surface area

If you wanted to gift-wrap a solid, or paint it, or know how much sheet metal a tank needs, you are asking for its surface area — the total area of the outer skin of a three-dimensional object. It is an area, so it lives in square units, even though the object itself is three-dimensional.

The reliable way to find it is to unfold the solid into its net and add up the areas of the flat pieces. A rectangular box (length l, width w, height h) has six rectangular faces in three matching pairs, giving total surface area 2(lw + lh + wh). A cylinder gives 2·pi·r^2 + 2·pi·r·h, a cone gives pi·r^2 + pi·r·L (where L is the slant height), and a sphere gives 4·pi·r^2. People often split this into the lateral surface area (the 'side' walls only, ignoring the top and bottom bases) and the total surface area (lateral plus the bases).

Keep surface area and volume firmly apart: surface area is the skin (square units), volume is the filling (cubic units). They scale differently too — under a scale factor k, surface area multiplies by k^2 while volume multiplies by k^3. This mismatch, the square-cube law, is why a mouse can shed heat easily but a whale needs blubber, and why large animals have proportionally thicker legs.