How to Find the hypotenuse length

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This article was co-written by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math teacher at City University of San Francisco and previously worked in the math department of Saint Louis University. She has taught math at the elementary, middle, high, and college levels. She holds a master’s degree in education from Saint Louis University, majoring in management and supervision in education.

There are 9 references cited in this article that you can view at the bottom of the page.

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Every right triangle has a right angle (90 degrees) and a hypotenuse is the side opposite the right angle or longest side of the right triangle. ^{[1] X Research Source} The hypotenuse is the longest side of a right triangle, and with several different methods the length of this side is also easy to find. The following article will show you how to find the length of the hypotenuse when knowing the lengths of the other two sides of a right triangle using the Pythagorean theorem. Next, you will be shown how to recognize the hypotenuse of a few special right triangles that often appear in tests. And finally, you will be familiar with how to find the hypotenuse length using the Sin theorem when only knowing the length of one side and the measure of an acute angle.

Table of Contents

Using Pythagorean Theorem

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Learn Pythagorean Theorem. The Pythagorean theorem describes the relationship between the sides of a right triangle. ^{[2] X Research Source} It says that for any right triangle with the lengths of two sides a and b respectively, the length of the hypotenuse is c, we have: a ² + b ² = c ² . ^{[3] X Research Sources}

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Make sure your triangle is a right triangle. The Pythagorean theorem only holds true for right triangles, and by definition only right triangles have a hypotenuse. If your triangle contains an angle that measures exactly 90 degrees, it is a right triangle and you can continue.

Right angles are often denoted in textbooks and in tests by a small right angle located at the corner of the angle. This particular sign means “90 degrees”.

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Assign variables a, b, and c to the sides of your triangle. The variable “c” is always used for the hypotenuse – the longest side. Choose one of the two remaining edges a and call the other edge b (which edge is a and which is b is not important, the calculation will give us the same result). Next, substitute the lengths of a and b into the formula, like the example below:

If your triangle has two right-angled sides 3 and 4, and you named those sides a = 3 and b = 4 respectively, then our equation would be: 3 ² + 4 ² = c ² .

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Find the squares of a and b. To find the square of a number, you simply multiply the number by itself, that is, a ² = axa . Find the squares of both a and b, and write in your formula.

If a = 3, a ² = 3 x 3, or 9. If b = 4, then b ² = 4 x 4, or 16.
Substituting the input values, we have the following equation: 9 + 16 = c ² .

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Add the values a ² and b ² together. Substituting into the equation, we get the value of c ² . Only one last step left, and you get the length of the hypotenuse!

In our example: 9 + 16 = 25 , so you can write 25 = c ² .

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Find the square root of c ² . Use the square root function in your calculator (or whatever you can remember from your multiplication table) to find the square root of c ² . The answer is the length of your hypotenuse!

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

Find the hypotenuse of the special right triangle

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Learn to recognize the Pythagorean Triangle. The lengths of the sides in a triangle Pythagorean triples are integers that satisfy the Pythagorean theorem. These special triangles frequently appear in geometry textbooks and standardized tests, such as the SAT or GRE. If you can memorize, especially the first two Pythagorean triples, you can save a lot of time when you do the test, because then, just by looking at the lengths of their right angles, you can instantly know the length of the hypotenuse of one of these triangles! ^{[4] X Research Sources}

The first Pythagorean triple is 3-4-5 (3 ² + 4 ² = 5 ² , 9 + 16 = 25). When you see a right triangle with sides 3 and 4, respectively, you can immediately determine without any calculations that it has a hypotenuse of 5.
The ratio of Pythagorean triples remains true, even when the edges are multiplied by another number. For example, a right triangle with sides of length 6 and 8 will have a hypotenuse of 10 (6 ² + 8 ² = 10 ² , 36 + 64 = 100). Same goes for 9-12-15 , or even 1.5-2-2.5 . Try to do the math and see for yourself!
The Pythagorean triple that often appears in tests is 5-12-13 (5 ² + 12 ² = 13 ² , 25 + 144 = 169). Also look out for multiples like 10-24-26 or 2.5-6-6.5 .

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Remember the ratio of the sides of a right triangle 45-45-90. Right triangle 45-45-90 is a triangle with three angles of 45, 45 and 90 degrees respectively, also known as Isosceles Right Triangle. Isosceles right triangles appear frequently in standardized tests and are very easy to solve. The side of this triangle has a ratio of 1:1:Sqrt(2) , which means that the two sides of the right angle are equal, and the length of the hypotenuse is simply equal to the length of the side of the right angle multiplied by the square root of two.

To calculate the hypotenuse of a triangle based on the length of a right angle, simply multiply the length of the right angle by Sqrt(2). ^{[5] X Research Sources}
Knowing this ratio can be extremely helpful, especially when a test or exercise question gives you the length of the side of the right angle in a variable instead of an integer.

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Learn the proportions of the sides of a right triangle 30-60-90. This is a triangle whose angles measure 30, 60, and 90 degrees, respectively, and this triangle appears when you divide an equilateral triangle in half. The sides of a right triangle 30-60-90 always keep the ratio 1:Sqrt(3):2 , or x:Sqrt(3)x:2x . If given the length of a right angle side of a right triangle 30-60-90 and asked to find the length of the hypotenuse, it would be a very easy problem: ^{[6] X Research Source}

If the problem tells you that the length of the side of the right angle is shorter (opposite the angle 30 degrees), you can simply double the length of that side to find the length of the hypotenuse. For example, if the length of the shorter side of the right angle is 4 , you know that the length of the hypotenuse must be 8 .
If the problem indicates the length of the side of the longer right angle (opposite the angle 60 degrees), multiply the length of that side by 2/Sqrt(3) to find the length of the hypotenuse. For example, if the length of the side of a right angle is longer than 4 , you know that the length of the hypotenuse must be 4.62 .

Find the hypotenuse using Sin .’s theorem

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Understand what “Sin” means. The terms “sin”, “cosine”, and “tan” are all used to refer to different ratios between the angles and/or sides of a right triangle. In a right triangle, the sine of an angle is determined by the length of the opposite side divided by the hypotenuse . In equations and calculators, sin is both denoted by sin . ^{[7] X Research Sources}

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Learn how to calculate sine. Even basic scientific calculators have sine functions. Look for the key with the symbol sin . To find the sine of an angle, you’ll usually have to press the sine key and then enter the angle measure in degrees. However, on some calculators, you will have to enter the degree measurement first and then the sin key. You will have to experiment on the computer or check the manual to determine which is the right way.

To find the sine of an 80 degree angle, you will have to press sin 80 then the equal sign or enter key or 80 sin (Answer is -0.9939).
You can also type “sine calculator” into a search engine and find tons of easy-to-use calculators that take the guesswork out of it. ^{[8] X Research Sources}

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Learn the Sine theorem. The Sin theorem is a useful tool in solving triangle problems. Specifically, it will help you find the hypotenuse of a right triangle when you know the length of one side of the right angle and the measure of another angle, besides the right angle. For every triangle with sides a , b , and c , and angles A , B , and C , the Sin theorem says that a / sin A = b / sin B = c / sin C . ^{[9] X Research Source}

The true Sin theorem can be used to solve any triangle, but only right triangles have hypotenuse.

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Assign variables a, b, and c to the sides of your triangle. The hypotenuse (longest) must be “c”. For simplicity, we set the known edge as “a” and the other side as “b”. Next, assign variables A, B, and C to the angles of the triangle. The right angle opposite the hypotenuse will be “C”. Opposite side “a” is angle “A” and opposite side “b” is “B”.

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Calculate the measure of the third angle. Since it’s a right triangle, you already know C = 90 degrees , and you also know the measure of A or B. Since the sum of the measures of the three interior angles of a triangle is always 180 degrees, you can easily calculate the measure of the third angle using the following formula: 180 – (90 + A) = B . You can also invert the equation like 180 – (90 + B) = A .

For example, if A = 40 degrees is known, then B = 180 – (90 + 40) . Reduce to B = 180 – 130 , and we can quickly determine B = 50 degrees .

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Test your triangle. By this point, you should know the measures of all three angles and the length of side a. Now it’s time to put the information into the Sin theorem equation to determine the length of the other two sides.

To continue the example, suppose the length of side a = 10. Angle C = 90 degrees, angle A = 40 degrees, and angle B = 50 degrees.

Steps

Using Pythagorean Theorem

Find the hypotenuse of the special right triangle

Find the hypotenuse using Sin .’s theorem

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