Pythagoras' theorem is one of the most famous and useful rules in all of mathematics. If you're studying for your GCSE maths exam (with AQA, Edexcel, or OCR), you must know how to apply it. This guide will walk you through everything you need to know, from the basic rule to tackling trickier problems.
What is Pythagoras' Theorem?
Pythagoras' theorem applies only to right-angled triangles. A right-angled triangle has one angle of 90°, marked with a small square.
The theorem states:
> In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's break down that statement:
Hypotenuse: The longest side of the triangle, which is always opposite the right angle.
- The other two sides: Often called the adjacent and opposite sides relative to a given angle, but for Pythagoras, we just call them the two shorter sides.
The Formula
The theorem is written as the famous formula:
a² + b² = c²
Where:
c is the length of the hypotenuse.
- a and b are the lengths of the other two sides.
flowchart TD
A["Identify the Right-Angled Triangle"] --> B["What side are you finding?"]
B --> C["The Hypotenuse (c)"]
B --> D["One of the shorter sides (a or b)"]
C --> E["Use formula: c = √(a² + b²)"]
D --> F["Use formula: a = √(c² - b²)"]
E --> G["Square, Add, Square Root"]
F --> H["Square, Subtract, Square Root"]
G --> I["State final answer with units"]
H --> I
Step-by-Step: Finding the Hypotenuse (Longest Side)
This is the most common type of question. You are given the two shorter sides and need to find c.
Example 1: A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.
Label the sides. a = 6, b = 8, c = ?
Write down the formula: a² + b² = c²
Substitute the known values: 6² + 8² = c²
Calculate the squares: 36 + 64 = c²
Add: 100 = c²
Solve for c by square rooting: c = √100
Calculate: c = 10
- State the answer with units: The hypotenuse is 10 cm.
Step-by-Step: Finding a Shorter Side
Sometimes the question gives you the hypotenuse and one shorter side, asking you to find the other shorter side.
Example 2: A ladder 5 m long leans against a wall. The foot of the ladder is 1.2 m from the wall. How far up the wall does the ladder reach? (This forms a right-angled triangle with the hypotenuse as the ladder).
Label the sides. Hypotenuse, c = 5. One shorter side, a = 1.2. The unknown side is the other shorter side, b = ?.
Write down the formula: a² + b² = c²
Substitute the known values: 1.2² + b² = 5²
Calculate the squares: 1.44 + b² = 25
Rearrange to isolate b²: b² = 25 - 1.44
Subtract: b² = 23.56
Solve for b by square rooting: b = √23.56
Calculate (using your calculator): b ≈ 4.8539...
- Round appropriately and state the answer with units: The ladder reaches approximately 4.85 m up the wall.
Key Skills and Common Pitfalls
Using Your Calculator Correctly
When finding a shorter side, the order of operations is crucial. For Example 2, to calculate b = √5² - 1.2² on your calculator, you must either:
Type: √5² - 1.2² or
- Type:
5² - 1.2² =and then press the square root button.
Never do √5² - √1.2² – this is wrong!
Leaving Your Answer in Exact Form (Surds)
For the Higher GCSE paper, you may be asked to leave your answer as a surd (an exact square root).
Example 3: Find the hypotenuse of a triangle with shorter sides 2 cm and 7 cm.
2² + 7² = c²
4 + 49 = c²
53 = c²
c = √53
The exact answer is √53 cm. You would only convert this to a decimal if the question asks for a rounded answer.
Is it a Right-Angled Triangle?
You can also use Pythagoras' theorem in reverse to check if a triangle is right-angled. If a² + b² = c² is true for the three sides (with c as the longest), then the triangle is right-angled. If it's false, it isn't.
Example 4: Is a triangle with sides 5 cm, 7 cm, and 9 cm right-angled?
Longest side is 9 cm, so this is c.
- Check: 5² + 7² = 25 + 49 = 74.
c² = 9² = 81.
- 74 ≠ 81, so the triangle is not right-angled.
Applying Pythagoras in Context
GCSE questions often embed Pythagoras in worded problems or diagrams. The key is to spot the right-angled triangle.
Diagonals of rectangles/squares: The diagonal splits the shape into two right-angled triangles.
Isosceles triangles: Drawing the perpendicular height from the apex creates two congruent right-angled triangles.
- 3D problems: Often require using Pythagoras twice – once in a 2D face of the 3D shape, and then again in the 3D space.
Example 5 (Problem Solving): A rectangular TV screen is labelled as 65" (inches). This measurement is the diagonal length. If the screen is 56.7" wide, how tall is it?
Spot the right-angled triangle: The width, height, and diagonal form the sides.
Diagonal is the hypotenuse, c = 65. Width is one side, a = 56.7. Height b = ?.
Apply the formula for a shorter side: b² = c² - a²
b² = 65² - 56.7² = 4225 - 3214.89 = 1010.11
b = √1010.11 ≈ 31.78
- The screen is approximately 31.8 inches tall.
Summary and Exam Tips
Identify: Is it a right-angled triangle? Look for the right-angle symbol or infer it from context (e.g., ladder against wall).
Label: Correctly identify the hypotenuse (longest side, opposite right angle) and the two shorter sides.
Choose Formula:
Finding hypotenuse: c = √a² + b²
Finding a shorter side: a = √c² - b²
Substitute & Calculate: Carefully substitute numbers, square them, then add or subtract, then square root.
Units: Always include units (cm, m, km) in your final answer.
- Exact or Decimal?: Leave your answer as a surd (e.g., √20) unless the question asks for a decimal rounding.
- By following these steps and practising plenty of questions, you'll turn Pythagoras' theorem from a daunting formula into a reliable tool for your GCSE maths success. Good luck!