 Home > College of Sciences > Institute of Fundamental Sciences > Maths First > Online Maths Help > Trigonometry > Pythagoras Theorem and Applications > The Pythagoras Theorem SEARCH MASSEY  # Pythagoras Theorem

## Pythagoras Theorem

A right triangle is a triangle that has one angle equal to 90 º. The triangles shown below are right triangles. The right angle (90º) is indicated by a square. The side opposite the right angle in a right triangle is called the hypotenuse. The hypotenuses of the right triangles above are shown in red. In any right angled triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides. That is, in the above triangle:

a2 + b2 = c2

### Example 1

Find the length of the missing side: The hypotenuse (side opposite the right angle) is the missing side. If this side is c, then by pythagoras theorem

c2 = 72 + 242
= 49 + 576
= 625

c   = √625 = 25

### Example 2

Find the length of the missing side: The hypotenuse is the side of length 10. If the missing side is a, then by pythagoras theorem

102 = a2 + 62     Rearrange to solve for a

a2 = 102 − 62
= 100 − 36
= 64

a = √64 = 8

### Example 3

Find the length of the missing side: The hypotenuse is the side of length 29. If the missing side is b, then by pythagoras theorem

292 = b2 + 212     Rearrange to solve for b

b2 = 292 − 212
= 841− 441
= 400

b = √400 = 20

### Example 4

Consider the right triangle below with sides a and b and hypotenuse c. Each time you click "Get New Example" a new example will be shown. The lengths of two sides will be shown. Click "Show third side" to see how the length of the third side is calculated given the lengths of the other two. In the solution, the notation √ = sqrt() will be used.

 a = c = b =

### Exercise 1

Now try some yourself. Consider the right triangle below with sides a and b and hypotenuse c. Find the length of the missing side. Enter your answer to 2 decimal places.

 a = c = b =