Measurement and Geometry
What to know
Cover perimeter, area, surface area, and volume for standard figures, unit conversion within and across systems, angle relationships (complementary, supplementary, vertical, parallel-line transversals), triangle properties, the Pythagorean theorem, similarity and congruence, and coordinate geometry (distance, midpoint, transformations). Circle formulas - circumference $C = 2\pi r$, area $A = \pi r^2$ - appear frequently.
Traps to watch
- Area vs. perimeter confusion in word problems; read what is being asked.
- Units: convert before computing, and remember area scales by the square of a linear factor while volume scales by the cube. Doubling a figure's dimensions quadruples area and multiplies volume by eight.
- Similar figures: corresponding sides are proportional, but angles are equal - not proportional.
- Pythagorean theorem applies only to right triangles; the hypotenuse is always the longest side and opposite the right angle.
- Angle pairs: vertical angles are equal; angles on a straight line sum to 180°; interior angles of any triangle sum to 180°.
Worked example
A right triangle has legs 9 cm and 12 cm. Hypotenuse: $c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ cm. If a similar triangle has a hypotenuse of 25 cm, its scale factor is $25/15 = 5/3$, so its legs are $9 \cdot \tfrac{5}{3} = 15$ cm and $12 \cdot \tfrac{5}{3} = 20$ cm. Rule of thumb: identify the corresponding parts first, then set up the proportion.