Painting a cylinder or wrapping a box needs the lateral area, not the total. This gives both.
Total surface area counts every face. Lateral counts only the sides, leaving out the top and bottom. The distinction matters more often than it is taught: labelling a tin needs the lateral area, painting a room’s walls needs lateral, and a cylindrical tank standing on the ground needs lateral plus one end rather than both.
| Solid | Total surface area |
|---|---|
| Cube | 6a² |
| Rectangular box | 2(lw + wh + lh) |
| Cylinder | 2πr(r + h) |
| Sphere | 4πr² |
| Cone | πr(r + l), where l = √(r² + h²) |
A cone’s curved surface depends on the slant height l — the distance from the rim to the apex along the surface — not the vertical height h. They are related by Pythagoras: l = √(r² + h²). A cone of radius 3 and height 4 has a slant height of exactly 5, giving a total surface area of 75.3982.
The surface area of a sphere, 4πr², is exactly four times the area of the circle you get by slicing through its middle. Archimedes proved it, considered it his finest result, and asked for the sphere-and-cylinder diagram on his tomb.
Whenever the ends are not part of the job. Labelling a tin, painting the walls of a room, or calculating the material for a pipe all use lateral area. A closed tank being painted inside and out uses total.
Because the curved surface is measured along the slope, not vertically. Unrolled, it forms a sector of a circle whose radius is the slant height. Using the vertical height instead understates the area.
A sphere has exactly two thirds the volume and two thirds the total surface area of the smallest cylinder that contains it. This is Archimedes’ theorem and it holds for any radius.
For a sphere or cube you can work backwards, since one dimension determines everything: find the radius or edge from the volume, then apply the surface formula. For a cylinder or cone you cannot — the same volume can be achieved by many different combinations of radius and height, each with a different surface area.
For the capacity rather than the skin, use the Volume Calculator. Flat shapes are handled by the Area Calculator and the Circle Calculator, and the slant-height step is just the Pythagorean theorem.