Geometry
What the domain tests
Geometry covers plane figures and their perimeters and areas, solids and their volumes and surface areas, measurement with unit conversion, similarity, the Pythagorean theorem, and the basic trigonometric ratios in a right-angled triangle.
Core rules
- Angles: the angles of a triangle sum to 180°, of a quadrilateral to 360°. With parallel lines cut by a transversal, alternate angles are equal and corresponding angles are equal. An exterior angle of a triangle equals the sum of the two opposite interior angles.
- Areas: triangle ½ah; parallelogram ah; trapezium ((a + b)/2)·h; circle πr² with circumference 2πr.
- Pythagoras: in a right-angled triangle a² + b² = c², where c is the hypotenuse.
- Similarity: corresponding lengths are in ratio k. Perimeters scale by k, areas by k², volumes by k³.
- Solids: prism and cylinder V = A_base·h; pyramid and cone V = A_base·h/3; sphere V = (4/3)πr³. Surface area is the sum of the areas of all faces, with the curved surface of a cylinder equal to 2πrh.
- Trigonometry: sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent. Use the inverse functions to find the angle.
- Units: 1 m = 100 cm, so 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³; 1 L = 1 dm³ = 1000 cm³.
Worked example
Two similar triangles have corresponding sides 3 cm and 5 cm. The smaller triangle has area 18 cm². Find the area of the larger one. The length ratio is k = 5/3, so areas are in ratio k² = 25/9. Larger area = 18 · 25/9 = 2 · 25 = 50 cm². Notice that multiplying 18 by 5/3 alone would give 30 cm², which is wrong because area scales with the square of the length ratio.
Traps that cost marks
- Using the diameter as the radius in circle, cylinder and sphere formulas.
- Scaling an area by k instead of k², or a volume by k instead of k³.
- Converting lengths correctly but forgetting that area and volume units convert by squares and cubes.
- Applying Pythagoras to a triangle that is not right-angled, or taking the wrong side as the hypotenuse.
- Leaving the calculator in the wrong angle mode, or rounding π so early that the final answer drifts.