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Geometry

Surface Area

Surface area formulas for 3D solids: cubes, prisms, spheres, and cylinders.

Surface area is the total area of all faces of a 3D solid, in square units. For prismatic solids it splits into a lateral component (the side walls) plus the area of the two bases. The formulas below cover cubes, rectangular and irregular prisms, spheres, and cylinders.

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Surface area is the sum of all outer face areas.

Cube

Cube

Six congruent square faces of edge a, each of area a². Surface area is six times that:

SA=6a2SA=6a^{2}

Rectangular Prism

Rectangular Prism

Edges a, b, c. Three pairs of congruent faces: two of area ab, two of bc, two of ac. Sum the three pairs:

SA=2ab+2bc+2acSA=2ab+2bc+2ac

Irregular Prism

Irregular Prism

B is the base polygon and L is the prism’s height along the axis. The lateral surface is the base perimeter times L. Add twice the base area for the two ends:

SA=(Perimeter of B)×L+2×(Area of B)SA=\left(Perimeter\text{ }of\text{ }B\right)\times L+2\times\left(Area\text{ }of\text{ }B\right)

Sphere

Sphere

Radius r. Surface area is four times the area of a great-circle cross-section (πr²):

SA=4πr2SA=4\pi r^{2}

Cylinder

Cylinder

Base radius r and height h. Two circular ends of area πr² plus a wrapped rectangle of height h and width 2πr (the base circumference) factor into:

SA=2πr(r+h)SA=2\pi r\left(r+h\right)