Sheet № 48 · Foundation + Higher · AQA · Edexcel · OCR
Pythagoras' Theorem
Pythagoras' theorem is one of the most frequently tested topics in GCSE Maths and one of the most reliable sources of marks on the exam. It appears on both Foundation and Higher papers across AQA, Edexcel and OCR, sometimes as a standalone question, sometimes embedded inside a larger problem involving coordinates, trigonometry or 3D shape
§Key definitions
Step 1:
c² = a² + b² = 5² + 12² = 25 + 144 = 169.
Step 2:
c = √169 = 13 cm.
§Formulas to memorise
a^2 + b^2 = c^2
d = \sqrt{l^2 + w^2 + h^2}
AC = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}
AG = \sqrt{AC^2 + 3^2} = \sqrt{52 + 9} = \sqrt{61} = 7.8 \text{ cm (1 d.p.)}
Add them: a² + b² = c².
Take the square root: c = √(a² + b²).
Rearrange: a² = c² − b².
Worked example
See example below.
A right-angled triangle has legs of 5 cm and 12 cm. Find the hypotenuse.
⚠ Common mistakes
- ✗Labelling the wrong side as the hypotenuse. Always check which side is opposite the right angle.
- ✗Adding when you should subtract. Finding the hypotenuse → add. Finding a shorter side → subtract. Get this the wrong way round and the answer is completely wrong.
- ✗Forgetting to square root at the end. The formula gives you c² — you must take the square root to find c.
- ✗Squaring incorrectly. 5² = 25, not 10. Take care with basic arithmetic.
- ✗Applying Pythagoras to non-right-angled triangles. The theorem only works for right-angled triangles. If the triangle does not have a 90° angle, you need the cosine rule instead.
✦ Exam tips
- →Draw a diagram if one is not provided. Label the right angle and the hypotenuse. This takes seconds and prevents errors.
- →Show every step — write the formula, show the substitution, show the addition or subtraction, then the square root. Each step can earn a method mark.
- →Use exact values where possible. If the question says "give an exact answer", leave as √52 rather than rounding.
- →In coordinate geometry, the distance formula is just Pythagoras applied to the horizontal and vertical differences.
- →For 3D questions, sketch the triangle you are using and label the sides clearly.