OneTaskBy Tim King · Updated
Surface Area Calculator
Total and Lateral Area of Cones, Cylinders, Spheres and Frustums, With Working
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How the Surface Area Calculator works
This surface area calculator finds the total outside area of eleven common solids, from cubes and boxes to cones, spheres, capsules and cone frustums. Where a solid has ends or a base, it also gives the lateral area, the curved or sloping sides alone, which is what you need for labels, cladding or paint that skips the top and bottom.
Surface area is the total area of every face of a solid, as if you unfolded it flat and measured the net. It is in square units, so centimetres in gives square centimetres out.
How the surface area calculator works
Flat-faced solids add up their faces: a box has three pairs of rectangles, so the formula is 2(lw + lh + wh). Curved solids unroll. A cylinder's side unrolls to a rectangle 2πr wide and h tall, and a cone's side to a sector with area πr × s, where s is the slant height. A sphere's area is 4πr², exactly four of its own great circles.
First example. A cylinder with radius 3 and height 10 has ends of 2π × 9 ≈ 56.55 and a side of 2π × 3 × 10 ≈ 188.50, for a total of 245.04 square units.
Second example. A square pyramid with a base of 6 and a height of 4 has a slant height of 5, so four faces of 2 × 6 × 5 = 60 plus a base of 36 make 96.
How many square metres of paint for a 2 m cube? Six faces of 4 m² is 24 m², or 20 m² without the base.
Surface area of a cube, and of a cylinder, sphere and cone that fit inside it, for sizes 1 to 24
| Size s | Cube (6s²) | Cylinder (d = h = s) | Sphere (d = s) | Cone (d = h = s) |
|---|---|---|---|---|
| 1 | 6 | 4.71 | 3.14 | 2.54 |
| 2 | 24 | 18.85 | 12.57 | 10.17 |
| 3 | 54 | 42.41 | 28.27 | 22.87 |
| 4 | 96 | 75.4 | 50.27 | 40.67 |
| 5 | 150 | 117.81 | 78.54 | 63.54 |
| 6 | 216 | 169.65 | 113.1 | 91.5 |
| 7 | 294 | 230.91 | 153.94 | 124.54 |
| 8 | 384 | 301.59 | 201.06 | 162.66 |
| 9 | 486 | 381.7 | 254.47 | 205.87 |
| 10 | 600 | 471.24 | 314.16 | 254.16 |
| 11 | 726 | 570.2 | 380.13 | 307.53 |
| 12 | 864 | 678.58 | 452.39 | 365.99 |
| 13 | 1,014 | 796.39 | 530.93 | 429.53 |
| 14 | 1,176 | 923.63 | 615.75 | 498.15 |
| 15 | 1,350 | 1,060.29 | 706.86 | 571.86 |
| 16 | 1,536 | 1,206.37 | 804.25 | 650.65 |
| 17 | 1,734 | 1,361.88 | 907.92 | 734.52 |
| 18 | 1,944 | 1,526.81 | 1,017.88 | 823.48 |
| 19 | 2,166 | 1,701.17 | 1,134.11 | 917.52 |
| 20 | 2,400 | 1,884.96 | 1,256.64 | 1,016.64 |
| 21 | 2,646 | 2,078.16 | 1,385.44 | 1,120.85 |
| 22 | 2,904 | 2,280.8 | 1,520.53 | 1,230.14 |
| 23 | 3,174 | 2,492.85 | 1,661.9 | 1,344.51 |
| 24 | 3,456 | 2,714.34 | 1,809.56 | 1,463.96 |
Tips
- Lateral or total? A label round a tin is lateral area; a box you wrap is total area.
- Slant height is not height. Cones and pyramids use the length down the sloping face.
- Ellipsoids are approximate. No exact simple formula exists; this one is within about 1%.
FAQ
How do you find the surface area of a cylinder?
Add the two circular ends to the curved side: SA = 2πr² + 2πrh. A tin with radius 4 cm and height 11 cm has ends of 2π × 16 ≈ 100.5 cm² and a side of 2π × 4 × 11 ≈ 276.5 cm², for 377 cm² in all. A cylinder with an open top drops one πr² term.
What is lateral surface area?
It is the area of the sides only, leaving out the base and top. For a cylinder it is 2πrh, for a cone πrs and for a prism the perimeter of the end times the length. It matters when the ends are not covered, such as a label, a pipe or the walls of a room.
Why does a sphere have the least surface area for its volume?
Of all shapes holding a given volume, a sphere encloses it with the smallest surface. That is why bubbles and water droplets are round: surface tension pulls them to the shape with the least area. A cube holding the same volume as a sphere has about 24% more surface area.
How do you find the slant height of a cone?
Use Pythagoras on the height and radius: s = √(r² + h²). A cone with radius 3 and height 4 has a slant height of 5. The calculator works it out from the height you enter and shows the value in its working, so you only need to measure straight up from the base.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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