What is Pythagoras' Theorem?

In any right-angled triangle, the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides.

a² + b² = c²

where c is the hypotenuse (the side opposite the right angle).

Labelling a Right-Angled Triangle

Hypotenuse (c) = longest side, opposite the right angle

Visual Proof: Square Areas

If a = 3 and b = 4, the squares built on each side show that the areas add up:

3²=9

a² = 9

+

4²=16

b² = 16

=

5²=25

c² = 25

9 + 16 = 25 ✓ So c = √25 = 5

Finding Unknown Sides

Finding the Hypotenuse

When you know both shorter sides:

c = √(a² + b²)

Add the squares, then square root.

Finding a Shorter Side

When you know the hypotenuse and one side:

a = √(c² − b²)

Subtract the squares, then square root.

Real-World Applications

📐

Construction

Checking if walls and floors are at right angles.

🗺

Navigation

Finding the shortest distance between two points on a map.

🚬

Ladders

Calculating safe ladder placement against a wall.

Key Vocabulary

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

Right Angle

An angle of exactly 90 degrees, shown by a small square symbol.

Square Root (√)

The inverse of squaring. √25 = 5 because 5² = 25.

Pythagorean Triple

A set of three whole numbers that satisfy a² + b² = c². Example: 3, 4, 5 and 5, 12, 13.

Worked Examples

1

Find the hypotenuse: a = 6, b = 8

Step 1: Write the formula: c² = a² + b²

Step 2: Substitute: c² = 6² + 8² = 36 + 64 = 100

Step 3: Square root: c = √100 = 10

Answer: c = 10

2

Find the missing side: c = 13, a = 5

Step 1: Rearrange: b² = c² − a²

Step 2: Substitute: b² = 13² − 5² = 169 − 25 = 144

Step 3: Square root: b = √144 = 12

Answer: b = 12

3

A ladder 10 m long leans against a wall. The base is 6 m from the wall. How high up the wall does it reach?

Step 1: The ladder is the hypotenuse (c = 10). The base distance is a = 6. We need b (height).

Step 2: b² = c² − a² = 10² − 6² = 100 − 36 = 64

Step 3: b = √64 = 8

Answer: The ladder reaches 8 m up the wall.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Find the hypotenuse of a right triangle with sides a = 3 and b = 4.

7 12 25 5

Question 2

A right triangle has a hypotenuse of c = 10 and one side a = 8. Find the other side b.

6 2 36 12.8

Question 3

Which side of a right-angled triangle is always the longest?

The base The hypotenuse The height It depends on the triangle

Question 4

Find the hypotenuse: a = 5, b = 12

17 169 13 14

Question 5

A rectangular field is 40 m long and 30 m wide. What is the diagonal distance across the field?

70 m 2500 m 35 m 50 m

Pythagoras' Theorem