Geometry and Trigonometry: shapes, triangles and bearings
This domain tests spatial reasoning with coordinates, solids and triangles: distance and midpoint, surface area and volume of prisms, cylinders, pyramids, cones and spheres, right-angled trigonometry, the sine and cosine rules, the area formula ½ab sin C, and applications such as bearings and angles of elevation and depression.
Core rules
- Distance = √((x₂−x₁)² + (y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
- Volume: prism Ah, cylinder πr²h, pyramid ⅓Ah, cone ⅓πr²h, sphere (4/3)πr³. Surface area formulas must be memorised; the AI SL formula booklet supplies only some.
- Right triangles: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
- Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. Area = ½ab sin C, where C is the included angle.
- Bearings are three-figure angles measured clockwise from north.
Worked example
Two markers are 8 km and 11 km from a boat, with a 62° angle between the lines of sight. Find the distance between them. The known angle is included between the two known sides, so use the cosine rule: a² = 8² + 11² − 2(8)(11)cos 62° = 64 + 121 − 176(0.46947) = 185 − 82.63 = 102.37, so a ≈ 10.1 km. The sine rule would be the wrong choice here because it needs a known side opposite a known angle.
Traps that cost marks
Leaving the GDC in radian mode; using the sine rule when the given angle is not opposite a given side; forgetting that the sine rule can give an ambiguous obtuse solution; substituting into ½ab sin C with a non-included angle; measuring bearings anticlockwise from north or writing two figures; mixing surface-area and volume units.