All right triangles have one right angle (90 degrees), and the hypotenuse is the side opposite that angle. The hypotenuse is the longest side of the triangle, and it's also very easy to find it using a few different ways. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem if you know the lengths of the other two sides of the triangle. Next, this article will teach you how to identify the hypotenuse of some special right triangles that appear frequently in exams. Finally, this article will teach you how to find the length of the hypotenuse using the Law of Sine if you only know the length of one side and the measurement of an angle other than a right angle.
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Method 1 of 3: Using the Pythagorean Theorem
Step 1. Learn the Pythagorean Theorem
The Pythagorean theorem describes the relationship between the sides of a right triangle. This theorem states that for any right triangle with sides along a and b, and a hypotenuse along c, a2 + b2 = c2.
Step 2. Make sure that your triangle is a right triangle
The Pythagorean theorem applies only to right triangles, and by definition, only right triangles have a hypotenuse. If your triangle has one angle that is exactly 90 degrees, it is a right triangle and you can move on.
Right angles are often denoted in textbooks and exams by a small square in the corner of the corner. This particular sign means "90 degrees"
Step 3. Assign variables a, b, and c to the sides of your triangle
The variable "c" will always be assigned to the hypotenuse, or longest side. Choose one of the other sides to be "a", and call the other side "b" (it doesn't matter which side is a or b; the calculation will remain the same). Then, plug the lengths of a and b into the formula, according to the following example:
If your triangle has sides of lengths 3 and 4, and you have assigned the letters to the sides so that a = 3 and b = 4, you would write your equation as: 32 + 42 = c2.
Step 4. Find the square of a and b
To find the square of a number, you simply multiply the number by itself, so that a2 = a x a. Find the squares of a and b, and plug them into your formula.
- If a = 3, a2 = 3 x 3, or 9. If b = 4, b2 = 4 x 4, or 16.
- When you plug those values into your equation, your equation should now look like this: 9 + 16 = c2.
Step 5. Add up the values of a2 and b2.
Plug the sum into your equation, and this will give you the value of c2. There is only one step left, and you will solve the hypotenuse!
In our example, 9 + 16 = 25, so you would write 25 = c2.
Step 6. Find the square root of c2.
Use the square root function on your calculator (or memory or your multiplication table) to find the square root of c2. The answer is the length of your hypotenuse!
In our example, c2 = 25. The square root of 25 is 5 (5 x 5 = 25, so Root(25) = 5). It means, c = 5, the length of our hypotenuse!
Method 2 of 3: Finding the Hypotenuse of a Special Right-angled Triangle
Step 1. Learn to recognize triangles with the Pythagorean Triple
The side lengths of the Pythagorean triple are integers according to the Pythagorean Theorem. These special triangles often appear in geometry textbooks and standardized exams such as the UN. If you remember especially the first 2 Pythagorean triples, you can save a lot of time on these tests because you will quickly find out the hypotenuse of one of these triangles just by looking at the side lengths!
- The first Pythagorean triple was 3-4-5 (32 + 42 = 52, 9 + 16 = 25). When you see a right triangle with legs of lengths 3 and 4, you will immediately believe that its hypotenuse is 5 without having to do any calculations.
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The Pythagorean triple ratio holds true even if the sides are multiplied by another number. For example, a right triangle with leg length
Step 6. da
Step 8. will have a hypotenuse
Step 10. (62 + 82 = 102, 36 + 64 = 100). The same goes for 9-12-15, and even 1, 5-2-2, 5. Try the calculations and see for yourself!
- The second Pythagorean triple that appears frequently in exams is 5-12-13 (52 + 122 = 132, 25 + 144 = 169). Also pay attention to the multiples like 10-24-26 and 2, 5-6-6, 5.
Step 2. Remember the ratio of the sides of a right triangle 45-45-90
A right triangle 45-45-90 has angles of 45, 45, and 90 degrees, and is also called an isosceles right triangle. This triangle appears frequently in standardized exams, and is a very easy triangle to solve. The ratio of the sides of this triangle is 1:1:Root(2), which means that the lengths of the legs are the same, and the length of the hypotenuse is simply the length of the legs times the square root of two.
- To calculate the hypotenuse of this triangle based on the length of one of its legs, simply multiply the length of the leg by Sqrt(2).
- Knowing these comparisons is helpful, especially when your exam or homework questions give the side lengths as variables instead of integers.
Step 3. Study the side ratios of a 30-60-90 right triangle
These triangles have angle measurements of 30, 60, and 90 degrees, and occur when you cut an equilateral triangle in half. The sides of a right triangle 30-60-90 always have the ratio 1:Root(3):2, or x:Root(3)x:2x. If you were given the length of one leg of a right triangle 30-60-90 and asked to find the hypotenuse, this problem would be very easy to do:
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If you are given the length of the shortest leg (opposite at a 30-degree angle), simply multiply the length of the leg by 2 to find the length of the hypotenuse. For example, if the length of the shortest leg is
Step 4., you know that the length of the hypotenuse must be
Step 8..
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If you are given the length of the longer leg (opposite an angle of 60 degrees), multiply that length by 2/Root(3) to find the length of the hypotenuse. For example, if the length of the longer leg is
Step 4., you know that the length of the definite hypotenuse is 4, 62.
Method 3 of 3: Finding the Hypotenuse Using the Law of Sine
Step 1. Understand the meaning of "Sine"
The terms "sine", "cosine", and "tangent" refer to the various ratios between the angles and/or sides of a right triangle. In a right triangle, sine an angle is defined as the length of the side opposite the angle divided by triangle hypotenuse. The abbreviation for sine in equations and calculators is sin.
Step 2. Learn how to calculate sine
Even basic scientific calculators have a sine function. Look for the button that says sin. To find the sine of an angle, you usually press the key sin and then enter the angle measurement in degrees. However, in some calculators, you must first enter the angle measurement and then press the button sin. You'll have to experiment with your calculator or check the manual to figure out which method to use.
- To find the sine of an 80 degree angle, you must enter sin 80 followed by an equal sign or Enter, or 80 sin. (The answer is -0, 9939.)
- You can also type "sine calculator" into a web search, and look for some easy-to-use calculators, which will take any guesswork out of the way.
Step 3. Learn the Law of Sine
The law of sines is a useful tool for solving triangles. In particular, this law can help you find the hypotenuse of a right triangle if you know the length of one side, and the measurement of one angle other than that right angle. For any triangle with sides a, b, and c, and angles A, B, and C, the Law of Sine states that a / sin A = b / sin B = c / sin C.
The law of sines can actually be used to solve any triangle, but only right triangles have a hypotenuse
Step 4. Assign variables a, b, and c to the sides of your triangle
The hypotenuse (longest side) must be "c". For convenience, label "a" for the side of known length, and label "b" for the other side. The right angle opposite the hypotenuse is "C". The angle opposite side "a" is angle "A", and the angle opposite side "b" is "B".
Step 5. Calculate the measurement of the third angle
Since it is a right angle, we already know that C = 90 degrees, and you also know the measurements A or B. Since the measurement of the inner degree of a triangle always equals 180 degrees, you can easily calculate the measurement of the angles of all three using the formula: 180 – (90 + A) = B. You can also reverse the equation to 180 – (90 + B) = A.
For example, if you know that A = 40 degrees, B = 180 – (90 + 40). Simplify this to B = 180 – 130, and you can quickly determine that B = 50 degrees.
Step 6. Check your triangle
In this step, you already know the measurements of the three angles, and the length of side a. Now it's time to plug this information into the Law of Sine equations to determine the lengths of the other two sides.
To continue our example, let's say the length of side a = 10. Angle C = 90 degrees, angle A = 40 degrees, and angle B = 50 degrees
Step 7. Apply the Law of Sine to your triangle
We just need to plug our numbers in and solve the following equation to find the length of hypotenuse c: side length a / sin A = side length c / sin C. This equation may look a little scary, but the sine of 90 degrees is always the same, and always equals 1! Thus, our equation can be simplified to: a / sin A = c / 1, or just a / sin A = c.
Step 8. Divide the length of side a with the sine of the angle A to find the length of the hypotenuse!
You can find it in two separate steps, first by calculating sin A and writing down the result, then dividing by a. Or you can enter everything into the calculator at the same time. If you're using a calculator, remember to put the parentheses after the division sign. For example, enter 10 / (sin 40) or 10 / (40 sin), depending on your calculator.
