acalculator

What is the surface area of a solid?

Pick a shape and type its measurements to get the total surface area, with the side and base areas shown separately.

Your numbers

Total surface area
150.796447

Cylinder: total surface area 150.796447.

Side (lateral) area
94.24778
Base area
28.274334

Total surface area: 150.796447. Cylinder: total surface area 150.796447.

How to calculate

Finds the total, side (lateral), and base surface area of a cube, rectangular box, sphere, cylinder, cone, or square pyramid from its dimensions.

Example with the default inputs (Shape Cylinder, Radius 3, Height 5): Cylinder: total surface area 150.796447.

Method: Cube 6a²; box 2(lw + lh + wh); sphere 4πr²; cylinder 2πr² + 2πrh; cone πr² + πrs with s = √(r² + h²); square pyramid a² + 2as with s = √(h² + (a/2)²).

  • Every length is in one unit, and each area is in that unit squared.
  • The cylinder and cone are right circular (the tip or top is straight above the centre of the base), and the pyramid is a right pyramid with a square base.
  • The total includes every base: both ends of a cylinder, the base of a cone or pyramid, and all six faces of a cube or box.
  • Each length must be greater than 0.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Shape Cylinder, Radius 3, Height 5 gives Total surface area 150.796447, Side (lateral) area 94.24778, Base area 28.274334.Source: OpenStax Prealgebra 2e, Example 9.53 (150.72 with π ≈ 3.14); hand calculation in content.mdx: 2π × 3² + 2π × 3 × 5 = 48π
  2. Shape Rectangular box, Length 14, Width 9, Height 17 gives Total surface area 1,034, Side (lateral) area 782, Base area 126.Source: OpenStax Prealgebra 2e, Example 9.47: 2 × 14 × 17 + 2 × 14 × 9 + 2 × 9 × 17 = 1,034
  3. Shape Cube, Edge length 2.5 gives Total surface area 37.5, Side (lateral) area 25, Base area 6.25.Source: OpenStax Prealgebra 2e, Example 9.49: 6 × 2.5² = 37.5
  4. Shape Sphere, Radius 6 gives Total surface area 452.389342.Source: OpenStax Prealgebra 2e, Example 9.51 (452.16 with π ≈ 3.14); hand calculation in content.mdx: 4π × 6² = 144π
  5. Shape Cone, Radius 3, Height 4 gives Slant height 5, Side (lateral) area 47.12389, Base area 28.274334, Total surface area 75.398224.Source: OpenStax Calculus Volume 2, section 2.4 (lateral area πrs); hand calculation in content.mdx: s = √(3² + 4²) = 5, π × 3 × 5 + π × 3² = 24π
  6. Shape Square pyramid, Edge length 6, Height 4 gives Slant height 5, Side (lateral) area 60, Base area 36, Total surface area 96.Source: hand calculation in content.mdx: slant = √(4² + 3²) = 5; 4 triangles of ½ × 6 × 5 = 60; 60 + 6² = 96

How it works

Pick the shape, then type its measurements in one unit. Each surface area is the sum of the areas of its faces:

  • Cube, edge a: side 4a², base a², total 6a².
  • Rectangular box, length l, width w, height h: side 2h(l + w), base lw, total 2(lw + lh + wh).
  • Sphere, radius r: total 4πr². A sphere has no flat base, so the total is not split.
  • Cylinder, radius r, height h: side 2πrh, base πr², total 2πr² + 2πrh.
  • Cone, radius r, height h: side πrs, base πr², total πr² + πrs.
  • Square pyramid, base edge a, height h: side 2as, base a², total a² + 2as.

The slant height s is found with the Pythagorean theorem:

  • cone: s = √(r² + h²), from the radius and the height
  • square pyramid: s = √(h² + (a/2)²), from the height and half the base edge. Each of the four side triangles has area ½ × a × s.

Assumptions

  • Every length is in one unit, and the areas are in that unit squared.
  • The cylinder and cone are right circular: the top or tip is straight above the centre of the base. The pyramid is a right pyramid with a square base.
  • The total counts every base: both ends of a cylinder, the one base of a cone or pyramid, and all six faces of a cube or box.
  • π is the full-precision value 3.141592653589793….
  • Each length is greater than 0.

Worked examples by hand

Cylinder, radius 3, height 5 (OpenStax Example 9.53). Side: 2π × 3 × 5 = 30π = 94.2478. Base: π × 3² = 9π = 28.2743. Total: 2 × 9π + 30π = 48π = 150.7964. With π ≈ 3.14 the textbook gets 150.72.

Rectangular box, 14 × 9 × 17 (OpenStax Example 9.47). Total: 2 × (14 × 9 + 14 × 17 + 9 × 17) = 2 × (126 + 238 + 153) = 1,034. Side: 2 × 17 × (14 + 9) = 782. Base: 14 × 9 = 126.

Cube, edge 2.5 (OpenStax Example 9.49). Total: 6 × 2.5² = 6 × 6.25 = 37.5. Side: 4 × 6.25 = 25. Base: 6.25.

Sphere, radius 6 (OpenStax Example 9.51). Total: 4π × 6² = 144π = 452.3893. With π ≈ 3.14 the textbook gets 452.16.

Cone, radius 3, height 4. Slant: √(3² + 4²) = √25 = 5. Side: π × 3 × 5 = 15π = 47.1239. Base: 9π = 28.2743. Total: 24π = 75.3982.

Square pyramid, base edge 6, height 4. Slant: √(4² + 3²) = 5. Side: 4 × ½ × 6 × 5 = 60. Base: 6² = 36. Total: 96.

Other questions people ask

What is surface area?

The total area of all the outside surfaces of a solid, as if you unfolded it flat and measured the paper. It is in square units, such as in² or cm², because it is an area.

What is the difference between lateral area and total surface area?

The lateral (side) area leaves out the flat ends. For a can, the lateral area is the label (2πrh) and the total adds the top and bottom circles (2πr²). A can with radius 3 cm and height 5 cm has a label of 94.25 cm² and a total of 150.80 cm².

How do I find the surface area of a cylinder?

Add the two circular ends, 2πr², to the side, 2πrh: A = 2πr² + 2πrh = 2πr(r + h). The side unrolls into a rectangle as tall as the cylinder and as wide as the circumference, 2πr.

How do I find the surface area of a cone?

Add the base circle, πr², to the side, πrs, where s is the slant height √(r² + h²). A cone with radius 3 and height 4 has s = 5, so A = 9π + 15π = 24π = 75.40.

How do I find the surface area of a square pyramid?

Add the square base, a², to four triangles. Each triangle has base a and height equal to the slant height s = √(h² + (a/2)²), so the four together are 4 × ½ × a × s = 2as. A pyramid with base 6 and height 4 has s = 5 and A = 36 + 60 = 96.

What is the difference between surface area and volume?

Surface area measures the outside (how much paint or wrapping it needs) in square units. Volume measures the space inside (how much it holds) in cubic units. The cylinder volume and cone volume calculators find the volume.

What units should I use?

Type every length in the same unit. The areas are in that unit squared: lengths in feet give square feet, and lengths in centimetres give square centimetres.