Pythagoras

What you'll learn

How to spot the hypotenuse in a right-angled triangle.
How to use $a^2 + b^2 = c^2$ to find a missing side.
When to add and when to subtract.
How to use Pythagoras in rectangles, composite shapes, and worded problems.

1. Start with the right angle

Pythagoras is only for right-angled triangles. So before doing any calculation, check that the triangle has a 90° angle.

Definition

Right-angled triangle

A right-angled triangle is a triangle with one right angle, which is an angle of 90°.

Definition

Hypotenuse

The hypotenuse is the side opposite the right angle. It is always the longest side in a right-angled triangle.

The hypotenuse is directly opposite the right angle and is the longest side of the right-angled triangle.

Example

Spotting the hypotenuse

A triangle $P QR$ has a right angle at $Q$ . Which side is the hypotenuse?

In triangle PQR, the side opposite the right angle at Q is PR, so PR is the hypotenuse.

Find the right angle. It is at $Q$ .
Look at the side opposite $Q$ .
The side opposite $Q$ is $P R$ , so the hypotenuse is $P R$ .

2. Squares, square roots, and the rule

To use Pythagoras, you need squares and square roots.

Definition

Squares and square roots

To square a number, multiply it by itself. For example, $x^2$ means $\times x$ .
A square root reverses squaring. $x\sqrt{x}$ means the positive number which squares to make $x$ .

Key Idea

Pythagoras' theorem

In any right-angled triangle, if $a$ and $b$ are the two shorter sides and $c$ is the hypotenuse, then $a^2 + b^2 = c^2$ .

The two shorter sides a and b meet at the right angle, and c is the hypotenuse opposite it.

Common Mistake

Right angles only

Pythagoras' theorem works only when you have a right angle. If there is no 90° angle, do not use it unless you can create one in the diagram.

3. Finding the hypotenuse: add the squares

If the missing side is the hypotenuse, square the two shorter sides, add them, then square root.

Example

Finding the longest side

A right-angled triangle has a right angle at $B$ . The two shorter sides are $\text{ cm}$ and $\text{ cm}$ . Find $A C$ .

AB and BC are the two shorter sides meeting at the right angle, so AC is the unknown hypotenuse.

The right angle is at $B$ , so the hypotenuse is $A C$ .
Use Pythagoras with $c$ as the hypotenuse:
$c2=5.42+7.22=29.16+51.84=81\begin{aligned} c^2 &= 5.4^2 + 7.2^2 \\ &= 29.16 + 51.84 \\ &= 81 \end{aligned}$
Square root to find the length:
$\sqrt{81} = 9$
So $\text{ cm}$ .

4. Finding a shorter side: subtract

If the hypotenuse is already given, the missing side must be one of the shorter sides. This time, subtract the square of the known shorter side from the square of the hypotenuse.

Key Idea

Add or subtract?

Add when you are finding the hypotenuse. Subtract when the hypotenuse is already known and you are finding a shorter side.

Example

Finding a shorter side

A right-angled triangle has hypotenuse $\text{ cm}$ and one shorter side $\text{ cm}$ . Find $E F$ to 1 decimal place.

DF is the given hypotenuse, so the missing side EF is one of the shorter sides.

The hypotenuse is $D F$ , so it goes first in the subtraction.
Work out the square of the missing side:
$EF2=162−72=256−49=207\begin{aligned} EF^2 &= 16^2 - 7^2 \\ &= 256 - 49 \\ &= 207 \end{aligned}$
Square root the answer:
$\sqrt{207} = 14.387\ldots$
To 1 decimal place, $\text{ cm}$ .

Common Mistake

Adding every time

Do not always add the squares. If the longest side is already given, you need to subtract to find the missing shorter side.

5. Pythagoras inside other shapes

Other shapes often hide right-angled triangles. A diagonal is a line joining two opposite corners of a shape. In a rectangle, the diagonal is the hypotenuse of a right-angled triangle.

A trapezium is a four-sided shape with one pair of parallel sides. In trapezium questions, look for right angles or draw in a height to create a right-angled triangle.

Example

Diagonal of a rectangle

A rectangle is 16 cm long and 9 cm wide. Find the length of its diagonal to 1 decimal place.

The diagonal of the rectangle forms the hypotenuse of a right-angled triangle with sides 16 cm and 9 cm.

The diagonal splits the rectangle into two right-angled triangles.
The diagonal is the hypotenuse, so add the squares of 16 and 9:
$d2=162+92=256+81=337\begin{aligned} d^2 &= 16^2 + 9^2 \\ &= 256 + 81 \\ &= 337 \end{aligned}$
Square root:
$\sqrt{337} = 18.357\ldots$
To 1 decimal place, the diagonal is 18.4 cm.

Tip

Isosceles triangles

An isosceles triangle has two equal sides. If you draw the perpendicular height from the top, it usually splits the base into two equal halves, making a right-angled triangle.

6. Using Pythagoras twice

Some questions contain two right-angled triangles joined together. You may need to find a shared side first, then use it in the second triangle.

Example

Two joined right-angled triangles

Two right-angled triangles share side $A C$ . In the first triangle, $\text{ m}$ , $\text{ m}$ , and the right angle is at $A$ . In the second triangle, $\text{ m}$ and the right angle is at $C$ . Find $A D$ to 3 significant figures.

The shared side AC is found from triangle ABC, then used as a shorter side in triangle ACD to find AD.

Start with triangle $A B C$ , because it has two known sides and lets you find $A C$ .
In triangle $A B C$ , $B C$ is the hypotenuse:
$AC^2 = 13^2 - 12^2 = 25$
Find $A C$ :
$\sqrt{25} = 5$
Now use triangle $A C D$ . The sides $A C$ and $C D$ meet at the right angle, so $A D$ is the hypotenuse:
$AD^2 = 5^2 + 6^2 = 61$
Square root and round:
$\sqrt{61} = 7.810\ldots$
To 3 significant figures, $\text{ m}$ .

7. Worded problems and units

Real-life questions often describe a right-angled triangle without drawing one. Vertical means straight up and down. Horizontal means flat, like the ground. Vertical and horizontal lines meet at 90°.

Tip

Sketch first

For journeys, “North then East” makes a right angle. For ladders, the wall and ground make a right angle, and the ladder is the hypotenuse.

Example

A ladder against a wall

A ladder reaches 2.4 m up a wall. Its base is 90 cm from the wall. Find the length of the ladder to 3 significant figures.

The wall and ground form the right angle, and the ladder is the unknown hypotenuse.

The wall and ground form a right angle, so the ladder is the hypotenuse.
Convert 90 cm into metres: 90 cm is 0.9 m.
Use Pythagoras:
$l2=2.42+0.92=5.76+0.81=6.57\begin{aligned} l^2 &= 2.4^2 + 0.9^2 \\ &= 5.76 + 0.81 \\ &= 6.57 \end{aligned}$
Square root:
$\sqrt{6.57} = 2.563\ldots$
To 3 significant figures, the ladder is 2.56 m long.

Exam technique

In the exam

Mark the right angle and label the hypotenuse before writing the formula.
Decide whether to add or subtract: find the hypotenuse means add; find a shorter side means subtract.
Keep full calculator values until the final line, then round to the accuracy asked for.

Self review

Check yourself

Can you identify the hypotenuse in any right-angled triangle?
If the hypotenuse is already given, do you know why you subtract?
In a worded problem, can you sketch the right-angled triangle and keep the units matching?

What you'll learn

How to spot the hypotenuse in a right-angled triangle.
How to use $a^2 + b^2 = c^2$ to find a missing side.
When to add and when to subtract.
How to use Pythagoras in rectangles, composite shapes, and worded problems.

1. Start with the right angle

Pythagoras is only for right-angled triangles. So before doing any calculation, check that the triangle has a 90° angle.

Definition

Right-angled triangle

A right-angled triangle is a triangle with one right angle, which is an angle of 90°.

Definition

Hypotenuse

The hypotenuse is the side opposite the right angle. It is always the longest side in a right-angled triangle.

The hypotenuse is directly opposite the right angle and is the longest side of the right-angled triangle.

Example

Spotting the hypotenuse

A triangle $P QR$ has a right angle at $Q$ . Which side is the hypotenuse?

In triangle PQR, the side opposite the right angle at Q is PR, so PR is the hypotenuse.

Find the right angle. It is at $Q$ .
Look at the side opposite $Q$ .
The side opposite $Q$ is $P R$ , so the hypotenuse is $P R$ .

2. Squares, square roots, and the rule

To use Pythagoras, you need squares and square roots.

Definition

Squares and square roots

To square a number, multiply it by itself. For example, $x^2$ means $\times x$ .
A square root reverses squaring. $x\sqrt{x}$ means the positive number which squares to make $x$ .

Key Idea

Pythagoras' theorem

In any right-angled triangle, if $a$ and $b$ are the two shorter sides and $c$ is the hypotenuse, then $a^2 + b^2 = c^2$ .

The two shorter sides a and b meet at the right angle, and c is the hypotenuse opposite it.

Common Mistake

Right angles only

Pythagoras' theorem works only when you have a right angle. If there is no 90° angle, do not use it unless you can create one in the diagram.

3. Finding the hypotenuse: add the squares

If the missing side is the hypotenuse, square the two shorter sides, add them, then square root.

Example

Finding the longest side

A right-angled triangle has a right angle at $B$ . The two shorter sides are $\text{ cm}$ and $\text{ cm}$ . Find $A C$ .

AB and BC are the two shorter sides meeting at the right angle, so AC is the unknown hypotenuse.

The right angle is at $B$ , so the hypotenuse is $A C$ .
Use Pythagoras with $c$ as the hypotenuse:
$c2=5.42+7.22=29.16+51.84=81\begin{aligned} c^2 &= 5.4^2 + 7.2^2 \\ &= 29.16 + 51.84 \\ &= 81 \end{aligned}$
Square root to find the length:
$\sqrt{81} = 9$
So $\text{ cm}$ .

4. Finding a shorter side: subtract

If the hypotenuse is already given, the missing side must be one of the shorter sides. This time, subtract the square of the known shorter side from the square of the hypotenuse.

Key Idea

Add or subtract?

Add when you are finding the hypotenuse. Subtract when the hypotenuse is already known and you are finding a shorter side.

Example

Finding a shorter side

A right-angled triangle has hypotenuse $\text{ cm}$ and one shorter side $\text{ cm}$ . Find $E F$ to 1 decimal place.

DF is the given hypotenuse, so the missing side EF is one of the shorter sides.

The hypotenuse is $D F$ , so it goes first in the subtraction.
Work out the square of the missing side:
$EF2=162−72=256−49=207\begin{aligned} EF^2 &= 16^2 - 7^2 \\ &= 256 - 49 \\ &= 207 \end{aligned}$
Square root the answer:
$\sqrt{207} = 14.387\ldots$
To 1 decimal place, $\text{ cm}$ .

Common Mistake

Adding every time

Do not always add the squares. If the longest side is already given, you need to subtract to find the missing shorter side.

5. Pythagoras inside other shapes

Other shapes often hide right-angled triangles. A diagonal is a line joining two opposite corners of a shape. In a rectangle, the diagonal is the hypotenuse of a right-angled triangle.

A trapezium is a four-sided shape with one pair of parallel sides. In trapezium questions, look for right angles or draw in a height to create a right-angled triangle.

Example

Diagonal of a rectangle

A rectangle is 16 cm long and 9 cm wide. Find the length of its diagonal to 1 decimal place.

The diagonal of the rectangle forms the hypotenuse of a right-angled triangle with sides 16 cm and 9 cm.

The diagonal splits the rectangle into two right-angled triangles.
The diagonal is the hypotenuse, so add the squares of 16 and 9:
$d2=162+92=256+81=337\begin{aligned} d^2 &= 16^2 + 9^2 \\ &= 256 + 81 \\ &= 337 \end{aligned}$
Square root:
$\sqrt{337} = 18.357\ldots$
To 1 decimal place, the diagonal is 18.4 cm.

Tip

Isosceles triangles

An isosceles triangle has two equal sides. If you draw the perpendicular height from the top, it usually splits the base into two equal halves, making a right-angled triangle.

6. Using Pythagoras twice

Some questions contain two right-angled triangles joined together. You may need to find a shared side first, then use it in the second triangle.

Example

Two joined right-angled triangles

The shared side AC is found from triangle ABC, then used as a shorter side in triangle ACD to find AD.

Start with triangle $A B C$ , because it has two known sides and lets you find $A C$ .
In triangle $A B C$ , $B C$ is the hypotenuse:
$AC^2 = 13^2 - 12^2 = 25$
Find $A C$ :
$\sqrt{25} = 5$
Now use triangle $A C D$ . The sides $A C$ and $C D$ meet at the right angle, so $A D$ is the hypotenuse:
$AD^2 = 5^2 + 6^2 = 61$
Square root and round:
$\sqrt{61} = 7.810\ldots$
To 3 significant figures, $\text{ m}$ .

7. Worded problems and units

Tip

Sketch first

For journeys, “North then East” makes a right angle. For ladders, the wall and ground make a right angle, and the ladder is the hypotenuse.

Example

A ladder against a wall

A ladder reaches 2.4 m up a wall. Its base is 90 cm from the wall. Find the length of the ladder to 3 significant figures.

The wall and ground form the right angle, and the ladder is the unknown hypotenuse.

The wall and ground form a right angle, so the ladder is the hypotenuse.
Convert 90 cm into metres: 90 cm is 0.9 m.
Use Pythagoras:
$l2=2.42+0.92=5.76+0.81=6.57\begin{aligned} l^2 &= 2.4^2 + 0.9^2 \\ &= 5.76 + 0.81 \\ &= 6.57 \end{aligned}$
Square root:
$\sqrt{6.57} = 2.563\ldots$
To 3 significant figures, the ladder is 2.56 m long.

Exam technique

In the exam

Mark the right angle and label the hypotenuse before writing the formula.
Decide whether to add or subtract: find the hypotenuse means add; find a shorter side means subtract.
Keep full calculator values until the final line, then round to the accuracy asked for.

Self review

Check yourself

Can you identify the hypotenuse in any right-angled triangle?
If the hypotenuse is already given, do you know why you subtract?
In a worded problem, can you sketch the right-angled triangle and keep the units matching?

What you'll learn

1. Start with the right angle

2. Squares, square roots, and the rule

3. Finding the hypotenuse: add the squares

4. Finding a shorter side: subtract

5. Pythagoras inside other shapes

6. Using Pythagoras twice

7. Worded problems and units

Recap questions

Review Pythagoras by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Pythagoras

What you'll learn

1. Start with the right angle

2. Squares, square roots, and the rule

3. Finding the hypotenuse: add the squares

4. Finding a shorter side: subtract

5. Pythagoras inside other shapes

6. Using Pythagoras twice

7. Worded problems and units

Recap questions

Review Pythagoras by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Pythagoras

What you'll learn

1. Start with the right angle

2. Squares, square roots, and the rule

3. Finding the hypotenuse: add the squares

4. Finding a shorter side: subtract

5. Pythagoras inside other shapes

6. Using Pythagoras twice

7. Worded problems and units

Recap questions

Review Pythagoras by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Pythagoras Revision Guide

Pythagoras

What you'll learn

1. Start with the right angle

2. Squares, square roots, and the rule

3. Finding the hypotenuse: add the squares

4. Finding a shorter side: subtract

5. Pythagoras inside other shapes

6. Using Pythagoras twice

7. Worded problems and units

Recap questions

Review Pythagoras by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Pythagoras Revision Guide