Surface Area Calculator

Surface area is the total area of all the outside surfaces of a three-dimensional object, measured in square units such as square feet or square meters. It tells you how much paint covers a tank, how much sheet metal makes a can, how much wrapping paper a box needs or how quickly an object exchanges heat with its surroundings. Volume measures how much a shape holds; surface area measures how much material encloses it.

For solids with flat faces, the surface area is the sum of the face areas, which can be found by unfolding the shape into a flat net. Curved shapes use formulas involving π. Each calculator above lets you choose a unit for every length, so a tank measured partly in feet and partly in inches is handled correctly; the result is reported in square units of the first length.

Sphere

A sphere, or ball, is the set of all points the same distance r from a center. Its surface area is 4πr². Archimedes proved this more than 2,200 years ago by showing that a sphere's surface equals the curved side of the cylinder that fits snugly around it, which has radius r and height 2r and therefore area 2πr × 2r. A ball with radius 1.2 ft has a surface of 4π × 1.44 ≈ 18.0956 ft². Doubling the radius multiplies the surface area by four. Among all solids with a given volume, the sphere has the smallest surface area, which is why soap bubbles and raindrops are round.

A = 4πr²

r

the radius of the sphere

A sphere's surface is exactly four times the area of a circle with the same radius.

Cone

A right circular cone has a circular base of radius r and a point, the apex, directly above the center of the base at height h. Its surface has two parts: the flat circular base, with area πr², and the curved lateral surface. To find the lateral area, use the slant height l, the distance from the apex to the edge of the base along the surface. By the Pythagorean theorem l = √(r² + h²).

If you cut the curved surface along a slant line and flatten it, it becomes a sector of a circle with radius l and arc length 2πr, whose area is πrl. So the total surface area is A = πr² + πrl = πr(r + √(r² + h²)). For a cone with a 1.2 ft radius and 2 ft height, the slant height is about 2.3324 ft, the lateral area about 8.7929 ft², the base about 4.5239 ft² and the total about 13.3168 ft².

If a cone is open, like a paper drinking cup or a funnel, only the lateral area counts. If you know the slant height directly, as when you measure a cone along its side, you can skip the square root and use πr(r + l).

Cube

A cube has six identical square faces, each with side a, so its surface area is 6a². A cube with 3-inch edges has six faces of 9 in², for 54 in² in all. Unlike the volume, which grows with the cube of the edge, surface area grows with its square. That difference explains why crushed ice melts faster than a single large block and why small animals lose body heat faster than large ones: smaller objects have more surface relative to their volume.

Cylindrical Tank

A closed cylinder has two circular ends of radius r and a curved side of height h. Unrolled, the side is a rectangle whose width is the circumference 2πr and whose height is h, so its area is 2πrh. Adding the two ends gives A = 2πr² + 2πrh = 2πr(r + h). A tank with a 1.2 ft radius and 3 ft height has a surface area of about 31.6673 ft². For an open-top tank, subtract one end, πr²; for a pipe with no ends, use only 2πrh.

Rectangular Tank

A rectangular tank, box or cuboid has six rectangular faces arranged in three matching pairs: top and bottom, front and back, and the two ends. With length l, width w and height h, the pairs have areas 2lw, 2lh and 2wh, so the total is A = 2(lw + lh + wh).

For a box 4 ft long, 3 ft wide and 2 ft high, the faces contribute 24, 16 and 12 ft², for a total of 52 ft². This is the amount of material needed if the box has a lid; an open-top box such as an aquarium or planter needs one fewer lw face. When painting a room, which is a rectangular box viewed from the inside, the walls alone are 2lh + 2wh, and the ceiling adds lw.

When the faces are made from sheets of material, add an allowance for seams and overlaps, and remember to measure inside or outside dimensions consistently. A cube is the special case where l = w = h.

Capsule

A capsule, sometimes called a stadium of revolution, is a cylinder with a hemisphere attached to each end. Pills, propane tanks, grain silos and many pressure vessels have this shape because the rounded ends spread internal pressure evenly and have no corners to crack.

Here h is the length of the straight cylindrical section, not including the rounded ends. The two hemispheres together form one full sphere of radius r, with area 4πr², and the cylindrical middle contributes 2πrh with no flat ends, since they are covered by the hemispheres. The total is A = 4πr² + 2πrh = 2πr(2r + h).

With r = 1.2 ft and h = 2 ft, the ends contribute about 18.0956 ft² and the middle about 15.0796 ft², for a total of about 33.1752 ft². If the overall length L is known instead, including the ends, use h = L − 2r. A capsule with h = 0 is simply a sphere.

Spherical Cap

A spherical cap is the piece cut from a ball by a single flat plane, like the top of a dome or a slice off the end of an orange. Three measurements describe it: the radius R of the ball it came from, the radius r of the flat circular base, and the height h from the base to the top of the dome. Any two determine the third through the relation r² = h(2R − h), which follows from the Pythagorean theorem in the cross-section.

The curved surface of a cap has the surprisingly simple area 2πRh. This is another result due to Archimedes: a slice of a sphere between two parallel planes has the same area as the matching band of the surrounding cylinder, so the area depends only on the slice's thickness, not on where it is taken. Adding the flat base πr² gives the total surface area.

With r = 1.2 ft and R = 3 ft, the height is 3 − √(9 − 1.44) ≈ 0.2505 ft, the curved surface is about 4.721 ft² and the total, including the base, is about 9.2449 ft². When h = R the cap is a hemisphere, with curved area 2πR², exactly half of the full sphere. The calculator returns the smaller cap when only r and R are given.

Conical Frustum

A conical frustum is what remains when the top of a cone is cut off by a plane parallel to its base. Lampshades, buckets, flower pots and drinking cups are frustums. It has a circular top of radius r, a circular bottom of radius R and a height h between them.

The slant height is s = √((R − r)² + h²), and the curved side has area π(r + R)s, which can be thought of as the average circumference times the slant height. The top and bottom add πr² and πR², so the total is A = π(r² + R² + (r + R)s).

For r = 1.2 ft, R = 2 ft and h = 3 ft, the slant height is about 3.1048 ft, the lateral area about 31.2132 ft² and the total about 48.3035 ft². A bucket has no top, so only the lateral area and the bottom count. When r = 0 the shape becomes a full cone, and when r = R it becomes a cylinder.

Ellipsoid

An ellipsoid is a stretched or squashed sphere, with three semi-axes a, b and c measured from the center along three perpendicular directions. When all three are equal it is a sphere; when two are equal it is a spheroid, like the slightly flattened Earth or an American football.

Unlike its volume, 4/3·πabc, the surface area of a general ellipsoid has no formula in elementary functions; the exact value requires elliptic integrals. This calculator uses the widely cited approximation by Knud Thomsen, A ≈ 4π((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) ÷ 3)^(1/p) with p ≈ 1.6075, which is within about 1.06% of the true value for any ellipsoid. For semi-axes 1, 2 and 3 ft, it gives about 48.97 ft², while the exact value is about 48.88 ft².

Square Pyramid

A square pyramid has a square base with side a and four triangular faces that meet at an apex directly above the center of the base, at height h. The Great Pyramid of Giza is the most famous example. The base contributes a², and each triangular face has a base of a and a height equal to the slant height s, measured from the middle of a base edge up the face to the apex.

The slant height comes from a right triangle formed by the height h and half of the base edge: s = √((a/2)² + h²). Each face has area as/2, so the four faces together have area 2as, and the total is A = a² + 2a√((a/2)² + h²).

For a pyramid with a 4 ft base and a 3 ft height, s = √(4 + 9) ≈ 3.6056 ft, the four faces cover about 28.8444 ft², and with the 16 ft² base the total is about 44.8444 ft². Do not confuse the slant height with the length of the sloping edge from a corner to the apex, which is longer.

Common Area Units

Frequently asked questions