Measurement and Geometry
Measurement and Geometry
This domain tests measurement reasoning, plane and solid geometry, coordinate geometry, and transformations. Expect area, perimeter, volume, surface area, angle relationships, similarity, congruence, and geometric justification.
Core rules and concepts
- Triangle angle sum is 180°; an exterior angle equals the sum of the two remote interior angles.
- Pythagorean theorem: a² + b² = c² for right triangles. Common triples include 3-4-5 and 5-12-13; 45-45-90 and 30-60-90 triangles have fixed side ratios.
- Circle formulas: circumference C = 2πr, area A = πr², arc length = (θ/360)(2πr), sector area = (θ/360)(πr²).
- Similar figures have equal corresponding angles and proportional sides; scale factor k changes lengths by k, areas by k², volumes by k³.
- Coordinate geometry: distance = √((x_2 - x_1)² + (y_2 - y_1)²), midpoint averages coordinates, parallel slopes are equal, perpendicular slopes are negative reciprocals.
- Volume and surface area formulas must match the solid; composite figures require adding or subtracting parts without double-counting shared boundaries.
Worked example
A sector has radius 6 cm and central angle 60°. Find its arc length and area.
Arc length: (60/360)(2π · 6) = (1/6)(12π) = 2π cm. Sector area: (60/360)(π · 6²) = (1/6)(36π) = 6π cm². Reasoning check: 60° is one-sixth of a full circle, so both results should be one-sixth of the full circumference 12π and full area 36π.
Traps that cost marks
- Using diameter in place of radius in circle formulas.
- Mixing degrees and radians in arc or sector formulas.
- Applying a linear scale factor directly to area or volume.
- Counting an internal shared edge as exterior perimeter in composite figures.
- Forgetting square units versus cubic units.