What this surface area calculator does
This calculator gives the total surface area of a solid: the sum of the areas of all its outside faces. Choose one of 11 solids (cube, rectangular box, sphere, hemisphere, cylinder, hollow tube, cone, truncated cone, square pyramid, triangular prism or ellipsoid), type its dimensions and read the result straight away.
Besides the total, it shows each part: the lateral surface, the bases, the curved surface, and useful lengths such as the slant height of a cone or the space diagonal of a box. A drawing with the same letters as the input boxes shows what each number means.
Use it to know how much paint, wallpaper, wrapping paper, sheet metal or insulation a job needs. For the space inside the solid, use the volume calculator; for flat shapes, use the area calculator.
How to use this calculator
- Pick the solid in the grid. The drawing changes to show its dimensions.
- Choose the length unit (cm by default). Every box uses that same unit.
- Type each dimension. The result updates as you type, with no Calculate button.
- Read the total surface in the big number, then the lateral surface, bases and other lengths in the receipt.
- Tap the button under the result to calculate the volume of the same solid with the same numbers.
If a box is empty or a value is not valid (zero, negative, an inner radius larger than the outer one), the calculator tells you which box to fix instead of showing a wrong answer. Switching solids keeps the values that still make sense, for example the radius.
The formulas behind the results
The total surface of a solid is always the sum of its parts. These are the main cases:
Cylinder: S = 2πr × (r + h) · Cone: S = πr × (r + s), s = √(r² + h²)
Sphere: S = 4πr² · Pyramid: S = a² + 2a × s
- A cylinder has a lateral surface 2πrh (a rectangle once unrolled) and two circular bases of area πr² each.
- A cone has a lateral surface πrs, where s is the slant height √(r² + h²), plus one circular base.
- A sphere has only a curved surface, 4πr², which is four times the area of its great circle.
- A pyramid with a square base has four triangles of area a × s ÷ 2 plus the square base a².
- An ellipsoid with two equal semi-axes (a spheroid, or a sphere) has an exact formula, which the calculator uses. With three different semi-axes there is no simple exact formula: the result uses Knud Thomsen's approximation, off by about 1% at most, and is marked with the sign ≈.
π is used at full calculator precision. Only the displayed numbers are rounded, to 4 decimals at most.
Worked examples
Example 1: painting a cylindrical water tank
- Choose Cylinder, unit m, and enter r = 0.5 and h = 1.2.
- Lateral surface: 2 × π × 0.5 × 1.2 ≈ 3.7699 m².
- Two bases: 2 × π × 0.5² ≈ 1.5708 m².
- Total: 3.7699 + 1.5708 ≈ 5.3407 m².
About 5.34 m² of surface. With two coats and a paint that covers 8 m² per litre, plan on about 1.34 L.
Example 2: wrapping a gift box of 30 × 20 × 10 cm
- Choose Rectangular box, unit cm, and enter L = 30, l = 20 and h = 10.
- Faces: 30 × 20 = 600, 30 × 10 = 300 and 20 × 10 = 200 cm², each counted twice.
- Total: 2 × (600 + 300 + 200) = 2,200 cm².
- Add 10% for overlaps and folds: 2,420 cm².
2,200 cm² of box surface, so about 2,420 cm² (0.24 m²) of wrapping paper.
Key concepts
Surface area versus volume
Surface area measures the outside of a solid and is written in square units (cm², m²). Volume measures the space inside and is written in cubic units (cm³, m³). Two solids can have the same volume and very different surfaces: a flat box and a cube of equal volume do not need the same amount of wrapping. Use the volume calculator for capacity.
Lateral and total surface
The lateral surface is the side only, without the bases. It is what you cover for a label on a can or the walls of a room. The total surface adds the bases, which is what a paint job on the whole object needs.
Slant height
In a cone or a pyramid, the slant height is measured along the face, not straight up. It is longer than the height, and the lateral surface depends on it.
Hollow solids
A tube has an outer wall, an inner wall and two ring-shaped ends. All of them are counted. This is the surface to paint or coat if the inside is reachable.
Tips and common mistakes
Practical tips
- Measure every length in the same unit. To convert first, use the unit converter.
- Remember that a radius is half the diameter. A pipe of 20 cm across has r = 10.
- Add 5 to 10% to paint, wrapping or sheet material for waste and overlaps.
- Remove openings when they are not covered: a tank with no lid does not need the top base.
Common mistakes
- Mixing up surface and volume: a surface in cm² is not a capacity in litres.
- Typing the diameter in the radius box, which multiplies the surface by four.
- Using the vertical height of a cone where a formula needs the slant height. The calculator finds the slant height for you.
- Forgetting that 1 m² is 10,000 cm², not 100 cm².
Frequently asked questions
How do I find the surface area of a cylinder?
Add the lateral surface 2πrh and the two bases 2πr², which gives 2πr(r + h). For r = 3 and h = 10, that is about 245.04.
What is the surface area of a sphere?
4πr². A sphere of radius 3 has a surface of about 113.10. A hemisphere adds its flat base, so its total is 3πr².
Why is the ellipsoid result sometimes marked with ≈?
When its three semi-axes are all different, the exact surface needs a complicated integral, so the calculator gives an approximation within about 1% of the true value. When two semi-axes are equal, the result is exact and has no ≈.
Does the result change with the unit?
The number follows the unit you pick: cm gives cm², m gives m². Changing the unit without changing the numbers means a different physical size.
Can I use it to estimate paint?
Yes. Divide the surface by the coverage printed on the tin, and multiply by the number of coats. Add a margin for waste.
How do I get the volume of the same solid?
Use the button under the result. It opens the volume calculator with the same solid and dimensions.