How this instrument works
This quick-reference page covers the volume formulas for four commonly needed solids at once — a cube (s³), a sphere ((4⁄3)πr³), a cylinder (πr²h), and a cone (⅓πr²h) — selected with a simple shape number rather than navigating to four separate dedicated pages.
Three of these four formulas share a visible family resemblance: a cylinder's volume is its circular base area times its height, a cone's is exactly one third of that same product (the standard pyramid-and-cone one-third relationship), and a sphere's works out to exactly two-thirds of the volume of the cylinder that tightly encloses it — three related solids, three related formulas.
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 volume 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 Volume: the sheet applies the matching formula automatically.
Worked example — a cube with side 4
Shape 1 (cube) with dim1 = 4 gives a volume of 4³=64, using the plain s³ formula automatically selected for that shape number.
Switch to shape 3 (cylinder) with dim1 = 3 (radius) and dim2 = 10 (height), and the sheet applies πr²h instead, giving a volume of 90π ≈ 282.74 — 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 volume formula to whatever dimensions are entered.
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.
Why is a cone's volume exactly one third of a cylinder's?
Because a cone tapers to a point while a cylinder keeps the same cross-section all the way up — exactly three cones sharing a cylinder's base and height can be assembled to fill that cylinder completely, a general geometric fact behind every pyramid-and-prism pairing too.
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.
Can this handle other solids, like a pyramid or a torus?
Not on this page — it's scoped to the four most commonly needed basic solids. Pyramids, tori, and other shapes each have their own dedicated pages elsewhere on this site.