two values together would give the area of the corresponding rectangle, and the triangle is half of that, the formula is:

area = ½ × base × height .

If you don't know the base or the height, you can find it using the Pythagorean theorem. Try the right triangle calculator to check your calculations or calculate the area of triangles with sides that have larger or decimal-value lengths.

Other considerations when dealing with a right triangle

Now we're gonna see other things that can be calculated from a right triangle using some of the tools available at Omni. The sides of a triangle have a certain gradient or slope. The formula for the slope is

slope = (y₂ - y₁)/(x₂ - x₁) .

So if the coordinates are (1,-6) and (4,8) , the slope of the segment is (8 + 6)/(4 - 1) = 14/3 . An easy way to determine if the triangle is right, and you just know the coordinates, is to see if the slopes of any two lines multiply to equal -1 .

There is an easy way to convert angles from radians to degrees and degrees to radians with the use of the angle conversion:

• If an angle is in radians – multiply by 180/π ; and
• If an angle is in degrees – multiply by π/180 .

Sometimes you may encounter a problem where two or even three side lengths are missing. In such cases, the right triangle calculator, hypotenuse calculator, and method on how to find the area of a right triangle won't help. You have to use trigonometric functions to solve for these missing pieces.

Special triangles

The right triangle is just one of the many special triangles that exist. These triangles have one or several special characteristics that make them unique. For example, as we have seen, the right triangle has a right angle and hence a hypotenuse, which makes it a unique kind of triangle. Aside from the right-angled triangle, there are other special triangles with interesting properties.

One of the most known special triangles is the equilateral triangle, which has three equal sides and all its angles are 60°. This makes it much simpler to make a triangle solver calculator evaluate different parameters of such a triangle.

Another of special triangles is the isosceles triangle, which has 2 sides of equal length , and hence two angles of the same size. As opposed to the equilateral triangle, isosceles triangles come in many different shapes.

There are many other special triangles. However, we will now take a look at a few very special right triangles that, besides being right-angled triangles, they have other unique properties that make them interesting.

Special right triangles

The so-called "45 45 90" triangle is probably the most special among all the special right triangles. This is a right-angled triangle that is also an isosceles triangle . Both its catheti are of the same length (isosceles), and it also has the peculiarity that the non-right angles are exactly half the size of the right angle that gives the name to the right triangle.

This right triangle is the kind of triangle that you can obtain when you divide a square by its diagonal . That is why both catheti (sides of the square) are of equal length. For those interested in knowing more about the most special of the special right triangles, we recommend checking out the 45 45 90 triangle calculator made for this purpose.

Another fascinating triangle from the group of special right triangles is the so-called "30 60 90" triangle. The name comes from having one right angle (90°), then one angle of 30°, and another of 60°. These angles are special because of the values of their trigonometric functions (cosine, sine, tangent, etc.). The consequences of this can be seen and understood with the 30 60 90 triangle calculator , but for those who are too lazy to click the link, we will summarize some of them here . Assuming that the shorter side is of length a , the triangle follows:

• The second length is equal to a√3 ;
• The hypotenuse is 2a ;
• The area is equal to (a²√3)/2 ; and
• The perimeter equals a(3 + √3) .

Right angled triangles and parallelograms

It might seem at first glance that a right triangle and a parallelogram do not have anything in common. How can a triangle solver help you understand a parallelogram? The reality is that any parallelogram can be decomposed into 2 or more right triangles . Let's take an example of the rectangle, which is the easiest one to see it.

Imagine a rectangle, any rectangle. Now draw a trace on one of the diagonals of this rectangle. If we separate the rectangle by the diagonal, we will obtain two right-angled triangles . Looking at the triangles, there is no need to use the right triangle calculator to see that both are equal, so their areas will be the same. This means that the area of the rectangle is double that of each triangle .

If we think about the equations, it makes sense since the area of a rectangle of sides a and b is exactly area = a × b , while for the right triangle is area = base × height / 2 which, in this case, would mean area = a × b /2 . This is precisely what we already saw by just cutting the rectangle by the diagonal.

It was a simple example of a rectangle, but the same applies to the area of a square. For other parallelograms, the process becomes a bit more complicated (it might involve up to 4 right triangles of different sizes). Still, with a bit of skill, you can use the same idea and calculate the area of a parallelogram using right-angled triangles. You can, of course, be even more efficient and just use our calculator.

Pythagorean triplets, triangles meet maths

Geometry and polygons, especially triangles, always come together. The properties of some triangles, like right triangles, are usually interesting and shocking, even for non-mathematicians. We will now have a look at an interesting set of numbers very closely related to right-angled triangles that mathematicians love, and maybe you will too.

These sets of numbers are called the Pythagorean triplets and are sets of 3 integers (let's call them a , b , and c ) and satisfy the Pythagorean theorem: a² + b² = c² . That is, they could form a right triangle with sides of length a , b , and c . The amount of numbers that satisfy this relationship is limited, but mathematicians find joy in searching for new ones.

Aside from the curiosity factor of this relationship, it has some interesting properties that are exploited in cryptography . Given the applications that one might find for such sets of numbers, mathematicians have explored even beyond, using 4, 5… and more sets of numbers that satisfy a similar relation in which the sum of the squares of all the numbers except for one, give the square of the number that's left.

Also very connected to these Pythagorean triplets is the infamous Fermat's last theorem in which the almost legendary cryptic mathematician Pierre Fermat stated that there could not be a set of three integer numbers that would satisfy the relation: aⁿ + bⁿ = cⁿ for n bigger than 2. This conjecture was only proven in 1995, more than 300 years after it was first formulated , and it's considered one of the most important mathematical problems of the century.

Shadows and right triangles (radius of the Earth)

We have talked a lot about triangles, particularly right triangles, and their applications in maths and geometry. What we haven't talked about yet is the usefulness of right triangles for calculating things in real life . It might seem like the applications outside of geometry are limited, but let's have a look at shadows.

Yes, shadows. The dark shade projected by an object when it is illuminated. If you were to look at the shape made by the shadow, the object, and the ground, you would notice that it is, in fact, a right-angled triangle! At least, it is when the object is perfectly vertical and the ground is horizontal. Most of the time, this is the case, or at least close enough. This means that we can use the right triangle calculator to find different pieces of information about objects under the sun. Let's see how.

Imagine that you have a building of which we want to know the height , but you cannot measure it directly because it's too high to drop a measuring tape from the top. What you can do is measure the length of the shadow on the street. Then, with the help of any angle-measuring tool and a piece of paper, you can find out the angle between the shadow and the ground. Knowing that the angle between the building and the ground is 90°, you can obtain the value of the height of the building.

Using this technique, you can measure the height of many objects as long as you have a bright sunny day or other light sources to illuminate the object. In fact, this used to be a very common measuring technique in the olden days. Probably the most interesting and mind-blowing use of right triangles is that of Eratosthenes, who managed to use right-angled triangles and shadows to measure the radius of the Earth , and now we are gonna explain how he did it.

Eratosthenes noticed that on the summer solstice there was a place on Earth where the wells did not have a shadow at midday, i.e., the sun shone straight down onto them. Noting this, he set up a column of a known height at a known distance from that well and measured the size of the shadow created by the pole at noon on the solstice day. Then using right-angled triangles and trigonometry, he was able to determine the angle going from the center of the Earth between the well and the pole, as well as the radius of the Earth , based on the known distance between these two points.

It was quite an astonishing feat that now you can do much more easily, by just using the Omni calculators that we have created for you .

Which side lengths form a right triangle?

Side lengths a , b , c form a right triangle if, and only if, they satisfy a² + b² = c² . We say these numbers form a Pythagorean triple .

Do 2, 3, and 4 make a right triangle?

We have 4² = 16 and 2² + 3² = 4 + 9 = 13 , so the sum of squares of the two smaller numbers is NOT equal to the square of the largest number. That is, 2, 3, and 4 do not form a Pythagorean triple; in other words, there is no right triangle with sides 2, 3, and 4.

How do I find the circumcenter of a right angle triangle?

For a right-angled triangle, the circumcenter, i.e., the center of the circle circumscribed on the triangle, coincides with the midpoint of the triangle's longest side (its hypotenuse ).

How do I find the orthocenter of a right angle triangle?

The orthocenter of a right-angled triangle, i.e., the point where the triangle's altitudes intersect, coincides with the triangle's vertex of the right angle.

A labelled right triangle.

© Omni Calculator

Hypotenuse (c)

• Math Formulas

Right Triangle Formula

The right triangle formula  includes the formulas of the area of a right triangle, along with its perimeter and length of the hypotenuse formula.

In geometry, you come across different types of figures, the properties of which, set them apart from one another. One common figure among them is a triangle. A triangle is a closed figure, a polygon , with three sides. It has 3 vertices and its 3 sides enclose 3 interior angles of the triangle. The sum of the three interior angles in a triangle is always 180 degrees.  

The most common types of triangles that we study are equilateral, isosceles, scalene and right-angled triangles. In this section, we will talk about the right triangle formula, also called the right-angled triangle formulas. 

Right Triangle

A right triangle is the one in which the measure of any one of the interior angles is 90 degrees. It is to be noted here that since the sum of interior angles in a triangle is 180 degrees, only 1 of the 3 angles can be a right angle.

Read more:   Right Angled Triangle

If the other two angles are equal, that is 45 degrees each, the triangle is called an isosceles right-angled triangle. However, if the other two angles are unequal, it is a scalene right-angled triangle.

The most common application of right-angled triangles can be found in trigonometry. In fact, the relation between its angles and sides forms the basis for trigonometry.

Area of a right triangle – Formula

The area of a right triangle is the region covered by its boundaries or within its three sides.

The formula to find the area of a right triangle is given by:

Where b and h refer to the base and height of the triangle, respectively.

Perimeter of a right triangle – Formula

The perimeter of a right triangle is a distance covered by its boundary or the sum of all its three sides.

The formula to find the perimeter of a triangle is given by:

Where a, b and c are the measure of its three sides.

Hypotenuse of a right triangle – Formula

A right triangle has three sides called the base, the perpendicular and the hypotenuse. The hypotenuse is the longest side of the right triangle.  Pythagoras Theorem defines the relationship between the three sides of a right-angled triangle. Thus, if the measure of two of the three sides of a right triangle is given, we can use the Pythagoras Theorem to find out the third side.

In the figure given above, ∆ABC is a right-angled triangle that is right-angled at B. The side opposite to the right angle, which is the longest side, is called the hypotenuse of the triangle. In ∆ABC , AC is the hypotenuse. Angles A and C are the acute angles. We name the other two sides (apart from the hypotenuse) as the ‘base’ or ‘perpendicular’ depending on which of the two angles we take as the basis for working with the triangle.

Derivation of Right Triangle Formula

Consider a right-angled triangle ABC which has B as 90 degrees and AC is the hypotenuse.

Now we flip the triangle over its hypotenuse such that a rectangle ABCD with width h and length b is formed.

You already know that area of a rectangle is given as the product of its length and width, that is, length x breadth.

Hence, area of the rectangle ABCD = b x h

As you can see, the area of the right angled triangle ABC is nothing but one-half of the area of the rectangle ABCD.

Thus, \(\begin{array}{l}Area ~of \Delta ABC = \frac{1}{2} Area ~of~ rectangle ABCD\end{array} \)

Hence, area of a right angled triangle, given its base b and height

\(\begin{array}{l}A= \frac{1}{2} bh\end{array} \)

Solved Examples on Right Triangle Formula

Question 1: The length of two sides of a right angled triangle is 5 cm and 8 cm. Find:

• Length of its hypotenuse
• Perimeter of the triangle
• Area of the triangle

Solution: Given,

One side a = 5cm

Other side b = 8 cm

• The length of the hypotenuse is,

Using Pythagoras theorem,

\(\begin{array}{l}Hypotenuse^{2} = Perpendicular^{2} + Base^{2}\end{array} \)

\(\begin{array}{l}c^{2} = a^{2} +b^{2}\end{array} \)

\(\begin{array}{l}c^{2} = 5^{2} +8^{2}\end{array} \)

\(\begin{array}{l}c= \sqrt{25+64}= \sqrt{89}= 9.43cm\end{array} \)

Perimeter of the right triangle = a + b + c = 5 + 8 + 9.43 = 22.43 cm

\(\begin{array}{l}Area ~of~ a~ right ~triangle = \frac{1}{2} bh\end{array} \)

Here, area of the right triangle =   \(\begin{array}{l}\frac{1}{2} (8\times5)= 20cm^{2}\end{array} \)

Question 2:   The perimeter of a right-angled triangle is 32 cm. Its height and hypotenuse measure 10 cm and 13cm respectively. Find its area.

Perimeter = 32 cm

Hypotenuse a= 13 cm

Height b= 10 cm

Third side, c=?

We know that perimeter = a+ b+ c

32 cm = 13 + 10 + c

Therefore, c = 32 – 23 = 9 cm

\(\begin{array}{l}Area = \frac{1}{2} bh = \frac{1}{2} (9\times10)= 45cm^{2}\end{array} \)

• What is the area of right triangle if base = 6 cm and height = 4 cm?
• Find the area of right triangle if base = 15 cm and height = 10 cm.
• Find the area of a right triangle if base = 6 cm and hypotenuse = 10 cm.
• What is the perimeter of a triange if base = 3 in, height = 4 in and hypotenuse = 5 in?

To solve more problems on the topic and for video lessons , download BYJU’S -The Learning App .

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Finding an Angle in a Right Angled Triangle

Angle from any two sides.

We can find an unknown angle in a right-angled triangle , as long as we know the lengths of two of its sides .

The ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

The answer is to use Sine, Cosine or Tangent !

But which one to use? We have a special phrase " SOHCAHTOA " to help us, and we use it like this:

Step 1 : find the names of the two sides we know

• Adjacent is adjacent to the angle,
• Opposite is opposite the angle,
• and the longest side is the Hypotenuse .

Example: in our ladder example we know the length of:

• the side Opposite the angle "x", which is 2.5
• the longest side, called the Hypotenuse , which is 5

Step 2 : now use the first letters of those two sides ( O pposite and H ypotenuse) and the phrase " SOHCAHTOA " to find which one of Sine, Cosine or Tangent to use:

In our example that is O pposite and H ypotenuse, and that gives us “ SOH cahtoa”, which tells us we need to use Sine .

Step 3 : Put our values into the Sine equation:

S in (x) = O pposite / H ypotenuse = 2.5 / 5 = 0.5

Step 4 : Now solve that equation!

sin(x) = 0.5

Next (trust me for the moment) we can re-arrange that into this:

x = sin -1 (0.5)

And then get our calculator, key in 0.5 and use the sin -1 button to get the answer:

x = 30°

But what is the meaning of sin -1 … ?

Well, the Sine function "sin" takes an angle and gives us the ratio "opposite/hypotenuse",

But sin -1 (called "inverse sine") goes the other way ... ... it takes the ratio "opposite/hypotenuse" and gives us an angle.

• Sine Function: sin( 30° ) = 0.5
• Inverse Sine Function: sin -1 ( 0.5 ) = 30°

On your calculator, try using sin and sin -1 to see what results you get!

Also try cos and cos -1 . And tan and tan -1 . Go on, have a try now.

Step By Step

These are the four steps we need to follow:

• Step 1 Find which two sides we know – out of Opposite, Adjacent and Hypotenuse.
• Step 2 Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question.
• Step 3 For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse or for Tangent calculate Opposite/Adjacent.
• Step 4 Find the angle from your calculator, using one of sin -1 , cos -1 or tan -1

Let’s look at a couple more examples:

Find the angle of elevation of the plane from point A on the ground.

• Step 1 The two sides we know are O pposite (300) and A djacent (400).
• Step 2 SOHCAH TOA tells us we must use T angent.
• Step 3 Calculate Opposite/Adjacent = 300/400 = 0.75
• Step 4 Find the angle from your calculator using tan -1

Tan x° = opposite/adjacent = 300/400 = 0.75

tan -1 of 0.75 = 36.9° (correct to 1 decimal place)

Unless you’re told otherwise, angles are usually rounded to one place of decimals.

Find the size of angle a°

• Step 1 The two sides we know are A djacent (6,750) and H ypotenuse (8,100).
• Step 2 SOH CAH TOA tells us we must use C osine.
• Step 3 Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333
• Step 4 Find the angle from your calculator using cos -1 of 0.8333:

cos a° = 6,750/8,100 = 0.8333