3D shape formulas

Sphere (radius r)

Surface Area: SA = 4πr²
Volume: V = (4/3)πr³

Cube (side a)

Surface Area: SA = 6a²
Volume: V = a³

Rectangular Prism (l × w × h)

Surface Area: SA = 2(lw + lh + wh)
Volume: V = l·w·h

Cylinder (radius r, height h)

Surface Area: SA = 2πr(r + h)
Volume: V = πr²h

Cone (base radius r, height h)

Slant height: l = √(r² + h²)
Surface Area: SA = πr(r + l)
Volume: V = (1/3)πr²h

Regular Tetrahedron (edge a)

Surface Area: SA = √3·a²
Volume: V = a³ / (6√2)

Tips

All measurements should be in the same unit (cm, m, in, ft, etc.). The resulting surface area will be in square units and volume in cubic units.

Common mistake to avoid: Mixing feet and inches in the same calculation produces incorrect results. Convert everything to one unit first - for example, convert 1 ft 6 in to 1.5 ft before entering values.

Why these formulas work

The sphere surface area SA = 4πr² can be derived by "unrolling" a sphere as a surface of revolution. The sphere volume V = (4/3)πr³ follows from Cavalieri's principle - comparing cross-sectional areas of a sphere and a cylinder-minus-cone stack at every height. The cylinder formula SA = 2πr(r + h) adds two circular caps (2πr²) to the lateral surface (2πrh).

Worked example

Shape	Input	Surface Area	Volume
Sphere	r = 5 cm	314.16 cm²	523.60 cm³
Cube	a = 4 cm	96.00 cm²	64.00 cm³
Cylinder	r = 3 cm, h = 10 cm	244.35 cm²	282.74 cm³
Cone	r = 3 cm, h = 4 cm	75.40 cm²	37.70 cm³

Additional shapes

Torus (major radius R, minor radius r)

Surface Area: SA = 4π²Rr
Volume: V = 2π²Rr²

Square Pyramid (base side a, height h)

Slant height: l = √(h² + (a/2)²)
Surface Area: SA = a² + 2al
Volume: V = (1/3)a²h

Frustum / Truncated Cone (radii R and r, height h)

Slant height: l = √(h² + (R − r)²)
Surface Area: SA = π(R² + r² + (R + r)l)
Volume: V = (πh/3)(R² + Rr + r²)

3D Shape Calculator - Volume & Surface Area

Results