Surface Area Calculator

Eleven calculators sit above this, one for each shape. Enter the measurements, choose the unit beside each field, and the surface area comes back with the substitution shown. Units can be mixed, so a radius in inches and a height in feet still gives a correct answer. The Volume Calculator covers the same eleven shapes for the space inside them, and the Area Calculator handles flat two-dimensional figures.

What surface area measures

Surface area is the total area of the outside of a solid, the amount you would have to cover to wrap it, paint it, or plate it. It is measured in squared units: m², ft², in². Volume is measured in cubed units and answers a different question, which is how much fits inside.

The distinction matters commercially because the two scale differently. Double every dimension of a solid and its surface area goes up four times while its volume goes up eight. A cube of edge 4 has a surface area of 96, and at edge 8 it is 384, four times as much, while the volume has gone up eightfold. This square-cube relationship is why large storage tanks are cheaper per litre than small ones, why crushed ice melts faster than a block, and why small animals lose body heat more quickly than large ones.

Most of the formulas below are built by adding simpler pieces. A cylinder is two circles plus a rectangle rolled into a tube. A capsule is a cylinder wall plus one whole sphere split into two ends. Where a shape has no closed formula, as with the ellipsoid, an approximation is used and the error is stated.

One decision runs through every formula here: whether the flat ends count. This calculator returns the total surface area, closed on all sides, for every shape. If you need the lateral area only, the curved part without the ends, subtract the ends yourself. The relevant subtraction is noted under each shape.

Sphere

A sphere is every point at a fixed distance r from a centre. It is the shape with the least surface area for a given volume, which is why bubbles and droplets pull themselves into it and why it is the most efficient shape for a pressure vessel.

$$\text{surface area}=4\pi r^2$$

A basketball of radius 0.12 m has a surface area of 4 × π × 0.12² = 0.181 m². Note the exponent: the radius is squared here, not cubed as it is for volume. Doubling the radius to 0.24 m gives 0.7238 m², exactly four times as much.

Cone

A cone tapers from a circular base to a single apex. This calculator handles the right circular cone, where the apex sits directly above the centre of the base. The height h is measured straight up from the base, not along the slope.

The slanted side needs the slant height, which is the hypotenuse of the right triangle formed by the radius and the height, or √(r² + h²). The curved part is π r times that slant, and the base adds π r².

$$\text{surface area}=\pi r(r+\sqrt{r^2+h^2})$$

A conical hat with a base radius of 1 ft and a height of 0.5 ft has a total surface area of π × 1 × (1 + √(1² + 0.5²)) = 6.654 ft². For a hat you would want the lateral area only, since the base is the opening, so subtract π r². Using the slant height where the vertical height belongs is the usual error and it overstates the answer.

Cube

A cube has six identical square faces meeting at right angles, so its surface area is simply six times the area of one face.

$$\text{surface area}=6a^2$$

A Rubik's cube with 4 inch edges has a surface area of 6 × 4² = 96 in². Because a single measurement drives the whole result, an error in it is doubled: measuring the edge 5% too long overstates the surface area by about 10%.

Cylinder

A right circular cylinder is two parallel circles joined by a curved wall. Unroll the wall and it is a rectangle whose width is the circumference 2 π r and whose height is h, giving a lateral area of 2 π r h. Add the two circular ends and the whole thing factors neatly.

$$\text{surface area}=2\pi r(r+h)$$

A cylindrical tank 5.5 ft tall with a radius of 3.5 ft has a surface area of 2 × π × 3.5 × (3.5 + 5.5) = 197.92 ft². For an open-topped tank subtract one π r²; for a pipe with both ends open subtract both. If you have the diameter, halve it first, since using diameter in place of radius roughly doubles the lateral term and quadruples the ends.

Rectangular tank

A rectangular tank, or cuboid, has six rectangular faces in three matching pairs. Add the area of one of each pair and double it.

$$\text{surface area}=2(lw+lh+wh)$$

A box measuring 3 ft by 4 ft by 5 ft has a surface area of 2 × (3×4 + 3×5 + 4×5) = 94 ft², which is the amount of wrapping paper it would take with no overlap. This is the formula behind most packaging and insulation estimates, and it is worth adding a margin for overlap and waste on any real job.

Capsule

A capsule is a cylinder capped with a hemisphere at each end. The two hemispheres together make one complete sphere, so the total is the sphere's 4 π r² plus the cylinder wall's 2 π r h. There are no flat ends to add, since the hemispheres cover them.

Watch the definition of h: it is the length of the straight cylindrical section only, not the overall length of the capsule. The overall length is h + 2r.

$$\text{surface area}=2\pi r(2r+h)$$

A medicine capsule with a radius of 0.05 in and a cylindrical section of 0.5 in has a surface area of 2 × π × 0.05 × (2×0.05 + 0.5) = 0.188 in². That figure is what a coating has to cover, which is how pharmaceutical coating quantities are estimated.

Spherical cap

A spherical cap is the piece left when a plane slices through a sphere. It is described by the ball radius R, the base radius r of the circular cut, and the cap height h, and any two of those give the third.

The curved part is 2 π R h, a result that is surprising and useful: it depends only on the ball radius and the cap height, not on where the slice sits. This calculator adds the flat circular base, π r², to give a closed solid.

$$\text{surface area}=2\pi Rh+\pi r^2$$

A cap cut from a ball of radius 1 ft with a base radius of 0.6 ft has a height of 1 − √(1² − 0.6²) = 0.2 ft, and a surface area of 2 × π × 1 × 0.2 + π × 0.6² = 2.388 ft². Note that two heights fit any given pair of radii, the shallow cap and the large piece that is most of the ball, so the calculator reports both. For a hollow dome, subtract the base term.

Conical frustum

A conical frustum is a cone with the top sliced off parallel to the base, leaving two circular ends of different radii. Buckets, lampshades, and drinking cups are frustums.

The slant of the sloping wall spans the difference in radii and the height, so it is √((R − r)² + h²). The lateral area is π (r + R) times that slant, and both circular ends are added.

$$\text{surface area}=\pi (r+R)\sqrt{(R-r{)}^2+h^2}+\pi r^2+\pi R^2$$

A frustum with a bottom radius of 1 ft, a top radius of 0.3 ft, and a height of 1.5 ft has a surface area of 10.185 ft². For a bucket you would drop the top end, since it is open, and often the bottom too if you only need the outside wall.

Ellipsoid

An ellipsoid is a sphere stretched by different amounts along three perpendicular axes a, b, and c, measured from the centre outward as semi-axes rather than full widths.

Unlike every other shape here, a general ellipsoid has no exact surface area formula in elementary functions; the true value needs elliptic integrals. This calculator uses Knud Thomsen's approximation with p = 1.6075, which stays within about 1.06% for any ellipsoid and is far closer than that for near-spherical ones. When the three axes are equal it reduces exactly to the sphere formula.

$$\text{surface area}=4\pi {\left(}^{\frac{a^p{b}^p+a^p{c}^p+b^p{c}^p}{3}},\text{p = 1.6075}$$

An elliptical slice with semi-axes of 0.1, 0.2, and 0.35 inches has an approximate surface area of 0.562 in². Because this is an approximation, quoting it to more than three or four significant figures claims a precision the method does not have.

Square pyramid

A square pyramid has a square base and four triangular faces meeting at an apex above the centre. Each triangular face needs the slant height, which runs from the apex down the middle of a face to the base edge, not down a corner edge.

That slant is √(h² + (a/2)²), using half the base edge because the face meets the base at its midpoint. The four faces together give 2a times the slant, and the base adds a².

$$\text{surface area}=a^2+2a\sqrt{h^2+\frac{a^2}{4}}$$

A pyramid with a 3 ft base edge and a height of 5 ft has a surface area of 3² + 2 × 3 × √(5² + 3²/4) = 40.321 ft². For the four faces alone, as when covering a roof, subtract the a² base. Confusing the slant height with the vertical height is the most common mistake here, and it always understates the area.

Tube

A tube is a hollow cylinder described by an outer diameter, an inner diameter, and a length. Its surface has four parts: the outer wall, the inner wall, and the two flat rings at the ends.

The two walls give π l (d₁ + d₂), and the two rings together give π (d₁² − d₂²) / 2.

$$\text{surface area}=\pi l(d_1+d_2)+\pi \frac{d_1^2-d_2^2}{2}$$

A pipe 4 ft long with an outer diameter of 3 ft and an inner diameter of 2 ft has a total surface area of 70.686 ft². For painting or galvanising only the outside, use π l d₁ on its own. Note that these inputs are diameters, not radii, which is the opposite of the cylinder above.

Common area units

Answers are shown in the unit you chose plus square meters and square feet. These are the conversions to one square meter:

Unit	In square meters
square kilometer	1,000,000
square centimeter	0.0001
square millimeter	0.000001
square micrometer	0.000000000001
hectare	10,000
square mile	2,589,988
square yard	0.83613
square foot	0.092903
square inch	0.00064516
acre	4,046.86

Area units square the length conversion, which trips people up. A yard is 3 feet, but a square yard is 9 square feet. A meter is about 3.28 feet, and a square meter is about 10.76 square feet. Converting a length factor when you needed an area factor is the most common unit error on this page, so let the calculator handle it where you can.