Geometry and Measurement
Geometry and Measurement tests angle relationships, triangle and circle theorems, coordinate geometry, and the calculation of perimeter, area, surface area, and volume in plane and solid figures.
Core rules
- Angles in a triangle sum to 180°; an exterior angle equals the sum of the two opposite interior angles.
- Similar figures have equal corresponding angles and proportional sides; areas scale by the square of the linear scale factor, volumes by its cube.
- Circle: circumference 2πr, area πr²; arc length = (θ/360°)·2πr for degrees; sector area = (θ/360°)·πr².
- Solids: cylinder volume πr²h; cone volume (1/3)πr²h, slant height l=√(r²+h²); sphere volume (4/3)πr³ and surface area 4πr².
- Coordinate geometry: distance √((x₂−x₁)²+(y₂−y₁)²); midpoint ((x₁+x₂)/2,(y₁+y₂)/2); parallel lines have equal gradients, perpendicular gradients multiply to −1.
Worked example
A cone has radius 6 cm and height 8 cm. Its slant height is l=√(6²+8²)=10 cm. Volume = (1/3)π(6²)(8)=96π cm³. Total surface area = πr²+πrl=36π+60π=96π cm². The equal numerical values are coincidental; keep units and square/cube dimensions distinct.
Traps
Use radius, not diameter. Convert units before calculating; 1 m²=10,000 cm², not 100 cm². In composite solids, do not double-count shared faces. For similar solids, a volume ratio is the cube of the side ratio. Angle-chasing answers need a stated theorem, not just a number.