Geometry and Measurement
Geometry and Measurement tests spatial reasoning and formula use: angles, triangles, circles, perimeter, area, surface area, volume, similarity, coordinate geometry, and unit conversions. Diagrams are described in words, so translate each condition into an equation before calculating.
Core rules
- Angles: a straight line is 180°, a triangle's interior angles sum to 180°, and vertical angles are equal.
- Pythagorean theorem: in a right triangle, a² + b² = c², where c is the hypotenuse. The converse identifies right triangles.
- Perimeter and circumference measure length; area measures square units; volume measures cubic units.
- Common formulas: rectangle A = lw; triangle A = ½bh; circle A = πr² and C = 2πr; cylinder V = πr²h; cone and pyramid V = ⅓ × base area × height.
- Similar figures have equal angles and proportional sides. If the scale factor is k, lengths scale by k, areas by k², and volumes by k³.
- Coordinate geometry: distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Worked example
A right triangle has legs 9 cm and 12 cm. Find the hypotenuse and the area. Hypotenuse: c² = 9² + 12² = 81 + 144 = 225, so c = 15 cm. Area: A = ½(9)(12) = 54 cm². If a similar triangle has a scale factor of 3, its corresponding area is 54 × 3² = 486 cm², not 54 × 3.
Traps that cost marks
- Using diameter as radius in circle formulas, or forgetting to halve it.
- Mixing perimeter and area, or reporting area in linear units.
- Applying a scale factor directly to area or volume instead of squaring or cubing it.
- Rounding π too early and losing precision; keep π symbolic until the final step.
- Converting units after calculating, when converting dimensions first is safer; label every unit.