Lesson 6.2 · 28 min

Pythagoras' Theorem

Finding a missing side of a right-angled triangle — and knowing whether to add or subtract.

In this lesson you will learn to…

Identify the hypotenuse of a right-angled triangle
Use $a^{2} + b^{2} = c^{2}$ to find the hypotenuse
Rearrange it to find a shorter side
Decide whether a triangle is right-angled
Apply the theorem to a real problem

Find the hypotenuse first

Before touching a formula, find the hypotenuse: the side opposite the right angle. It is always the longest side of the triangle.

This one habit removes almost every error in the topic, because it immediately tells you which version of the calculation you need:

The hypotenuse is the missing side → square and add.
The hypotenuse is one of the given sides → square and subtract.

Use the interactive triangle above to get a feel for it: as you drag the corners, notice that $c$ is always bigger than either short side, but always smaller than their sum.

The method

Label the right angle and identify the hypotenuse.
Decide: adding or subtracting?
Square the two sides you know.
Add or subtract as appropriate.
Square root — this is the step people forget, leaving the answer as $c^2$ .
Check it is sensible: the hypotenuse must be the largest of the three sides.

The sensible-answer check

This catches most slips in seconds. If you find a hypotenuse of $3\ \text{cm}$ for a triangle with a $4\ \text{cm}$ side, you have subtracted when you should have added.

Equally, a shorter side must come out smaller than the hypotenuse. If it does not, go back.

Real problems

Pythagoras appears in disguise whenever there is a right angle:

a ladder against a wall (ground, wall, ladder)
the diagonal of a rectangle or a TV screen
the straight-line distance between two points on a grid
the height of an isosceles triangle, by cutting it in half

The first move is always the same: sketch it, mark the right angle, and label the hypotenuse. Once you have done that, the rest is arithmetic.

Pythagorean triples. Some triangles have whole-number sides: $3, 4, 5$ ; $5, 12, 13$ ; $8, 15, 17$ ; $7, 24, 25$ . Their multiples work too, so $6, 8, 10$ and $9, 12, 15$ are also triples. Spotting one saves you the calculator — and examiners use them often.

Key rules and formulas

Pythagoras' theorem
$a^{2} + b^{2} = c^{2}$
For a right-angled triangle only, where $c$ is the hypotenuse and $a$ and $b$ are the two shorter sides.
The hypotenuse
The longest side, always opposite the right angle, never one of the two sides that form it. Find it first, every time, before writing anything down.
Finding the hypotenuse — add
$c = a^{2} + b^{2}$
Square both short sides, add, square root.
Finding a shorter side — subtract
$a = c^{2} - b^{2}$
Square the hypotenuse, subtract the other square, square root. Always the big square minus the small one.
Is it right-angled?
Test whether $a^{2} + b^{2} = c^{2}$ using the longest side as $c$ . If the two sides are equal, the triangle is right-angled; if not, it is not.

Try it

Drag the triangle

Height, a (cm)3

Base, b (cm)4

a² + b²: 3² + 4² = 9 + 16 = 25
c², so c: √25 = 5 cm

Drag the two red corners to change the short sides, and watch $a^{2} + b^{2}$ and $c$ update. Try $a = 3, b = 4$ and then $a = 6, b = 8$ — both give whole-number hypotenuses. These are called Pythagorean triples.

Worked examples

Worked example 1

Finding the hypotenuse

A right-angled triangle has shorter sides

3 cm

and

4 cm

. Find the hypotenuse.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Finding a shorter side

A right-angled triangle has hypotenuse

12 cm

and one shorter side

4 cm

. Find the other shorter side, to 2 decimal places.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Testing for a right angle

Is a triangle with sides

8 cm

15 cm

and

17 cm

right-angled?

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

A real problem

A ladder

5 m

long leans against a wall with its foot

1.5 m

from the base. How far up the wall does it reach, to 2 decimal places?

Try it yourself first, then reveal the solution one step at a time.

Practice: Pythagoras' theorem

10 questions

Ten questions, including two where you click on the triangle and one where you drag its corners.

Question 1 of 10Click the diagramFoundation
Click the hypotenuse of this right-angled triangle.
Click the right part of the diagram. You can also tab to each part and press Enter.
Question 2 of 10Click the diagramFoundation
Click the right angle in this triangle.
Click the right part of the diagram. You can also tab to each part and press Enter.
Question 3 of 10Type the answerFoundation
A right-angled triangle has shorter sides $3 cm$ and $4 cm$ . Find the hypotenuse.
Your answercm
Question 4 of 10Type the answer
A right-angled triangle has hypotenuse $12 cm$ and one shorter side $4 cm$ . Find the other shorter side, to 2 decimal places.
Your answercm
Question 5 of 10Explore and match
Drag the triangle's corners (or use the sliders) to make one with short sides $6 cm$ and $8 cm$ . Look at what the hypotenuse comes out as.
Move the controls until you match the question, then press Check.
Height, a (cm)3
Base, b (cm)4
a² + b²
3² + 4² = 9 + 16 = 25
c², so c
√25 = 5 cm
Target hypotenuse
0 cm
Question 6 of 10Multiple choice
A triangle has hypotenuse $13 cm$ and one short side $5 cm$ . Which calculation finds the other short side?
Question 7 of 10True or false
A triangle with sides $8 cm$ , $15 cm$ and $17 cm$ is right-angled.
Question 8 of 10Step by stepChallenge
A real problem, one step at a time.
A ladder $5 m$ long leans against a wall. Its foot is $1.5 m$ from the base of the wall. How far up the wall does it reach?
1. Which side of the right-angled triangle is the ladder?
Question 9 of 10Put in order
Put the steps for finding a missing side into the right order.
Put these in the right order. Drag them, or use the arrow buttons.
1. Find the hypotenuse — the side opposite the right angle
2. Check the hypotenuse is the longest side
3. Square the two sides you know
4. Decide whether to add or subtract the squares
5. Take the square root
6. Add or subtract them
Question 10 of 10Type the answerChallenge
A rectangle is $9 cm$ by $12 cm$ . Find the length of its diagonal.
Your answercm

In this lesson you will learn to…

Find the hypotenuse first

The method

The sensible-answer check

Real problems

Key rules and formulas