Calculate the total surface area, lateral surface area, and base areas of 10 different 3D geometric shapes with instant multi-unit conversions and step-by-step mathematical proofs.
• Select any 3D shape from the tabs below (Sphere, Cone, Cube, Cylinder, Tank, Capsule, Cap, Frustum, Ellipsoid, Pyramid).
• Enter the required dimensions, choose your preferred measurement units, and click Calculate.
Enter shape dimensions on the left and click "Calculate" to view surface area metrics and multi-unit conversions.
Definitions, derivations, lateral area breakdowns, and mathematical formulas for 10 common 3D geometric solids.
The surface area of a three-dimensional solid is the total measurement of all outer surfaces (faces, curved boundaries, and bases) enclosing the object. While volume represents 3D space capacity inside an object (m³), surface area represents the 2D skin area required to wrap or coat the exterior (m²).
| 3D Solid Shape | Lateral Area (LSA) | Total Surface Area (TSA) Formula |
|---|---|---|
| Sphere / Ball | — | 4πr² |
| Cone | πr√(r² + h²) | πr(r + s) |
| Cube | 4a² | 6a² |
| Cylindrical Tank | 2πrh | 2πr(r + h) |
| Rectangular Tank / Prism | 2h(l + w) | 2(lw + lh + wh) |
| Capsule | 2πrh | 2πr(2r + h) |
| Spherical Cap | 2πRh | π(2Rh + r²) |
| Conical Frustum | π(R + r)s | π(R² + r² + (R + r)s) |
| Ellipsoid | — | 4π × [((ab)^p + (ac)^p + (bc)^p)/3]^(1/p) |
| Square Pyramid | 2as | a² + 2as |
Lateral surface area (LSA) measures only the side faces or curved mantle of a solid, excluding its top and bottom base faces. Total surface area (TSA) includes both the lateral surfaces and all base faces.
For a right circular cone, the slant height is the hypotenuse formed by radius r and height h: s = √(r² + h²). For a square pyramid with base edge a, the slant height is s = √(h² + (a/2)²).
Because the exact surface area of a tri-axial ellipsoid cannot be expressed using elementary functions (it requires incomplete elliptic integrals), Knud Thomsen proposed an accurate approximation with exponent p ≈ 1.6075 that maintains an error of less than 1.061%.
A capsule consists of two identical hemisphere ends and a central cylindrical body. The two hemispheres combine to form a full sphere (4πr²), and the cylinder side has area 2πrh, giving TSA = 4πr² + 2πrh = 2πr(2r + h).