GCSE Maths Geometry & Measures Notes
Geometry and measures cover angles, shape, area, volume and the theorems that connect them. Clear diagrams and known angle facts win steady marks.
Key concepts
- Angle facts
- Angles on a line sum to 180°, around a point 360°, and in a triangle 180°.
- Pythagoras' theorem
- In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
- Trigonometry (SOH CAH TOA)
- Relates the sides and angles of right-angled triangles using sin, cos and tan.
- Area and volume
- Area is measured in square units, volume in cubic units — check units match.
- The hypotenuse is always the longest side, opposite the right angle.
- Parallel-line angles: alternate (Z) angles are equal, co-interior (C) angles sum to 180°.
- The interior angles of a polygon sum to (n − 2) × 180°.
Key formulas
Pythagoras
a² + b² = c²
Area of a circle
A = πr²
Circumference
C = πd = 2πr
Volume of a prism
V = cross-section area × length
Worked example
A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.
- 1Use Pythagoras: c² = 6² + 8².
- 2Calculate: c² = 36 + 64 = 100.
- 3Square root: c = √100 = 10.
Answer: 10 cm
Common mistakes
✗ Using Pythagoras on a non-right-angled triangle.
✓ a² + b² = c² only works when there is a right angle.
✗ Mixing area and volume units.
✓ Area uses cm²/m²; volume uses cm³/m³ — always include the correct unit.
✗ Forgetting to give reasons in angle questions.
✓ State the angle fact used, e.g. 'alternate angles are equal', for the reasoning mark.
Quick check
Try these, then reveal the answers.
Angles in a triangle sum to what?Reveal answer
180°
Find the area of a circle with radius 5 cm (to 1 dp).Reveal answer
78.5 cm²
In a² + b² = c², which side is c?Reveal answer
The hypotenuse.
Exam tips
- Always give a reason with every angle you find — reasons carry marks.
- Label diagrams as you work; add lengths and angles you calculate.
- Decide first whether a problem needs Pythagoras (sides only) or trigonometry (an angle involved).
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Frequently asked questions
When do I use Pythagoras and when do I use trigonometry?+
Use Pythagoras when you only involve the three sides; use trigonometry when an angle is given or required.
How do I find the interior angle sum of a polygon?+
Use (n − 2) × 180°, where n is the number of sides.
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