Surface Area Calculator
Select a 3D shape and enter its dimensions to calculate the surface area.
Surface Area Formulas & Reference
All shapes at a glance
| Shape | Total Surface Area | Lateral SA | Variables |
|---|---|---|---|
| Cube | SA = 6s² | 4s² | s = side |
| Rectangular Box | SA = 2(lw + lh + wh) | 2(lh + wh) | l, w, h |
| Sphere | SA = 4πr² | 4πr² | r = radius |
| Cylinder | SA = 2πr² + 2πrh | 2πrh | r, h |
| Cone | SA = πr² + πrl | πrl | r, h, l = slant = √(r²+h²) |
What is surface area?
Surface area is the total area of all faces (or surfaces) of a 3D object. It is measured in square units (cm², m², in², ft²). Surface area determines how much material is needed to cover or paint an object and is critical in packaging, heat transfer, and engineering design.
Worked example — Cylinder
A tin can has radius r = 4 cm and height h = 10 cm.
- Lateral SA = 2πrh = 2π × 4 × 10 = 80π ≈ 251.33 cm²
- Two circular ends = 2πr² = 2π × 16 = 32π ≈ 100.53 cm²
- Total SA = 80π + 32π = 112π ≈ 351.86 cm²
Worked example — Cone
An ice cream cone has base radius r = 3 cm and height h = 8 cm.
- Slant height l = √(r² + h²) = √(9 + 64) = √73 ≈ 8.544 cm
- Lateral SA = πrl = π × 3 × 8.544 ≈ 80.50 cm²
- Base = πr² = π × 9 ≈ 28.27 cm²
- Total SA ≈ 108.77 cm²
Worked example — Sphere
A ball has radius r = 5 m.
- SA = 4πr² = 4π × 25 = 100π ≈ 314.16 m²
Total vs. Lateral surface area
Total SA includes every face (top, bottom, sides). Lateral SA includes only the side surfaces — useful for calculating the amount of paint needed on the side of a tank, for example.
References
- Weisstein, E.W. Surface Area. MathWorld — A Wolfram Web Resource.
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. (Chapter 15: Multiple Integrals — Surface Area)
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics (14th ed.). Pearson. (Appendix: Geometric Properties of Solids)
