Solving right triangles is easy with the Pythagorean formula. If you know the length of any two sides of a right triangle, you can find the length of the third side.
Learn the Theorem
The Pythagorean Theorem, named after the Greek mathematician Pythagoras, is a formula used to calculate the length of the sides of a right triangle. The theorem states:
- In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
A right triangle is a triangle containing a right angle (90 degrees). The hypotenuse is the longest side of a right triangle, the one opposite the right angle. The legs are the two shorter sides, which meet at the right angle. By convention, the legs are usually labeled a and b and the hypotenuse is labeled c. Therefore, the Pythagorean theorem formula is as follows:
Pythagorean Theorem Examples: Solving Right Triangle Problems
Pythagorean theorem problems start by giving you the length of two of the sides of a right triangle. Using the Pythagorean formula, it is possible to calculate the length of the third side. Because you are using squares and square roots, you may need the help of a calculator.
Example 1: Find the hypotenuse
A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?
How to solve:
Name the legs a and b and the hypotenuse c. We are given that a = 3 and b = 4. Add these values to the formula and solve for c:
3² + 4² = c²
9 + 16 = c²
25 = c²
Now take the square root of both sides of the equation to solve for c. The square root of 25 is 5.
c = 5
Example 2: Find one of the legs
A right triangle has a leg of length 5 and a hypotenuse of length 13. What is the length of the other leg?
How to solve:
Name the known leg a, the unknown leg b, and the hypotenuse c. We are given that a = 5 and c = 13. Add these values to the formula and solve for b:
5² + b² = 13²
25 + b² = 169
Using algebra, subtract 25 from both sides:
b² = 144
Now take the square root of both sides to solve for b. The square root of 144 is 12.
b = 12
Pythagorean Theorem Word Problems
Word problems using the Pythagorean theorem require you to draw or imagine a right triangle where two of the sides are of known length. When solving these problems, remember the following:
- The diagonal across a rectangle creates two congruent right triangles.
- The angle between two walls, and the angle between the floor and a wall, is usually a right angle.
Example 3: TV Screen Width
You want to purchase a 30" television. You know that TV and monitor screens are measured across the diagonal, and you also know that this screen is 18" high. How wide is the screen horizontally?
How to solve:
The TV screen is a rectangle, and the diagonal is 30". The screen's height and width make up two legs of a right triangle with the diagonal as the hypotenuse. The problem gives us the length of the hypotenuse and one of the legs. Therefore, name the height a, the width b, and the diagonal c. Add these values to the formula and solve for b:
18² + b² = 30²
324 + b² = 900
Subtract 324 from both sides:
b² = 576
Now find the square root of both sides to solve for b (use a calculator to find the square root of large numbers). The square root of 576 is 24.
The screen is 24" wide.