Cube
Every one of a cube's six faces is an identical square, so a single measurement, the side length a, is all you need:
Surface Area = 6a²
Cube with side 4 cm:
Volume = 4³ = 64 cm³
Surface Area = 6 × 16 = 96 cm²
Cuboid (Rectangular Prism)
A rectangular box defined by three separate measurements: length l, width w, and height h:
Surface Area = 2(lw + lh + wh)
Box 5 × 3 × 2 cm:
Volume = 30 cm³
Surface Area = 2(15 + 10 + 6) = 2(31) = 62 cm²
Cylinder
A cylinder is built from a circular base of radius r, extruded up to a height h:
Surface Area = 2πr² + 2πrh = 2πr(r + h)
Cylinder r = 3 cm, h = 7 cm:
Volume = π × 9 × 7 ≈ 197.9 cm³
Surface Area = 2π × 3 × (3 + 7) = 60π ≈ 188.5 cm²
Sphere
A perfectly round solid, defined entirely by its radius r:
Surface Area = 4πr²
Sphere r = 5 cm:
Volume = (4/3)π × 125 ≈ 523.6 cm³
Surface Area = 4π × 25 ≈ 314.2 cm²
Cone
A cone combines a circular base of radius r, a vertical height h, and a slant height l running from the base edge up to the tip:
Surface Area = πr² + πrl = πr(r + l)
Slant height l = √(r² + h²)
Cone r = 3 cm, h = 4 cm:
l = √(9 + 16) = 5 cm
Volume = (1/3)π × 9 × 4 ≈ 37.7 cm³
Surface Area = π × 3 × (3 + 5) = 24π ≈ 75.4 cm²
Pyramid (Square Base)
A pyramid with a square base of side a, standing at height h:
Surface Area = a² + 2a × l (where l = slant height of triangular face)
l = √((a/2)² + h²)
Pyramid a = 6 cm, h = 4 cm:
l = √(9 + 16) = 5 cm
Volume = (1/3) × 36 × 4 = 48 cm³
Surface Area = 36 + 2 × 6 × 5 = 96 cm²
Formulas at a Glance
| Shape | Volume | Surface Area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l, w, h) | lwh | 2(lw + lh + wh) |
| Cylinder (r, h) | πr²h | 2πr(r + h) |
| Sphere (r) | (4/3)πr³ | 4πr² |
| Cone (r, h) | (1/3)πr²h | πr(r + l) |
| Square pyramid (a, h) | (1/3)a²h | a² + 2al |
Where People Slip Up
Keep the units straight: volume comes out in cubic units like cm³ or m³, while surface area comes out in square units like cm² or m². It's also easy to forget that a cylinder has two circular ends, not one - that's why its formula includes 2πr² rather than just πr². A cone, by contrast, only has a single base circle, so that term never gets doubled.
Calculating It Automatically
SolverCalc's geometry calculator is built for flat, 2D shapes, handling area and perimeter. For the solids on this page, plug the formulas above into SolverCalc's scientific calculator and you'll get an accurate result in seconds, without working through the exponents by hand.