Year 11 General Shape And Measurement

Pythagoras Theorem

Q: What is Pythagoras' theorem?

Pythagoras' theorem states that for any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared plus b squared equals c squared.

Q: What are Pythagorean triples?

Sets of three whole numbers that satisfy a squared plus b squared equals c squared. Common ones are 3-4-5, 5-12-13, 8-15-17, and 7-24-25, plus any whole number multiples.

Q: Does Pythagoras' theorem work for any triangle?

No. Pythagoras only works for right-angled triangles. For triangles without a right angle, use the cosine rule instead.

13 practice questions 2 video lessons Theory + worked examples

Theory

Pythagoras' theorem relates the three sides of a right-angled triangle: $a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse (the longest side, opposite the right angle). Add the squares to find the hypotenuse; subtract to find a shorter side. Recognising Pythagorean triples like 3-4-5 and 5-12-13 saves calculation time.

Pythagoras' theorem relates the three sides of a right-angled triangle. The longest side, opposite the right angle, is called the hypotenuse. The theorem says: the square of the hypotenuse equals the sum of the squares of the other two sides.

If the hypotenuse is $c$ and the two shorter sides are $a$ and $b$ , then $a^{2} + b^{2} = c^{2}$ . To find the hypotenuse, add the two squares and take the square root. To find a shorter side, subtract from the hypotenuse squared.

Pythagorean triples are sets of three whole numbers that satisfy the theorem. The most common are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any multiples also work, so 6-8-10 and 9-12-15 are triples too. Spotting one saves a calculation.

Pythagoras only works for right-angled triangles. If there's no right angle, you need the cosine rule instead.

c is opposite the right angle

For 3-4-5: areas 9 + 16 = 25

Pythagoras' theorem (for any right-angled triangle):

a^{2} + b^{2} = c^{2}

a^{2} + b^{2} = c^{2}

To find the hypotenuse $c$ :

c = \sqrt{a^{2} + b^{2}}

c = \sqrt{a^{2} + b^{2}}

To find a shorter side $a$ :

a = \sqrt{c^{2} - b^{2}}

a = \sqrt{c^{2} - b^{2}}

Quick reference:

To find	Use
The hypotenuse $c$	$c = \sqrt{a^{2} + b^{2}}$
A shorter side $a$	$a = \sqrt{c^{2} - b^{2}}$

Common Pythagorean triples: 3-4-5 (and 6-8-10, 9-12-15…), 5-12-13, 8-15-17, 7-24-25. Spotting one means no square roots needed.

How to solve any Pythagoras problem

Identify the hypotenuse — it's opposite the right angle and the longest side.
Decide what you're finding: the hypotenuse (add the squares) or a shorter side (subtract from the hypotenuse squared).
Substitute and solve. Take the positive square root.
Check the units and round as the question requires.

Always sketch the triangle for word problems. The sketch tells you which side is the hypotenuse and which is the unknown.

Example 1 — Find the hypotenuse

A right-angled triangle has shorter sides of

9

cm and

12

cm. Find the hypotenuse.

Solution

Add the two squares and take the square root.

$c^{2}$	$=$	$9^{2} + 12^{2}$
$c^{2}$	$=$	$81 + 144 = 225$
$c$	$=$	$\sqrt{225} = 15$ cm

c = 15

The hypotenuse is $15$ cm (a 9-12-15 triple).

Example 2 — Find a shorter side

A right-angled triangle has hypotenuse

13

m and one shorter side of

5

m. Find the other shorter side.

Solution

Subtract from the hypotenuse squared.

$a^{2}$	$=$	$c^{2} - b^{2}$
$a^{2}$	$=$	$13^{2} - 5^{2} = 169 - 25$
$a^{2}$	$=$	$144$
$a$	$=$	$\sqrt{144} = 12$ m

a = 12

The other shorter side is $12$ m (a 5-12-13 triple).

Example 3 — Ladder against a wall

5

m ladder leans against a wall. Its base is

1.5

m from the wall. How high up the wall does it reach (2 dp)?

Solution

The ladder is the hypotenuse; find the height $h$ .

$h^{2}$	$=$	$5^{2} - {1.5}^{2}$
$h^{2}$	$=$	$25 - 2.25 = 22.75$
$h$	$=$	$\sqrt{22.75}$
$h$	$\approx$	$4.77$ m

h \approx 4.77

The ladder reaches about $4.77$ m up the wall.

Example 4 — Diagonal of a rectangle

A rectangular field measures

40

m by

75

m. Find the length of its diagonal.

Solution

The diagonal is the hypotenuse of a right triangle with sides 40 and 75.

$d^{2}$	$=$	$40^{2} + 75^{2}$
$d^{2}$	$=$	$1,600 + 5,625$
$d^{2}$	$=$	$7,225$
$d$	$=$	$\sqrt{7,225} = 85$ m

d = 85

The diagonal is exactly $85$ m.

Common pitfalls

Wrong direction of subtraction. Finding a shorter side, the formula is

a^{2} = c^{2} - b^{2}

, with the hypotenuse squared first.

b^{2} - c^{2}

gives a negative number that has no real square root.

Picking the wrong side as the hypotenuse. The hypotenuse is always opposite the right angle, and it's the longest side. Sketch first, mark the right angle, and you won't mix this up.

Forgetting to take the square root. The formula gives

c^{2}

first — you need one more step to get

c

itself.

Using Pythagoras without a right angle. If the triangle has no right angle, Pythagoras doesn't apply. Use the cosine rule.

Frequently asked questions

What is Pythagoras' theorem?

For any right-angled triangle: $a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse (the longest side, opposite the right angle).

What is the hypotenuse of a triangle?

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

How do you find the hypotenuse using Pythagoras?

$c = \sqrt{a^{2} + b^{2}}$ . Square the two shorter sides, add them, then take the square root.

How do you find a shorter side using Pythagoras?

$a = \sqrt{c^{2} - b^{2}}$ . Square the hypotenuse, subtract the square of the known shorter side, then take the square root.

What are Pythagorean triples?

Sets of three whole numbers that satisfy $a^{2} + b^{2} = c^{2}$ . The most common are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Whole-number multiples (like 6-8-10) also work.