What is the formula for Pythagoras’ theorem?

The formula is $a^2+b^2=c^2$. The letters $a$ and $b$ represent the two shorter sides of a right-angled triangle, while $c$ is the hypotenuse opposite the right angle.

How do I know whether to add or subtract in Pythagoras’ theorem?

Add the squares when finding the hypotenuse: $c=\sqrt{a^2+b^2}$. When finding a shorter side, subtract the known shorter side squared from the hypotenuse squared.

Does Pythagoras’ theorem work on every triangle?

No. It applies only to **right-angled triangles**. For a triangle without a right angle, a different method may be needed.

How can I identify the hypotenuse?

Find the right angle and look directly opposite it. That side is the hypotenuse. It is also the longest side of a right-angled triangle.

Is three-dimensional Pythagoras included in GCSE Maths?

Three-dimensional applications are generally studied at Higher Tier. They often require two connected Pythagoras calculations, first across a face and then through the solid.

Should I round during a Pythagoras calculation?

Keep the full calculator value during intermediate stages and round only the final answer. Follow the requested number of decimal places or significant figures, and include the correct unit.

Pythagoras’ Theorem GCSE Maths Revision Guide

Pythagoras’ theorem GCSE Maths questions involve finding or checking side lengths in right-angled triangles. The central rule is $a^2+b^2=c^2$ , where $c$ is always the hypotenuse, the side opposite the right angle.

This guide explains how to identify the correct sides, rearrange the formula, present complete working and tackle common exam applications. It includes Pythagoras’ theorem worked examples and exam-style questions with answers for both Foundation and Higher revision.

What is Pythagoras’ theorem?

Pythagoras’ theorem describes the relationship between the three side lengths of a right-angled triangle:

a^2+b^2=c^2

In this formula:

$a$ and $b$ are the two shorter sides that meet at the right angle
$c$ is the hypotenuse, which is opposite the right angle
$c$ is always the longest side

The theorem works only for right-angled triangles. A triangle may be turned, reflected or included inside another shape, so do not assume that the hypotenuse is always drawn horizontally or diagonally. Locate the right-angle symbol first, then identify the side directly opposite it.

On common GCSE specifications, applying Pythagoras’ theorem in two-dimensional figures forms part of Foundation content. Higher Tier questions can also involve three-dimensional shapes, several connected triangles and problems that combine Pythagoras with trigonometry or other geometry.

How to use Pythagoras’ theorem step by step

The correct method depends on whether the unknown side is the hypotenuse or one of the shorter sides.

Finding the hypotenuse

When the two shorter sides are known, add their squares:

c^2=a^2+b^2

Then take the positive square root:

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

Lengths are positive, so only the positive square root is relevant in a GCSE geometry problem.

Finding a shorter side

When the hypotenuse and one shorter side are known, subtract the smaller square from the hypotenuse squared. For example, if $a$ is unknown:

a^2=c^2-b^2

Therefore:

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

A useful decision rule is:

finding the hypotenuse means add the two smaller squares
finding a shorter side means subtract from the hypotenuse squared

A reliable exam method

Use this process whenever a question involves a right-angled triangle:

Mark or locate the right angle.
Identify the hypotenuse opposite the right angle.
Label the hypotenuse $c$ .
Substitute the known lengths into $a^2+b^2=c^2$ .
Rearrange before using the calculator if a shorter side is unknown.
Take the square root only after completing the addition or subtraction.
Round at the end and include the correct unit.

Writing the substitution clearly can earn method marks even if a later calculation is incorrect.

Pythagoras’ theorem worked examples

Worked example 1: finding the hypotenuse

A right-angled triangle has shorter sides of $6$ cm and $8$ cm. Find the hypotenuse.

Start with the theorem:

a^2+b^2=c^2

Substitute the two shorter sides:

6^2+8^2=c^2

36+64=c^2

100=c^2

Take the square root:

c=\sqrt{100}=10

The hypotenuse is $10$ cm.

Worked example 2: finding a shorter side

A right-angled triangle has a hypotenuse of $13$ cm and another side of $5$ cm. Find the missing side $x$ .

The hypotenuse must be used as $c$ :

x^2+5^2=13^2

x^2+25=169

x^2=169-25

x^2=144

x=\sqrt{144}=12

The missing side is $12$ cm.

This calculation uses subtraction because the unknown is not the hypotenuse.

Worked example 3: finding the diagonal of a rectangle

A rectangle is $9$ cm long and $12$ cm wide. Find its diagonal.

The diagonal divides the rectangle into two right-angled triangles. It is the hypotenuse of either triangle, so:

d^2=9^2+12^2

d^2=81+144=225

d=\sqrt{225}=15

The diagonal is $15$ cm.

This illustrates why you may need to identify or construct a right-angled triangle before using the theorem.

Worked example 4: three-dimensional Pythagoras

A cuboid measures $3$ cm by $4$ cm by $12$ cm. Find the length of its space diagonal.

First find the diagonal across the $3$ cm by $4$ cm base:

d^2=3^2+4^2=25

d = 5

The base diagonal, vertical edge and space diagonal form another right-angled triangle. If the space diagonal is $s$ :

s^2=5^2+12^2

s^2=25+144=169

s = 13

The space diagonal is $13$ cm.

Three-dimensional questions often require Pythagoras twice. Keep the exact result from the first calculation where possible, rather than rounding it prematurely. Higher Tier students can revise this further using the 3D Pythagoras and trigonometry lesson.

Pythagoras’ theorem exam questions

Embedded exam-style question

A ladder is $6.5$ m long. Its base is $2.4$ m from a vertical wall, and its top touches the wall. Calculate the height reached by the ladder. Give your answer to $3$ significant figures.

The ladder is the hypotenuse, so if the height is $h$ :

h^2+2.4^2=6.5^2

h^2=6.5^2-2.4^2

h^2=42.25-5.76=36.49

h=\sqrt{36.49}=6.040\ldots

To $3$ significant figures, the ladder reaches $6.04$ m up the wall.

Try 4 more Pythagoras practice questions, with worked solutions and mark-scheme guidance.

Further exam-style practice

Question 1

The perpendicular sides of a right-angled triangle are $7.2$ cm and $9.5$ cm. Calculate the hypotenuse to $3$ significant figures.

Answer:

c=\sqrt{7.2^2+9.5^2}=\sqrt{142.09}=11.920\ldots

The hypotenuse is $11.9$ cm to $3$ significant figures.

Question 2

A triangle has side lengths $8$ cm, $15$ cm and $17$ cm. Determine whether it is right-angled.

Answer:

The longest side is $17$ cm, so compare its square with the sum of the other two squares:

8^2+15^2=64+225=289

17^2=289

Since $8^2+15^2=17^2$ , the triangle is right-angled.

Question 3

A rectangle has a diagonal of $20$ cm and a width of $12$ cm. Calculate its area.

Answer:

Let the length be $x$ :

x^2+12^2=20^2

x^2=400-144=256

x = 16

The area is:

16\times12=192

The area is $192$ cm $^2$ .

More topic practice is available in the Pythagoras revision lesson, the OCR Pythagoras revision resources and the GCSE exam booklets.

Checking whether a triangle is right-angled

The converse of Pythagoras’ theorem can be used to test a triangle. First identify the longest side and call it $c$ . Then compare $c^2$ with $a^2+b^2$ .

If $a^2+b^2=c^2$ , the triangle is right-angled.
If the values are not equal, the triangle is not right-angled.

The longest side must be used on its own. Testing the wrong arrangement can produce an invalid conclusion even when the arithmetic is accurate.

Common mistakes in Pythagoras’ theorem questions

Using the theorem on a non-right-angled triangle

Pythagoras’ theorem applies only when there is a right angle. Do not use it simply because three side lengths are shown.

Misidentifying the hypotenuse

The hypotenuse is opposite the right angle, not merely the side that looks longest in a diagram. GCSE diagrams are not always drawn accurately.

Adding when a shorter side is missing

If the unknown is a shorter side, subtract its known partner’s square from the hypotenuse squared. Adding would produce an answer longer than the hypotenuse, which is impossible.

Forgetting to square the lengths

The theorem is not $a + b = c$ . Each length must be squared before the values are added or subtracted.

Taking the square root too early

For example, calculate $132−52\sqrt{13^2-5^2}$ , not $13 - 5$ . The subtraction happens between the squared values.

Rounding during an intermediate step

Keep the full calculator value until the final answer. Early rounding is particularly risky in multi-stage Higher Tier problems.

Omitting units or rounding instructions

Lengths use units such as cm or m, while areas use square units such as cm $^2$ . If the question requests a particular degree of accuracy, state the answer in that form.

How marks are awarded

A typical Pythagoras question awards marks for a valid method and an accurate answer. Although exact mark schemes vary, clear working should normally include:

a correct Pythagoras equation
correct substitution of the side lengths
a correct rearrangement where necessary
evaluation of the square root
appropriate rounding and units

Do not write only the calculator answer. A visible equation protects method marks and makes it easier to check whether you identified the hypotenuse correctly. The MathsGenie worked solutions show how marks are attached to stages of the method.

Revision advice for Pythagoras’ theorem GCSE Maths

Begin with straightforward triangles, then practise rectangles, isosceles triangles, coordinate problems and contextual questions. Higher Tier students should progress to three-dimensional shapes and multi-stage problems.

When checking an answer, compare it with the known sides. A calculated hypotenuse must be longer than either shorter side, while a missing shorter side must be less than the hypotenuse.

You should also practise recognising when Pythagoras is required without being told directly. Words such as diagonal, shortest distance, straight-line distance and right-angled are useful clues, but the geometry must still support the method.

Use the GCSE Maths topics by exam board and tier to place this topic within your full revision plan.

Conclusion

Pythagoras’ theorem links the three sides of a right-angled triangle through $a^2+b^2=c^2$ . Successful solutions depend on identifying the hypotenuse first, choosing addition or subtraction correctly, showing complete working and rounding only at the end.

MathsGenie provides revision lessons, exam-style questions, printable booklets, worked solutions and mark schemes. After learning the method, complete several mixed questions without notes and use the solutions to correct both mathematical errors and gaps in presentation.

Sources and referenced URLs

What is Pythagoras’ theorem?

Pythagoras’ theorem describes the relationship between the three side lengths of a right-angled triangle:

a^2+b^2=c^2

In this formula:

$a$ and $b$ are the two shorter sides that meet at the right angle
$c$ is the hypotenuse, which is opposite the right angle
$c$ is always the longest side