Calculate the total and lateral surface area of common 3D shapes: sphere, cube, cylinder, cone, pyramid, and more.
| Component | Value |
|---|---|
| Lateral surface | 251.327412 |
| Base(s) | 100.530965 |
| Total Surface Area | 351.858377 |
| Shape | Formula |
|---|---|
| Sphere | SA = 4πr² |
| Cube | SA = 6a² |
| Rectangular Box | SA = 2(lw + lh + wh) |
| Cylinder | SA = 2πr(r + h) |
| Cone | SA = πr(r + l), l = √(r² + h²) |
| Capsule | SA = 2πr(2r + h) |
| Conical Frustum | SA = π(R + r)l + π(R² + r²), l = √((R−r)² + h²) |
| Square Pyramid | SA = a² + 2al, l = √(h² + (a/2)²) |
Archimedes proved that the surface of a sphere is exactly four times the area of its largest circle, which is where 4πr² comes from. Cicero reported that he wanted a sphere inside a cylinder carved on his tomb, because the sphere's volume and total surface are each two-thirds of the enclosing cylinder's. This calculator applies the same kind of reasoning to eight solids and, for most of them, splits the total into lateral and base parts.
Sphere: 4πr². Cube: 6a². Rectangular box: 2(lw + lh + wh). Cylinder: 2πr(r + h). Cone: πr(r + l), where the slant height l = √(r² + h²). Capsule: 2πr(2r + h). Conical frustum: π(R + r)l + π(R² + r²), with l = √((R − r)² + h²). Square pyramid: a² + 2al, with l = √(h² + (a/2)²).
Take a cone with radius 3 and height 4. The slant height is l = √(3² + 4²) = √25 = 5. The lateral surface is π × 3 × 5 = 47.12389, the circular base is π × 3² = 28.274334, and the total is 75.398224 square units. The breakdown table on the page lists all three along with the slant height. Results are rounded to six decimals.
Now the limits. The calculator treats every solid as an infinitely thin shell, so wall thickness, seams, overlaps and waste are not included. Cylinder and cone totals count every flat end, so an open-top tub needs the top circle subtracted by hand. The pyramid is a right pyramid on a square base, with h measured straight up from the centre of the base, not along a face. The capsule's h is only its straight middle section. Paint or fabric coverage depends on the product and the number of coats, so treat the area as a starting point, not an order quantity.
Calculate the closed cylinder, then subtract one circular end, πr². For r = 4 and h = 10 the closed total is 351.858377 and the open-top figure is 301.592895. The same adjustment applies to any shape with a missing face.
The page expects the vertical height and calculates the slant itself. For r = 3, entering 5 instead of 4 gives a slant of 5.830952 and a total of 83.229761, instead of 75.398224. That is an overstatement of more than 10 percent.
Yes for the box, cylinder, cone, frustum and pyramid, since the Breakdown table lists a lateral surface for them. For a cylinder it is 2πrh, which is 251.327412 for r = 4 and h = 10. The sphere, cube and capsule show only a total.
It multiplies by four, whereas volume multiplies by eight. A cube with side 2 has area 24 and one with side 4 has area 96. This is why small objects lose heat or dry out faster relative to their size.