Author:
Roger Morrison
Date Of Creation:
20 September 2021
Update Date:
8 May 2024
Content
- To step
- Method 1 of 3: Using the Pythagorean Theorem
- Method 2 of 3: Determining the hypotenuse of special right triangles
- Method 3 of 3: Calculate the hypotenuse with the law of sine
All right triangles have a right angle (of 90 degrees), and the hypotenuse is the side opposite that angle. The hypotenuse is the longest side of the triangle, and it is also very easy to find using a number of different methods. This article will teach you how to determine the length of the hypotenuse using the Pythagorean Theorem, provided you know the length of the other two sides of the triangle. Then you will learn how to calculate the hypotenuse of a number of special right triangles that you will often encounter in tests. And finally you will learn how to calculate the length of the hypotenuse using the Law of Sine, when only the length of one side and the magnitude of one of the angles is known.
To step
Method 1 of 3: Using the Pythagorean Theorem
Learn Pythagorean Theorem. The Pythagorean Theorem describes the relationship between the sides of a right triangle. It states that for any right triangle with sides a and b, and hypotenuse with length c, this holds a + b = c. 
Make sure your triangle is a right triangle. The Pythagorean Theorem only works with right triangles, and by its definition, only a right triangle can have a hypotenuse. If your triangle has an angle of exactly 90 degrees, it is a right triangle and you can proceed. - Right angles are often indicated in textbooks and tests with a small square in the right angle. This special marking means "90 degrees."
Assign variables a, b, and c to the sides of your triangle. The variable "c" is always assigned to the hypotenuse, or the longest side. Choose one of the other sides for the a, and name the remaining side b (It doesn't matter which one, the math remains the same). Then incorporate the lengths of a and b into the formula, as shown in the following example: - If your triangle has sides of 3 and 4, and you assigned letters to those sides, such that a = 3 and b = 4, then you evaluate the equation as: 3 + 4 = c.
Find the squares of a and b. To find the square of a number, you just multiply that number by itself, so a = a x a. Find the squares of both a and b and include them in the formula. - If a = 3, a = 3 x 3, or 9. If b = 4, then b = 4 x 4, or 16.
- When you include those values in your equation, it should now look like this: 9 + 16 = c.
Count the values of a and b together. Work this into your equation, and it will give you the value of c. Now there is only one more step to go, and you have determined the hypotenuse! - In our example, 9 + 16 = 25, write it down 25 = c.
Find the square root of c. Use the square root function on your calculator (or with the multiplication tables, if you know them by heart) to find the square root of c. The answer is the length of your hypotenuse! - In our example, c = 25. The square root of 25 is 5 (5 x 5 = 25, So Sqrt (25) = 5). And that means c = 5, the length of our hypotenuse!
Method 2 of 3: Determining the hypotenuse of special right triangles
Learn to recognize the triangles of a Pythagorean triad. The side lengths of a Pythagorean triad are integers that obey the Pythagorean Theorem. You will encounter these special triangles regularly in geometry textbooks and in standard tests such as the SAT and the GRE. In particular, if you memorize the first 2 Pythagorean triples, you can save yourself a lot of time on those tests, because you immediately know the hypotenuse of one of these triangles just by looking at the length of the sides. ! - The first Pythagorean triad is 3-4-5 (3 + 4 = 5, 9 + 16 = 25). When you see a right triangle with sides that are 3 and 4 in length, you immediately know for sure that the hypotenuse will be 5 without having to make any calculations.
- The ratio of a Pythagorean triad remains the same even when the sides are multiplied by another number. For example: a right triangle with sides 6 and 8 will have a hypotenuse of 10 (6 + 8 = 10, 36 + 64 = 100). The same applies to 9-12-15, even 1,5-2-2,5. Do the calculations and you will see!
- The second Pythagorean trio that you will often encounter during tests is 5-12-13 (5 + 12 = 13, 25 + 144 = 169). Also pay attention to multiples of it, like 10-24-26 and 2,5-6-6,5.
Memorize the ratio of the sides of a right triangle with angles 45-45-90. A 45-45-90 right triangle has angles of 45, 45, and 90 degrees, and is also called an isosceles right triangle. You will find this regularly on standard tests, and it is a very easy triangle to solve. The ratio between the sides of this triangle is 1: 1: Sqrt (2)meaning that the length of the legs is equal, and the length of the hypotenuse is then simply the length of the legs multiplied by the square root of two. - You calculate the hypotenuse of this triangle based on the length of one of its legs, and multiply its length by Sqrt (2).
- It is useful to know this ratio, especially if your test or homework assignments give you the length of the sides in the form of variables rather than integers.
Learn the proportions of a 30-60-90 right triangle. This triangle has angles of 30, 60 and 90 degrees and can be made by dividing an equilateral triangle in half. The sides of the right angle 30-60-90 triangle are always in proportion 1: Sqrt (3): 2, or x: Sqrt (3) x: 2x. When given the length of one leg of the right-angled 30-60-90 triangle and asked to determine the hypotenuse, it is very easy to do: - Given the length of the shortest leg (the one opposite the 30 degree angle), multiply the length of that leg by 2 to get the length of the hypotenuse. For example, as the length of the shortest leg 4 then you know that the hypotenuse 8 must be.
- If you know 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 example, as the length of the shortest leg 4 then you know that the hypotenuse 4,62 must be.
Method 3 of 3: Calculate the hypotenuse with the law of sine
Understand what the "Sine" means. The terms "sine," "cosine" and "tangent" all refer to specific ratios between the angles and / or sides of a right triangle. In a right triangle, the sinus of an angle defined as the length of the side opposite the corner divided by the hypotenuse of the triangle. The abbreviation for sine as it is on your calculator sin. 
Learn how to calculate the sine. Even a simple scientific calculator does have a sine function. Look at the key marked sin. To find the sine of an angle, you will usually tap the sin-key and then enter the angle in degrees. However, on some calculators you have to enter the angle in degrees first and then enter the sinbutton. You will have to experiment with your calculator or check the manual to determine which one it is. - To find the sine of an angle of 80 degrees, you will sin 80 followed by the equal sign or Enter, or 80 sin. (The answer is -0.9939.)
- You can also search for "sin calculator" with your browser for some easy to use calculators so you don't have to guess anymore.
Learn the rule of sine. The rule of sine is a useful tool for solving triangle problems. You can mainly use it to find the hypotenuse of a right triangle, if you know the length of one side and one of the angles other than the right angle. For any triangle with sides a, b and c and the corners a, B. and C., the rule of sine states that a / sin a = b / sin B. = c / sin C.. - The rule of sine can in fact be used for solving each triangle, but only a right triangle will have a hypotenuse.
Assign the variables a, b, and c to the sides of your triangle. The hypotenuse (the longest side) must always be the "c". For simplicity, label the side with the known length as "a" and others "b". Then assign the variables A, B, and C to the angles of the triangle. The right angle opposite the hypotenuse then becomes "C". The angle opposite "a" is angle "A" and the angle opposite side "b" is "B". 
Calculate the third angle. Since it is a right angle you already know that C = 90 degrees and you also know some angle a or B. is. Since the sum of all angles in a triangle must always equal 180 degrees, it is easy to calculate the third angle with the formula: 180 - (90 + A) = B. You can also reverse this equation so that 180 - (90 + B) = A. - For example, if you know that A = 40 degrees, than B = 180 - (90 + 40). Simplify this to B = 180-130 and you can quickly deduce that B = 50 degrees.
Examine your triangle. At this point you know all angles of the triangle and the length of side a. Now it is time to process these values in the Sine Law, to calculate the length of the other two sides. - To continue with our example, let's say the length of side a = 10. Angle C = 90 degrees, angle A = 40 degrees, and angle B = 50 degrees.
Apply the Sine Rule to your triangle. We just need to process our numbers and solve the following equation to find the length of hypotenuse c: length of side a / sin A = length of side c / sin C.. This may still look a little intimidating, but the 90 degree sine is a constant, and always equal to 1! Our equation can therefore be simplified to: a / sin A = c / 1, or simpler a / sin A = c. 
Divide the length of side a by the sine of angle a to find the length of the hypotenuse! You can calculate this in two separate steps, by doing first sin Calculate A and write it down, then divide it by a. Or enter it all in your calculator at the same time. If you do this, don't forget the brackets after the division sign. For example, key in 10 / (sin 40) or 10 / (40 sin)depending on your calculator. - Based on our example, we see that sin 40 = 0.64278761. To find the value of c, we divide the length of a by this number and see that 10 / 0,64278761 = 15,6, the length of our hypotenuse!
