Mensuration
What this domain tests
Mensuration tests surface areas and volumes of solids, including the frustum of a cone. AP SSC expects you to select the correct formula, use the correct radius, and keep units consistent.
Core rules
- Cuboid: total surface area = 2(lb + bh + hl); volume = lbh.
- Cube: total surface area = 6a²; volume = a³.
- Cylinder: curved surface area = 2πrh; total surface area = 2πr(r + h); volume = πr²h.
- Cone: slant height l = √(r² + h²); curved surface area = πrl; total surface area = πr(l + r); volume = (1/3)πr²h.
- Sphere: surface area = 4πr²; volume = (4/3)πr³.
- Hemisphere: total surface area = 3πr²; volume = (2/3)πr³.
- Frustum: slant height l = √[h² + (r₁ − r₂)²]; volume = (1/3)πh(r₁² + r₁r₂ + r₂²).
Worked example
A cone has radius 7 cm and height 24 cm. Find its slant height and volume. l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. V = (1/3) × (22/7) × 7² × 24 = (1/3) × 22 × 7 × 24 = 1232 cm³. The slant height is used for surface area; the height is used for volume.
Traps
- Using the diameter as the radius.
- Swapping slant height and perpendicular height in a cone.
- Using curved surface area when total surface area is requested.
- Writing square units for volume or cubic units for area.
- Forgetting the top and bottom circles of a cylinder.