Measurement and Geometry
Measurement and geometric reasoning
Know perimeter, area, surface area, and volume formulas; angle relationships; congruence and similarity; triangle theorems, including the Pythagorean theorem, triangle inequality, and exterior angle theorem; circle theorems, including inscribed angle, tangent-radius, and chord relationships; and coordinate geometry, including distance, midpoint, slope, and transformations. Use units consistently and distinguish exact from approximate values. Understand the structure of a proof: given information, definitions, postulates, theorems, and a logical chain of justification.
Traps
- Scaling a figure by factor k multiplies lengths by k, areas by k², and volumes by k³.
- Slant height and perpendicular height differ in cones and pyramids.
- An inscribed angle is half its intercepted arc; a central angle equals its arc.
- The ambiguous SSA case can produce zero, one, or two triangles.
- Arc length and sector area require radians unless converted deliberately.
Worked example
Two similar solids have surface areas in ratio 9:25. The linear scale factor is √9:√25 = 3:5, so the volume ratio is 3³:5³ = 27:125. If the smaller volume is 54 cm³, the larger is 54·125/27 = 250 cm³. Rule of thumb: reduce every similarity problem to the linear scale factor first.