Surface Area Calculator

Surface area is the total area of all the faces and curved surfaces that enclose a solid, measured in square units. Add up the area of each face, or use a shortcut: SA = 6a² for a cube, SA = 4πr² for a sphere and SA = 2πr² + 2πrh for a cylinder. A can 6 inches across and 7 inches tall has a total surface area of 188.50 in².

This page measures the outside of a solid. To find what it holds, use the Volume Calculator, or the Cylinder Calculator, which also works a cylinder out backwards from its volume or area. Flat shapes belong in the Area Calculator, a single circle in the Circle Calculator, and the walls of a room in the Paint Calculator.

The surface area formula for each shape

Every formula below is the same idea, the sum of the faces, written out for one shape. Prisms and cylinders have two identical ends plus a side, cones and pyramids have one base plus sloping faces, and spheres and capsules have no flat face at all.

Here s is the slant height, measured along the sloping surface, and A is the area of the triangular end. The calculator finds it from the three sides with Heron's formula, A = √(k(k − a)(k − b)(k − c)), where k is half the perimeter.

Worked example 1: total and lateral area of a cylinder

Take a round canister 6 inches across and 7 inches tall. The radius is half the diameter, 3 in. Each lid is a circle of π × 3² = 28.27 in², so the two lids add up to 56.55 in². The side unrolls into a rectangle as tall as the can and as long as its circumference, which makes the lateral area 2 × π × 3 × 7 = 131.95 in². Adding the parts gives the total:

A paper label that wraps the side needs the lateral 131.95 in² (0.92 ft²) before any overlap, while the whole canister is 1.31 ft² or 1,216.10 cm². The Canister 6 in across button in the calculator loads these exact numbers.

Worked example 2: a cone roof from height or slant height

A round tower has a conical roof with a 6 ft radius that rises 8 ft to its tip. Roofing covers only the curved surface, so the lateral area is the number you need. The slant height comes from the right triangle formed by the radius, the rise and the roof line:

The flat circle under the roof would add π × 6² = 113.10 ft² for a closed total of 301.59 ft², but it is not roofed. If you measured the roof line itself, 10 ft from eave to tip, switch to Slant height and enter 10: the result is identical. Roofing is sold in squares of 100 ft², so this roof is 1.88 squares before waste. The Pythagorean Theorem Calculator shows the slant-height step, and the Roofing Calculator handles pitched roofs.

Worked example 3: paint for a cylindrical tank

A water tank 8 ft in diameter and 12 ft tall stands on a concrete pad, so you will paint the side and the top but not the bottom. In the calculator choose Cylinder, ft, Diameter 8 and Height 12, then Side + one end in the estimate card:

At 350 sq ft per gallon, two coats need 2.01 gallons, which rounds up to 2 gallons plus 1 quart. Painted all over (402.12 ft²), the same tank would take 2.30 gallons, or 2 gallons plus 2 quarts. Landing just past a whole gallon is common, so the estimate shows the purchase in quarts.

How to use this calculator

Pick a shape and a length unit. For round shapes, say whether your number is a radius or a diameter; for cones and pyramids, whether you measured the vertical or the slant height. Results update as you type: the large figure is the total surface area, the tiles show the side and base areas, and the steps card writes each formula out with your numbers. The units table converts the answer, and the estimate card turns any surface into gallons of paint or sheet material with a waste allowance.

Which surface do you need: total, lateral or base?

Decide which faces actually get covered before you multiply anything:

• Labels, sleeves and pipe insulation: the lateral area only, because the ends stay bare.
• Open containers: an open box, a straight-sided planter or a can with no lid is the lateral area plus one base. A can 10 in across and 12 in tall with no lid has 2 × π × 5 × 12 + π × 5² = 455.53 in².
• Objects on the ground: tanks and columns on a slab skip the bottom face.
• Gift wrap and cartons: the total area, since every face is covered. A 12 × 8 × 4 in box has 352 in², and with a 10% overlap allowance you need about 387.2 in² of paper.
• Tents and roofs: the sloping faces only; a floor is a separate base.

Same volume, different surface area

Shape decides how much material it takes to wrap a given volume. Every row below encloses exactly 1,000 cm³ (one liter). Dimensions are in centimeters, rounded to four decimals, and typing them into the calculator reproduces each area to the tenth.

No shape beats the sphere. Within each family there is a best proportion: the cheapest cylinder is as tall as it is wide, and the cheapest cone has a slant height three times its radius. Squashing a box to 4 : 2 : 1 costs 16.7% more surface than the cube that holds the same volume.

Size changes surface area by the square

Scale every length by a factor k and surface area changes by k², volume by k³. The table compares three solids of the same width s: a cube with edge s, a cylinder whose diameter and height both equal s, and a sphere with diameter s, in square inches.

The sphere always has 52.36% of the cube's area (π ÷ 6), and exactly two thirds of the area of the cylinder that just contains it, the figure Archimedes is said to have wanted on his tomb. Doubling the cube from 6 to 12 inches quadruples its surface from 216 to 864 in², which is exactly 6 ft². Units scale the same way: 1 ft² is 144 in², not 12, and 1 m² is about 10.7639 ft².

Tips for paint, wrap and sheet-material estimates

Surface area is the starting point for an order, not the order itself:

• Convert before you multiply. Paint coverage is quoted per square foot, so an area in square inches must be divided by 144 first. The estimate card does this for you.
• Use the coverage printed on your product. Product data sheets list a theoretical coverage rate per gallon alongside the solids content and the recommended film thickness. Rough and porous surfaces take more.
• Know the geometric ceiling. One US gallon is 231 cubic inches, so spread 1 mil (0.001 in) thick it covers 231,000 in², which is 1,604.17 ft². At a 4-mil wet film the limit is 401.04 ft² per gallon, and real coverage is lower.
• Add overlap for sheet goods. Wrap, fabric and sheet metal need seams and offcuts. The default is 10%; cones and capsules usually need more, because flat material does not bend cleanly into compound curves.

Who this calculator is for

Students check geometry homework against the steps card. Painters and maintenance crews price coatings for tanks, columns and domes. Makers cut fabric or sheet metal for cylinders and cones, packaging designers compare shapes that hold the same volume, and plumbing and HVAC installers size insulation for pipes and round ducts from the lateral area.

Key terms

• Lateral surface area: the area of the sides only, without the top and bottom faces.
• Base: a flat end face. Prisms and cylinders have two, cones and pyramids have one.
• Slant height (s): the distance along a sloping surface from the edge of the base to the tip.
• Net: the flat pattern you get by unfolding a solid. Its area equals the surface area.
• Heron's formula: a triangle's area from its three sides.

Limitations

The formulas assume ideal shapes with sharp edges and no thickness. Real objects have rims, seams and walls, and the inside and outside of a hollow object differ, so measure the face you will cover. Frustums such as a tapered bucket, ellipsoids, tori and freeform shapes are not covered; break them into simpler pieces.

Related calculators

• To solve a cylinder from any two of its measurements, including volume and area, use the Cylinder Calculator.
• For the space inside these shapes, use the Volume Calculator.
• For rectangles, triangles, trapezoids and other flat shapes, use the Area Calculator.
• For the radius, circumference and area of one circle, use the Circle Calculator.
• For the walls and ceiling of a room, use the Paint Calculator.
• For the sides and angles behind a triangular prism, use the Triangle Calculator.

Sources

Surface area formulas are exact geometry, so the results need no outside data. The unit and coverage figures rest on these primary sources:

• National Institute of Standards and Technology, NIST Special Publication 811, Appendix B.8: the exact factors 1 in = 2.54 cm, 1 in² = 6.4516 cm² and 1 ft² = 0.09290304 m².
• National Institute of Standards and Technology, NIST Handbook 44 (2026), Appendix C, General Tables of Units of Measurement: the US gallon is exactly 231 cubic inches, the basis of the 1,604.17 ft² per gallon per mil figure.
• U.S. Bureau of Reclamation, Guide to Protective Coatings, Inspection, and Maintenance (2012): product data sheets list the theoretical coverage rate per gallon, the solids by volume and the recommended film thickness.
• Worked examples and tables: computed with the formulas on this page, using the same code the calculator runs.