How this instrument works
This quick-reference page covers the total surface-area formulas for four commonly needed solids at once — a cube (6s²), a sphere (4πr²), a cylinder (2πrh+2πr², the curved side plus both circular caps), and a cone (πr·slant+πr², the curved side plus its one circular base) — selected with a simple shape number rather than navigating to four separate dedicated pages.
The cone option computes its own slant height internally, from the radius and the perpendicular height entered, via the Pythagorean theorem — one less measurement to track separately, since the slant height (needed for the cone's curved surface) is a different length from the perpendicular height most people measure directly.
Each of these four formulas has its own dedicated, more detailed page elsewhere on this site, with fuller derivations and worked examples specific to that one solid. This page trades that depth for speed — a single stop for whichever basic solid's surface area happens to be needed right now.
- Enter the shape number into the Shape field: 1 for cube, 2 for sphere, 3 for cylinder, 4 for cone.
- Enter the first dimension (side or radius) into Dimension 1.
- Enter the second dimension (height, cylinder or cone only) into Dimension 2, if needed.
- Read Total surface area: the sheet applies the matching formula automatically, computing a cone's slant height internally.
Worked example — a cube with side 4
Shape 1 (cube) with dim1 = 4 gives a total surface area of 6×16=96, using the plain 6s² formula automatically selected for that shape number.
Switch to shape 4 (cone) with dim1 = 3 (radius) and dim2 = 4 (height), and the sheet computes the slant height internally (5, from the 3-4-5 right triangle) and applies πr·slant+πr², giving a total surface area of about 75.40 — the same page handling a completely different solid and formula just by changing the shape number.
Questions
How does the shape selector work?
Enter 1 for a cube, 2 for a sphere, 3 for a cylinder, or 4 for a cone, and the sheet automatically applies that solid's own surface-area formula to whatever dimensions are entered.
Does the cone option need the slant height entered separately?
No — it's computed internally from the radius and the perpendicular height, via the Pythagorean theorem, so only those two more commonly measured dimensions need entering directly.
What does a cylinder's surface area include?
The full exterior: the curved lateral side (2πrh) plus BOTH flat circular caps, top and bottom (2πr² together) — the entire enclosed exterior, not just the curved side alone.
Do I need to fill in both dimension fields?
Only for a cylinder or cone, which need both a radius and a height. A cube only needs its side, and a sphere only needs its radius; the second field is ignored for those two.
Where can I find a more detailed explanation for one specific solid?
Each of these four solids has its own dedicated page elsewhere on this site, with a fuller derivation and worked examples — this page trades that depth for a quick, single-stop reference across all four at once.