Surface Area Calculator: Cube, Sphere, Cylinder, Cone, Prism
Free surface area calculator for cube, rectangular prism, sphere, cylinder, and cone. Enter dimensions to get the surface area with the formula and steps.
What surface area measures
Surface area is the total area of all the faces (or the curved skin) of a three-dimensional solid, the amount of “wrapping paper” you would need to cover it completely. It is measured in square units: square inches, square feet, square meters. A cube with 3-inch sides has 6 faces of 9 square inches each, so its surface area is 54 square inches. Paint coverage, packaging material, heat loss through walls, and the fabric in a tent are all surface area problems in disguise.
This calculator covers the five most-encountered solids: cube, rectangular prism (a box), sphere, cylinder, and cone. Pick the shape, enter its dimensions, and get the surface area with the formula written out and every substitution shown. All dimensions must be positive numbers in the same unit, and the answer comes back in that unit squared.
How to use the surface area calculator
Choose the shape from the dropdown. The input fields change automatically: a cube needs only its side length, a box needs length, width, and height, a sphere needs its radius, and cylinders and cones need radius and height. Press Calculate to see the formula, the substituted values, and the final surface area.
Keep all measurements in one unit. Mixing inches and feet is the classic error: a box measured as 2 ft by 18 in by 6 in must become 2 ft by 1.5 ft by 0.5 ft (or all in inches) before any formula sees it. The calculator assumes consistent units and labels the answer in square units so the meaning stays clear.
The five formulas
- Cube: SA = 6s². Six identical square faces, each s by s.
- Rectangular prism: SA = 2(lw + lh + wh). Three pairs of rectangles: top/bottom, front/back, left/right.
- Sphere: SA = 4πr². Exactly four times the area of a great circle, a beautiful result first proved by Archimedes.
- Cylinder: SA = 2πr² + 2πrh. Two circular ends plus the curved side, which unwraps into a rectangle h tall and 2πr wide.
- Cone: SA = πr² + πrl, where the slant height l = √(r² + h²). The circular base plus the curved side, which unwraps into a sector of a circle of radius l.
The cone’s slant height deserves a note: it is the hypotenuse of the right triangle formed by the radius and the vertical height, found with Pythagoras. For a cone with r = 3 and h = 4, the slant height is √(9 + 16) = 5, one of the famous 3-4-5 triangles.
Example: surface area of a cylinder with r = 2, h = 5
Step 1: Formula: SA = 2πr² + 2πrh.
Step 2: Substitute: SA = 2π(2)² + 2π(2)(5) = 8π + 20π = 28π.
Step 3: Evaluate: 28 × 3.14159… = 87.96 square units.
Sanity check: the two circular ends contribute 2π(4) ≈ 25.1 and the side contributes 2π(2)(5) ≈ 62.8; the side dominates for a tall thin cylinder, as expected.
More examples: cube s = 3 → 54; box 2 × 3 × 4 → 52; sphere r = 5 → 314.16; cone r = 3, h = 4 → 75.40.
Common mistakes
- Using diameter instead of radius: the formulas want r, half the diameter. A sphere “10 across” has r = 5; plugging in 10 quadruples the answer.
- Forgetting the cylinder’s two ends: 2πrh alone is only the curved side (the label of a can). The full surface area adds both circular lids: 2πr².
- Using vertical height for the cone’s side: the curved area needs the slant height l = √(r² + h²), not h. For short wide cones the difference is large.
- Reporting square units as plain units: surface area is an area. “54 square inches” is correct; “54 inches” is a length and means something entirely different.
Where surface area shows up
Surface area governs anything that happens at a surface: paint and coating coverage, heat transfer (which is why small animals lose heat faster, they have more surface per unit volume), packaging cost, sail and tent sizing, and the drag on a falling object. In biology it explains why cells stay microscopic and why lungs fold into millions of alveoli. For the space inside these same shapes, the companion volume calculators (such as the cylinder volume calculator) complete the picture, and the unit converter keeps your dimensions consistent.
6s², where s is the side length. A cube has 6 identical square faces, so you square the side and multiply by 6. A 3-unit cube has surface area 6 × 9 = 54 square units.
Use SA = 2πr² + 2πrh: the two circular ends (2πr²) plus the curved side (2πrh). For r = 2, h = 5, that is 8π + 20π = 28π ≈ 87.96 square units.
The slant height l is the distance from the tip of the cone to the edge of its base, measured along the sloping side. By Pythagoras, l = √(r² + h²). The curved surface area is πrl, so you need l before you can finish the calculation.
Surface area measures the outside skin (square units), volume measures the space inside (cubic units). Doubling every dimension multiplies surface area by 4 but volume by 8, which is why the distinction matters in scaling problems.