Surface Area Calculator
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Results
Surface area of the cube: 54 units², of which 36 is the lateral area
How to use the surface area calculator
- Choose the solid and enter its dimensions in one unit (radius is half the diameter; heights are perpendicular).
- Read the total surface area, which includes every face, and the lateral area, which leaves out the flat ends or base.
- Pick the unit to label the results in square units. Imperial units also show square meters, and metric units square feet.
Surface area formulas
Cone: πr(r + l), side πrl, slant l = √(r² + h²)
Sphere: 4πr² · Hemisphere: 3πr² (curved part 2πr²) · Capsule: 2πr(2r + h)
Square pyramid: s² + 2sl, slant l = √(h² + (s ÷ 2)²)
Cone frustum: π(R + r)l + π(R² + r²), l = √((R − r)² + h²) · Spherical cap: 2πRh + πh(2R − h)
Surface area is the total area of a solid's outside, in square units. For solids with flat faces it is simply the sum of the faces. For curved solids, imagine unrolling the surface: a cylinder's side unrolls into a rectangle 2πr wide and h tall, and a cone's side into a sector of a circle whose radius is the slant height l, which is why cones, pyramids and frustums need l rather than the vertical height. A sphere's surface is exactly four times the area of its largest circle, 4πr².
Lateral area means the sides only. Use it when the ends are open or not covered: the label on a can, the walls of a round tank, the roof of a pyramid. Use the total when every face counts, such as wrapping a gift or coating a part.
Worked examples
- Painting a tank. A cylindrical tank 1.5 m in radius and 4 m tall has a side area of 2π × 1.5 × 4 = 37.7 m². Adding the top (7.07 m²) but not the base standing on the ground gives 44.77 m² to paint; with both ends it is 51.84 m².
- Wrapping a box. A 30 × 20 × 10 cm box has 2 × (600 + 300 + 200) = 2,200 cm² of surface. Cut the paper a little larger to allow for overlaps.
- A cone. With base radius 3 and height 4, the slant height is √(9 + 16) = 5, so the curved side is π × 3 × 5 = 15π ≈ 47.12 and the total with the base is 24π ≈ 75.4.
- A ball. A sphere of radius 2 has a surface of 16π ≈ 50.27 square units.
The ellipsoid: exact, not approximate
There is no elementary formula for the surface of an ellipsoid with three different semi-axes. A common shortcut is Knud Thomsen's approximation, which can be off by up to about 1%. This one evaluates Legendre's exact formula with elliptic integrals. As a check, the WGS 84 model of the Earth (equatorial radius 6,378.137 km, polar radius 6,356.752 km) comes out at 510,065,622 km², matching the published value for that ellipsoid; a sphere with the Earth's mean radius gives 510,065,881 km².
Surface areas at the default sizes
| Solid | Dimensions | Lateral area | Total surface area |
|---|---|---|---|
| Cube | s = 3 | 36 | 54 |
| Rectangular prism (box) | l = 4, w = 3, h = 2 | 28 | 52 |
| Cylinder | r = 3, h = 5 | 94.2478 | 150.7964 |
| Cone | r = 3, h = 4 | 47.1239 | 75.3982 |
| Sphere | r = 2 | – | 50.2655 |
| Hemisphere | r = 3 | 56.5487 | 84.823 |
| Capsule | r = 1, h = 2 | – | 25.1327 |
| Square pyramid | s = 6, h = 4 | 60 | 96 |
| Cone frustum | R = 4, r = 2, h = 3 | 67.963 | 130.7949 |
| Ellipsoid | a = 3, b = 2, c = 1 | – | 48.8821 |
| Spherical cap | R = 5, h = 2 | 62.8319 | 113.0973 |
Tips for using surface area
- Paint and coatings: divide the area by the coverage printed on the tin (for example square meters per liter) and allow for a second coat.
- Scale effect: doubling every dimension multiplies surface area by 4 but volume by 8, which is why small objects cool faster than large ones.
- Units are squared: 1 m² = 10,000 cm² and 1 ft² = 144 in². Convert lengths before calculating.
- Openings: subtract windows, doors or holes from the total if they will not be covered.
For the space inside the same solids, see the volume calculator. For flat shapes such as a single face or a floor plan, use the area calculator.
Frequently asked questions
How do I find the surface area of a cylinder?
Add the two circular ends (2πr²) to the curved side (2πrh), which gives 2πr(r + h). For a radius of 3 and a height of 5, that is 48π, about 150.80 square units; the side alone is 30π, about 94.25.
What is the difference between lateral and total surface area?
Lateral area covers only the sides, leaving out the flat ends or base. Total surface area includes every face. A can label is lateral area; the metal to make the whole can is total area.
How do I find the surface area of a sphere?
Use A = 4πr². A sphere with a radius of 2 has a surface area of 16π, about 50.27 square units: exactly four times the area of a circle with the same radius.
How do I find the surface area of a cone?
Find the slant height l = √(r² + h²), then the curved side is πrl and the total is πr(r + l). With r = 3 and h = 4, l = 5, the side is 15π ≈ 47.12 and the total is 24π ≈ 75.40.
How do I find the surface area of a box?
Add the areas of the six faces in three pairs: 2(lw + lh + wh). A box 4 × 3 × 2 has 2 × (12 + 8 + 6) = 52 square units of surface.