Geometry and Measurement
Core content
Geometry and measurement questions test angle relationships, triangle properties, similarity and congruence, the Pythagorean theorem, perimeter, area, surface area, volume, circles, and measurement conversion. Know that the interior angles of a triangle sum to 180 degrees, that similar figures have proportional sides, and that the Pythagorean theorem applies only to right triangles. Be able to decompose composite figures and to choose the correct formula: circumference = pi times diameter, area of a circle = pi times radius squared, and volume of a prism = base area times height. Coordinate geometry connects to distance and midpoint.
Traps
The biggest traps are using diameter as radius, using slant height where perpendicular height is required, and mixing units. For similar figures, a scale factor k multiplies lengths, k^2 multiplies areas, and k^3 multiplies volumes; students often apply k to area. In composite figures, subtracting an unshaded region is often easier than adding pieces. Check whether a question asks for surface area or volume, and whether units should be squared or cubed. Do not assume a triangle is right because it looks right.
Worked example
A cylinder has radius 3 cm and height 10 cm. Volume = pi r^2 h = pi(3^2)(10) = 90pi cubic cm, about 283 cubic cm. If a question asks for surface area instead, use 2pi r^2 + 2pi rh = 18pi + 60pi = 78pi square cm.
Rule of thumb
When a figure is scaled by factor k, multiply lengths by k, areas by k^2, and volumes by k^3.