Geometry and Measurement
Domain overview
Geometry and Measurement covers plane and solid geometry, coordinate geometry, transformations, congruence and similarity, right-triangle trigonometry, and measurement reasoning. The exam tests both formula use and conceptual understanding of why formulas work, including unit conversions and precision.
Core rules
- Angle relationships: vertical angles are equal; angles on a line sum to 180°; interior angles of a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles.
- Similar figures: corresponding angles are equal and side lengths are proportional; scale factor k makes areas scale by k^2 and volumes by k^3.
- Circles: circumference C = 2πr, area A = πr^2; central angle/360° gives arc length and sector area fractions.
- Solids: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr^3.
- Right triangles: a^2 + b^2 = c^2; sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Coordinate geometry: distance and midpoint formulas; parallel lines have equal slopes, perpendicular slopes are negative reciprocals.
Worked example
A cone has radius 3 cm and height 8 cm. Find its volume. V = (1/3)πr^2h = (1/3)π(3^2)(8) = (1/3)π(9)(8) = 24π ≈ 75.4 cm^3. The reasoning is to identify the base area πr^2 first, then apply the one-third factor that distinguishes a cone from a cylinder with the same base and height. The answer is in cubic units because volume measures three-dimensional space.
Traps
- Using slant height instead of perpendicular height in cone volume.
- Reporting square units for volume or cubic units for area.
- Applying a scale factor directly to area instead of squaring it.
- Mixing degrees and radians in trigonometric calculations.
- Rounding intermediate values and losing precision in the final answer.