Surface Area Calculator

What is the formula for surface area?

There is no single "surface area formula" — it depends on the shape. Surface area is always the sum of the areas of every face or curved surface that forms the outside of a 3D solid. For flat-faced solids (cubes, prisms, pyramids) you add up the area of each polygon face; for curved solids (spheres, cylinders, cones) you use a formula derived from calculus or, in the sphere's case, an identity known since antiquity (Archimedes): a sphere's surface area equals exactly 4 times the area of its largest cross-sectional circle.

The most commonly asked version, SA = 4 × π × r², is specifically the sphere formula. For a radius of 3, that's SA = 4 × π × 3² = 36π ≈ 113.097 square units. Pick your shape in the calculator above to get the formula, the substituted numbers, and the final answer for the one you actually need — see the shape-by-shape formulas below.

Surface area of a 12×12 room (walls + ceiling)

A common practical question is the surface area of a room — usually to figure out how much paint, wallpaper or drywall to buy. For a 12 ft × 12 ft room with an 8-ft ceiling (a common US ceiling height, since "12×12" only gives the floor plan): the four walls total 384 ft² (4 identical 12 ft × 8 ft walls), plus the 144 ft² ceiling — 528 ft² of paintable surface in total. Add the 144 ft² floor if you're counting every interior surface of the box (not just what gets painted): 672 ft² overall.

A room is really a rectangular prism with one "face" (the floor, or sometimes the ceiling) usually excluded depending on what you're estimating — this is exactly the open/closed distinction the calculator above makes explicit for cylinders, cones and frustums. For a full multi-room floor plan with irregular shapes, use Calcuris's Square Footage Calculator — it adds several 2D areas (floor plan) together, while this tool computes the full 3D exterior surface of a single solid.

Total vs. lateral vs. base surface area

"Total surface area" means every exterior face added together. "Lateral surface area" means only the side faces — excluding the top and bottom (or the base, for a pyramid or cone). This distinction matters in real problems: a can of soup needs its total surface area printed with a label (lateral only, since the label doesn't cover the top/bottom disks), while a water tank's total surface area determines how much steel to weld.

Most calculators only give you the total. This one shows the total, the lateral surface area, and the base surface area(s) separately for every shape where the distinction applies (hemisphere, cylinder, cone, square pyramid, capsule, conical frustum, triangular prism) — so you can pull out exactly the number your specific problem needs.

Formula reference: all 11 shapes

Sphere — SA = 4πr². Example: r = 3 → SA = 113.097.

Hemisphere — curved SA = 2πr², base SA = πr², total = 3πr². Example: r = 3 → curved 56.549, base 28.274, total 84.823.

Cube — SA = 6a². Example: a = 4 → SA = 96.000.

Rectangular prism — SA = 2(lw + lh + wh). Example: l=2, w=3, h=4 → SA = 52.000.

Cylinder — lateral = 2πrh, each base = πr², closed total = 2πr(r+h). Example: r=2, h=5, closed → total 87.965, lateral 62.832.

Cone — slant s = √(r²+h²), lateral = πrs, closed total = πr(r+s). Example: r=3, h=4 → s=5, total 75.398.

Square pyramid — base = a², lateral = a√(a²+4h²), total = base + lateral. Example: a=4, h=6 → total 66.596.

Capsule — end caps (one full sphere) = 4πr², lateral band = 2πrh, total = both combined. Example: r=2, h=5 → total 113.097.

Ellipsoid — no exact elementary formula exists; approximated (see below). Example: a=4, b=3, c=2 → total ≈ 111.604.

Conical frustum — slant s = √((R−r)²+h²), lateral = π(R+r)s, closed total adds both circular ends. Example: R=5, r=3, h=4 → total 219.211.

Triangular prism — base triangle area via Heron's formula from 3 side lengths, lateral = h × perimeter, total = 2×base + lateral. Example: a=3, b=4, c=5, h=6 → total 84.000.

The ellipsoid approximation — and its error bound

Unlike every other shape on this page, a general triaxial ellipsoid (three unequal semi-axes a, b, c) has no exact closed-form surface area formula — computing it exactly requires elliptic integrals. Every practical calculator, including this one, uses Knud Thomsen's approximation: SA ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p), with p ≈ 1.6075.

This calculator uses p = 1.6075 specifically, giving a maximum relative error of ≤1.061% across all possible ellipsoid shapes — the value documented by the wider mathematical literature on Thomsen's formula. Some competing calculators default to the simpler p = 1.6 (which is only optimal for near-spherical ellipsoids and has a looser ≤1.178% error bound) without stating either exponent or bound. When the three semi-axes are equal (a = b = c), the approximation collapses to the exact sphere formula with zero error — a useful sanity check.

Open vs. closed: why the toggle matters

Cylinders, cones and conical frustums are often not fully closed solids in real life: a pipe is a cylinder open at both ends, a party hat is a cone with no base, a lampshade is a frustum with both circular openings uncovered. None of the leading calculators offer this as a toggle by default — they assume a closed solid and leave you to manually subtract a base area yourself.

This calculator exposes the toggle directly: choose closed, open at one end, or open at both ends for a cylinder (and closed/open for a cone or frustum), and the total surface area — plus its lateral/base breakdown — updates immediately, with the formula used shown in the steps.