Measurement and Geometry
Measurement and Geometry tests perimeter, area, volume, surface area, angle relationships, triangle properties, similarity, the Pythagorean theorem, and unit conversions in both metric and customary systems. Many items combine two ideas, such as similarity plus area.
Core rules
- Circle: C = 2πr = πd and A = πr². The diameter is twice the radius.
- Triangle area = ½bh; trapezoid area = ½(b₁ + b₂)h; parallelogram area = bh.
- Volume: prism or cylinder V = Bh (B is base area); cone or pyramid V = ⅓Bh; sphere V = (4/3)πr³.
- Interior angles of an n-gon sum to (n − 2) × 180°. Complementary angles total 90°, supplementary angles 180°.
- Similar figures have equal corresponding angles and proportional sides. If the scale factor is k, areas scale by k² and volumes by k³.
- Pythagorean theorem: a² + b² = c², where c is the side opposite the right angle.
Worked example
A circular pond of radius 7 m is surrounded by a path 2 m wide. Find the area of the path. The path is an annulus, so subtract the inner circle from the outer circle. Outer radius = 7 + 2 = 9 m. Path area = π(9²) − π(7²) = 81π − 49π = 32π ≈ 100.5 m². The width is added to the radius, not to the diameter, and it is never doubled.
Traps that cost marks
- Substituting diameter for radius (or the reverse) in circle and sphere formulas.
- Scaling area by k instead of k² when a figure is enlarged.
- Calling a triangle right because two sides look Pythagorean: 3, 4, 6 is not right, since 3² + 4² = 25 ≠ 36.
- Mixing centimeters and meters inside one formula without converting first.
- Forgetting that a 2 m wide path on both sides changes the diameter by 4 m, not 2 m.