Geometry and Measurement
Geometry and Measurement - what the exam tests
This domain tests angle relationships, triangle properties, circles, area and volume, coordinate geometry, similarity, and unit conversion. Many items combine a described figure with a measurement conversion or a ratio between similar figures.
Core rules
- Angles: a straight line is 180° and a triangle is 180°; vertical angles are equal; with parallel lines cut by a transversal, alternate interior and corresponding angles are equal.
- Right triangles: a² + b² = c². The 45-45-90 triangle has sides in ratio 1:1:√2; the 30-60-90 triangle has sides 1:√3:2.
- Similar figures: if lengths scale by k, areas scale by k² and volumes by k³.
- Circles: circumference C = 2πr, area A = πr²; arc length = (θ/360)·2πr and sector area = (θ/360)·πr² when θ is in degrees.
- Volume: rectangular prism lwh; cylinder πr²h; cone (1/3)πr²h; sphere (4/3)πr³.
- Conversion: multiply by ratios equal to 1 and cancel units; convert first, then calculate.
Worked example
A cone has radius 3 and height 5. Volume = (1/3)π(3²)(5) = 15π ≈ 47.1 cubic units. If every dimension is doubled, the new volume is 2³ = 8 times as large, about 376.8 cubic units-area would grow only 4 times.
Traps that cost marks
- Using the diameter where the formula needs the radius.
- Mixing degrees and radians in arc or sector formulas.
- Confusing height with slant height in cone problems.
- Reporting perimeter when the question asks for area, or leaving square and cubic units incorrect.
- Assuming similar figures have equal areas rather than areas in ratio k².