Solid Geometry: 3D Shapes and Their Formulas

What a Solid Is

A solid, according to the entry Treccani devotes to the term in its Enciclopedia della Matematica, is a three-dimensional shape in ordinary space, made up of a closed and bounded set of points. The boundary of that shape is its surface, which can be flat, curved, or a mix of both, depending on the solid. When the surface is made entirely of polygons joined along their edges, the solid is called a polyhedron.

The volume of a solid is the intuitive measure of the space it takes up. Working it out means matching the volume to measurements of a few defining elements of the solid — edges, heights, radii — different for every kind of shape. Solid geometry runs on the same basic tools as Euclidean geometry: points, lines, and planes, applied to three-dimensional space instead of a flat sheet.

Polyhedra and the Five Platonic Solids

A polyhedron, per Wolfram MathWorld, is a three-dimensional solid made up of a collection of polygons joined along their edges. Some polyhedra are regular: that happens when both their faces and their vertex figures are regular polygons, meaning faces and vertices are both equivalent to one another.

There are exactly five convex regular polyhedra, the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Add their concave, stellated counterparts, the four Kepler-Poinsot polyhedra, and the total number of regular polyhedra rises to nine.

Among the most common polyhedra in everyday use are shapes that are not regular but just as useful: the cuboid (a box with three pairs of opposite rectangular faces), the prism (two matching, parallel polygonal bases joined by side faces), and the pyramid (one polygonal base with triangular faces meeting at a single vertex, the apex).

Surface Area and Volume Formulas for Polyhedra

Formulas swap numbers for letters — the same symbolic notation at the core of algebra — because they need to hold for any measurement of a given type of solid.

Solid	Surface Area	Volume
Cube (edge a)	S = 6a²	V = a³
Cuboid (edges a, b, c)	S = 2(ab + bc + ca)	V = abc
Right prism (base area A, base perimeter p, height h)	lateral = p × h	V = A × h
Right pyramid (base area A, base perimeter p, slant height a, height h)	lateral = (p × a) / 2	V = (A × h) / 3

Two things hold across the whole table. First: a prism and a cylinder share the same volume logic, base area times height, because the solid’s cross section stays identical from bottom to top. Second: a pyramid and a cone cut that logic down to a third, because the cross section shrinks on the way to the apex — Wolfram MathWorld notes that this holds for any shape of base, since the area of a cross section scales quadratically with height.

A prism’s lateral surface area, base perimeter times height, is the same rule that applies to a cylinder too (see below): both solids have a lateral surface that unrolls flat into a rectangle.

Cylinders, Cones, and Spheres, the Solids of Revolution

Cylinders, cones, and spheres are not polyhedra: their surfaces are curved, not made of polygons. They are called solids of revolution because they form when a flat shape spins 360 degrees around a line, the axis of rotation. A rectangle spun around one of its sides generates a cylinder; a right triangle spun around one leg generates a cone; a semicircle spun around its diameter generates a sphere.

This Recap stops at the definition and the formulas for these three common solids of revolution. Working out the volume of a general solid of revolution, formed by spinning an arbitrary curve instead of a basic shape, belongs to calculus.

Surface Area and Volume Formulas for Solids of Revolution

Solid	Surface Area	Volume
Cylinder (radius r, height h)	lateral = 2πrh · total = 2πr(r+h)	V = πr²h
Cone (radius r, height h, slant height a = √(r²+h²))	lateral = πra · total = πr(r+a)	V = (πr²h) / 3
Sphere (radius r)	S = 4πr²	V = (4/3)πr³

The formulas for cylinder, cone, and sphere show up, in the same form, on both Wolfram MathWorld and the math.it solid-geometry formula reference: two independent sources confirming each other. The sphere formulas in particular have a long history — MathWorld traces both to Archimedes, first derived in his treatise On the Sphere and Cylinder, around 225 BCE.

Practical example: A cylinder has a base radius of 3 cm and a height of 7 cm. Its volume comes from V = πr²h: 3 squared is 9, 9 times 7 is 63, so V = 63π cm³, about 197.9 cm³ using π ≈ 3.14159 — the same multiplication at the heart of arithmetic, applied to a solid instead of isolated numbers. The lateral surface area, meanwhile, is 2πrh: 2 times π times 3 times 7 is 42π cm², about 131.9 cm².

How 3D Shapes Are Classified, and How They Differ from Plane Shapes

Plane geometry, as Treccani defines it, studies the properties of shapes set in an ordinary two-dimensional Euclidean space: triangles, quadrilaterals, circles, and every other shape you can draw on a flat sheet. Solid geometry sits, by symmetry, in three-dimensional Euclidean space: it adds depth, and with it a surface to measure on top of a volume (the space it occupies).

Plane shapes and solid shapes are not rivals: they are often the same shape seen through a different number of dimensions. A square is a plane shape; six matching squares joined along their edges form a cube, a polyhedron. A circle is a plane shape; a circle swept along an axis, for the height of a cylinder, generates a solid of revolution. Telling whether a problem is about a plane shape or a solid is the first step toward picking the right formula: area formulas (in two dimensions) and surface-area-and-volume formulas (in three) are not interchangeable.