Geometry
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.
Archimedes (c. 287–212 BCE) considered his greatest achievement to be the discovery that a sphere's surface area and volume are always exactly two-thirds of those of the smallest cylinder that fits snugly around it — a result he valued so highly that he asked for a sphere inscribed in a cylinder to be carved on his tombstone. Surface area is not just an abstract measurement either: it directly drives real engineering decisions, since packaging designers try to minimise a container's surface area (to save material and cost) while still holding a fixed volume of product, which is exactly why cans and bottles tend toward efficient cylindrical shapes. Biology depends on the same relationship in reverse — small animals have a much higher surface-area-to-volume ratio than large ones, which is a major reason mice lose body heat far faster than elephants and must eat proportionally far more to stay warm.
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.