Surface Area Calculator (3D)

Surface area of solids

Surface area is how much you'd have to cover to wrap the outside of a 3D solid in paper. The standard formulas are worth memorizing: cube 6L²; cuboid 2(ab + bc + ac); sphere 4πr²; cylinder 2πr² + 2πrh, which is the two bases plus the side; cone πr² + πrg, where g = √(r² + h²) is the slant height; and a square pyramid at base + N·(L·apothem/2). Take a sphere with r = 5 cm: its area works out to 4π·25 ≈ 314 cm². That same geometry shows up in the volume-to-surface ratio, which is why small animals shed heat so quickly — they carry relatively more surface per unit of mass — and why droplets pull themselves into spheres, the shape that gives the least surface for a fixed volume.

Applications and context

You lean on surface area all the time. It sets paint estimates — acrylic covers roughly 10–12 m²/L, so dividing wall area by that yield tells you how many liters to buy. It also governs tiling and cladding, thermal insulation, how fast something evaporates, and plenty of biology; the alveoli in a human lung add up to something like 70 m² of gas-exchange surface, all of it folded into the chest.

FAQ

What's the difference between surface area and lateral area? Total surface area counts the bases as well, while lateral area is just the sides. For a cylinder, the lateral part is 2πrh and the total is 2πrh + 2πr².

Why is the slant height needed for cones? When you unroll the lateral face of a cone, it flattens into a circular sector whose radius is the slant height g = √(r² + h²), not the vertical height h.

How do I estimate paint for a room? Start with the wall area as perimeter × height, take out the doors and windows, divide by the paint's yield in m²/L, then multiply by however many coats you plan on, which is usually 2.