Surface Area – The Outer Skin of 3D Shapes
Surface area is the total area of all the faces (flat or curved) that cover the outside of a three-dimensional solid. It tells you how much material is needed to wrap or coat the outside of an object.
What Is Surface Area?
Surface area is measured in square units (cm², m²). To find it, calculate the area of every face of the 3D shape and add them all together. Imagine cutting the solid open and flattening it into a net — the area of the net equals the surface area.
Surface Area Formulas
| Solid | Formula | Variables |
|---|---|---|
| Cube | SA = 6s² | s = side length |
| Cuboid | SA = 2(lw + lh + wh) | l, w, h = length, width, height |
| Cylinder | SA = 2πr² + 2πrh | r = radius, h = height |
| Sphere | SA = 4πr² | r = radius |
| Cone | SA = πr² + πrl | r = radius, l = slant height |
| Triangular prism | SA = 2 × (triangle area) + 3 × (rectangle areas) | Depends on cross-section |
Worked Examples
SA = 6 × 5² = 6 × 25 = 150 cm².
SA = 2(8×3 + 8×4 + 3×4) = 2(24 + 32 + 12) = 2 × 68 = 136 cm².
SA = 2π(9) + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 cm².
SA = 4π(36) = 144π ≈ 452.39 cm².
SA = π(16) + π(4)(7) = 16π + 28π = 44π ≈ 138.23 cm².
The Slant Height of a Cone
If you know the radius r and the perpendicular height h of a cone, find the slant height l using Pythagoras: l = √(r² + h²).
Key Takeaways
- Surface area = total area of all outer faces or surfaces.
- Cube: 6s². Cuboid: 2(lw + lh + wh). Cylinder: 2πr² + 2πrh.
- Sphere: 4πr². Cone: πr² + πrl (where l is slant height).
- Use Pythagoras to find the slant height of a cone from radius and perpendicular height.
Practice Questions
- Find the surface area of a cube with side 9 cm.
- Find the surface area of a cuboid 12 × 5 × 6 cm.
- A cylinder has radius 5 cm and height 12 cm. Find its surface area in terms of π.
- A sphere has surface area 100π cm². Find its radius.
- A cone has radius 6 cm and perpendicular height 8 cm. Find its slant height and then its surface area.