Geometry
The Geometry domain covers plane figures, solids, and coordinate geometry. You are expected to compute perimeter, area, volume, and surface area, apply angle relationships, use the Pythagorean theorem and special right triangles, and work with slope, distance, and midpoints on the coordinate plane.
Angles and triangles
- The angles of a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles.
- Parallel lines cut by a transversal: alternate interior angles are equal and consecutive interior angles are supplementary.
- Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x.
- Similar triangles have proportional corresponding sides and equal corresponding angles.
Circles and coordinates
- Circumference = 2πr and area = πr²; arc length = (θ/360)(2πr) and sector area = (θ/360)(πr²).
- Distance = √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint averages the coordinates; slope = (y₂ − y₁)/(x₂ − x₁).
- Parallel lines share a slope; perpendicular slopes are negative reciprocals.
- A circle with center (h, k) and radius r satisfies (x − h)² + (y − k)² = r².
Solids and scaling
- Prism and cylinder: V = Bh. Pyramid and cone: V = (1/3)Bh. Sphere: V = (4/3)πr³ and SA = 4πr².
- If all lengths scale by factor k, areas scale by k² and volumes by k³.
Worked example
A right cone has radius 3 and height 4. Its slant height is √(3² + 4²) = 5. Volume = (1/3)π(3²)(4) = 12π. Surface area = πr² + πrl = 9π + 15π = 24π. A similar cone with every dimension doubled has volume multiplied by 2³ = 8, giving 96π.
Traps
- Substituting the diameter where the formula calls for the radius.
- Using the vertical height instead of the slant height in the cone's lateral area πrl.
- Mixing units: a length is in cm, an area in cm², a volume in cm³.
- Applying a length scale factor directly to an area or volume instead of squaring or cubing it.