Geometry and Measurement
This domain spans plane and solid geometry: angle relationships, triangle congruence and similarity, the Pythagorean theorem, circle properties, area and volume formulas, and coordinate geometry. You should be comfortable with proofs and with converting between units, including dimensional analysis.
Traps to watch
- Similar vs. congruent: similarity preserves angles and ratios of sides; scale factors apply to lengths, their square to areas, and their cube to volumes.
- Inscribed angle theorem: an inscribed angle is half its intercepted arc; an angle inscribed in a semicircle is right.
- Volume vs. surface area: doubling a dimension multiplies volume by 8, not 2.
- Unit conversion: mixed units (cm and m, minutes and hours) are a classic source of error-convert before computing.
Worked example / rule of thumb
Two similar cylinders have heights in ratio 2:3. If the smaller has volume $40\ \text{cm}^3$, the larger volume is $40\cdot(3/2)^3 = 40\cdot 27/8 = 135\ \text{cm}^3$. Rule of thumb: for similar figures, set up the ratio of corresponding lengths and raise it to the power matching the quantity-1 for length, 2 for area, 3 for volume. For coordinate geometry, use the distance and midpoint formulas and remember that perpendicular slopes are negative reciprocals. Always label units and check that answers are dimensionally consistent.