Stereometry
Core concepts
Know the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; the relationships between similar solids; and the geometry of lines and planes in space, including dihedral angles and the angle between a line and a plane. Master the formulas V = (1/3)·B·h for pyramids and cones, V = (4/3)πr³ for spheres, and S = 4πr² for sphere surfaces.
Traps
- Forgetting the 1/3 factor for pyramids and cones.
- Confusing slant height with perpendicular height in cones and pyramids.
- Mixing up surface area and lateral area (excluding bases).
- For similar solids, applying the linear ratio to volume without cubing.
- Misidentifying the angle between a line and a plane; it is measured between the line and its projection on the plane.
Worked example
A cone has radius 3 and height 4. Its slant height is √(3² + 4²) = 5. Volume = (1/3)π·3²·4 = 12π. Total surface area = πr² + πrl = 9π + 15π = 24π.
Rule of thumb
For any solid, sketch the right triangle formed by the height, radius, and slant height - the Pythagorean theorem then links all three.