Covering a solid
Surface area is the total area of all the surfaces of a 3D figure. Think of it as the amount of wrapping paper needed to cover the shape exactly.
It’s measured in square units (cm², m²).
Use a net
A net is a flat pattern that folds up into the solid. Surface area = total area of the net.
A cube’s net is six squares. A cylinder’s net is two circles and a rectangle.
Formulas
| Solid | Surface area |
|---|---|
| Cube (edge s) | 6s² |
| Rectangular prism | 2(lw + lh + wh) |
| Triangular prism | 2 × triangle area + 3 rectangles |
| Cylinder | 2πr² + 2πrh |
| Square pyramid | base² + 4 × ½ × base × slant height |
| Cone | πr² + πrs |
| Sphere | 4πr² |
Rectangular prism
Example: 5 cm × 3 cm × 2 cm
- 2(5 × 3) = 30 (top and bottom)
- 2(5 × 2) = 20 (front and back)
- 2(3 × 2) = 12 (sides)
- Total: 62 cm²
Cylinder
The curved side unrolls into a rectangle: its width is the circumference (2πr) and its height is h.
Example: r = 3 cm, h = 10 cm
- Two circles: 2π(9) = 18π
- Side: 2π(3)(10) = 60π
- Total: 78π ≈ 245 cm²
Pyramids and cones: slant height
The slant height (s) runs up the middle of a face, not the vertical height.
Square pyramid: base 6 m, slant height 5 m
- Base: 36
- Four triangles: 4 × ½ × 6 × 5 = 60
- Total: 96 m²
If you know the vertical height h, find s with Pythagoras: s = √(h² + (base/2)²).
Lateral area
The lateral area is the area of the sides only (no bases). Use it for things like a can’s label or the walls of a room.
- Can label (cylinder side): 2πrh
Sphere
Example: r = 5 → 4π(25) = 100π ≈ 314 units²
Open shapes
A box with no lid has only 5 faces. A pipe (open cylinder) has no circular ends. Read problems carefully.
Worked examples
Example 1: Cube with edge 4 m: 6 × 16 = 96 m².
Example 2: Label for a can, r = 4 cm, h = 11 cm: 2π(4)(11) = 88π ≈ 276 cm².
Example 3: Cone r = 3, s = 5: 9π + 15π = 24π ≈ 75.4 units².
Common mistakes
- Using vertical height instead of slant height.
- Forgetting faces (count them using the net).
- Using volume units (cm³) for surface area.
Try it yourself
- Rectangular prism 10 × 4 × 3
- Cylinder r = 2, h = 7 (exact answer)
- Square pyramid base 10 cm, slant height 13 cm
- Sphere r = 3 (exact answer)
- Open box (no top) 6 × 4 × 2
Answers: 1) 164 units² 2) 36π 3) 360 cm² 4) 36π 5) 64 units²
Quick recap
- Surface area is the total area of a solid’s surfaces.
- Use a net and add each face.
- Use slant height for pyramids and cones; lateral area excludes the bases.