Surface area formulas: cubes, boxes, cylinders, spheres and cones

Surface area is the total area of all the outside faces of a 3D shape: what you would paint, wrap or cut from sheet material. The main formulas are 6a² for a cube, 2(lw + lh + wh) for a box, 2πr(r + h) for a cylinder, 4πr² for a sphere and πr(r + l) for a cone, where l is the slant height. The surface area calculator I built covers these and more, and this guide explains each formula with an example.

What surface area means

The easiest way to understand surface area is to imagine unfolding the shape flat, into what is called a net. A cube unfolds into six squares; a cylinder into a rectangle and two circles. The surface area is the total area of the net, so the answer is always in square units: cm², m² or square feet. Surface area is different from volume, which measures how much the shape holds in cubic units. The volume calculator handles that side.

Cubes and cuboids

A cube with side a has six identical square faces, so its surface area is 6a². A 10 cm cube has 6 × 100 = 600 cm². A cuboid (a box) with length l, width w and height h has three pairs of matching rectangles: top and bottom (lw), front and back (lh), and the two ends (wh). Its surface area is 2(lw + lh + wh). A gift box 30 cm by 20 cm by 10 cm needs 2(600 + 300 + 200) = 2,200 cm² of wrapping, before you add anything for overlaps.

Cylinders

A cylinder has two circular ends, each πr², and a curved side. Unroll the side and it becomes a rectangle whose height is h and whose width is the circumference 2πr, so its area is 2πrh. Together: 2πr² + 2πrh = 2πr(r + h). For r = 3 and h = 5, the side is about 94.25, the two ends about 56.55, and the total about 150.80 square units. The side alone, the lateral area, is what a label on a can covers. The circle calculator gives the circle parts, and The area of a circle: why it is πr², with worked examples explains why each end is πr².

Spheres

A sphere of radius r has surface area 4πr², exactly four times the area of a circle through its centre. A ball of radius 2 has 4π × 4 ≈ 50.27 square units; one of radius 11 cm, about the size of a football, has about 1,520.5 cm². Archimedes discovered a beautiful link: a sphere has the same surface area as the curved side of the cylinder that just fits around it, since 2πr × 2r = 4πr², and the sphere's total surface is two-thirds that of the enclosing cylinder with its ends.

Of all shapes holding a given volume, the sphere has the smallest surface. A cube holding 1,000 cm³ has 600 cm² of surface; a sphere holding the same volume has a radius of about 6.2 cm and only about 483.6 cm². That is why soap bubbles are round, and why compact shapes lose heat more slowly.

Cones and pyramids

A cone has a circular base, πr², and a curved side of area πrl, where l is the slant height from the rim to the tip. If you know the vertical height h instead, use Pythagoras: l = √(r² + h²). The total is πr(r + l). A cone with r = 3 and h = 4 has l = 5, a curved side of about 47.12 and a total of about 75.40 square units; an ice-cream cone with no base is only the curved part. A square pyramid with base side a and slant height s of each triangular face has surface area a² + 2as.

Using surface area for real jobs

• Open containers. A cylindrical tank with no lid, radius 0.5 m and height 1.2 m, needs the base plus the side: π × 0.25 + 2π × 0.5 × 1.2 ≈ 4.56 m².
• Painting. Work out the area, subtract windows and doors for rooms, and multiply by the number of coats; the paint calculator does this for walls.
• Scaling. Doubling every dimension multiplies surface area by 4 and volume by 8, so bigger objects have less surface per unit of volume.
• Units. Keep every measurement in the same unit before you start; mixing centimetres and metres is the most common error, as Metric vs imperial: how we got two systems notes for imperial and metric too.

For flat shapes on their own, the area calculator handles rectangles, triangles and circles.