How to Figure Out the Length of the Hypotenuse
In geometry, a right triangle contains a 90-degree angle, two shorter sides and a long side called a hypotenuse. One of those three values can be calculated using the Pythagorean theorem if the other two are known. The Pythagorean theorem states a^2 + b^2 = c^2, where "a" and "b" are sides and "c" is the hypotenuse. The hypotenuse can also be found through trigonometric functions when only one side and an angle is known. In a right triangle, the sine of an angle in degrees is opposite side / hypotenuse, cosine = adjacent / hypotenuse and tangent = opposite/adjacent.
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Instructions
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Suppose you have an equation in which two sides are given and you need to find the hypotenuse. If you're given side measurements of 6 and 3, write out the formula: 6^2 + 3^2 = c^2 or 36 + 9 = c^2 or 45 = c^2. Take the square root of both sides to eliminate the exponent, leaving you with an answer of c = √45, or 3√5.
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Find the hypotenuse of a right triangle with an angle of 35 degrees and an opposite side equal to 2.8. Use the trigonometric function sine. Set up the equation: sin 35 degrees = 2.8/h. Multiply both sides by "h": (sin 35 degrees)h = 2.8. Divide both sides by "(sin 35 degrees)" and use your calculator to solve: h = 4.88 or h = 4.9 (rounded to tenths).
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Find the length of a hypotenuse using the tangent function by first using it to find the length of an unknown opposite side. Set up the equation using an angle of 40 degrees and an adjacent side length of 10: tangent 40 degrees = x / 10. Multiply both sides by 10 and solve using your calculator: x = 8.39 or 8.4 rounded. Use the two sides you now know to find the hypotenuse: 8.4^2 + 10^2 = c^2 or 70.56 + 100 = c^2. So 170.56 = c^2 or √170.56 = c.
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Tips & Warnings
Sine and cosine solve the same way when finding the hypotenuse, so if you can do one, you can do the other. Be sure you input the right trig function into the calculator, though.
