Surface Area Calculator

How each shape's surface area is calculated

This calculator computes the total exterior area of four common 3D shapes. Each has its own structure, so each requires a different approach. Whether you're wrapping a box, painting a sphere, or covering a cylindrical tank, the calculator handles the geometry for you.

The formulas

Box: SA = 2(lw + lh + wh)

Sphere: SA = 4πr²

Cylinder: SA = 2πr² + 2πrh

Cone: SA = πr² + πrl (where l is slant height)

Worked example

Box with dimensions 5 × 3 × 4 cm

1. Identify the three dimensions: length = 5 cm, width = 3 cm, height = 4 cm
2. Calculate each pair of opposite faces:
   • Top and bottom: 5 × 3 = 15 cm² each → 2 × 15 = 30 cm²
   • Front and back: 5 × 4 = 20 cm² each → 2 × 20 = 40 cm²
   • Left and right: 3 × 4 = 12 cm² each → 2 × 12 = 24 cm²
3. Add all six faces: 30 + 40 + 24 = 94 cm²

Sphere with radius 6 cm

1. Note the radius: r = 6 cm
2. Apply the formula: SA = 4π(6)²
3. Calculate: 4 × 3.14159 × 36 = 452.39 cm²

Cylinder with radius 4 cm and height 10 cm

1. Identify radius = 4 cm, height = 10 cm
2. Find the two circular bases: 2π(4)² = 2 × 3.14159 × 16 = 100.53 cm²
3. Find the curved side: 2π(4)(10) = 2 × 3.14159 × 40 = 251.33 cm²
4. Total: 100.53 + 251.33 = 351.86 cm²

Cone with radius 3 cm and slant height 8 cm

1. Note radius = 3 cm, slant height = 8 cm
2. Find the base: π(3)² = 3.14159 × 9 = 28.27 cm²
3. Find the lateral surface: π(3)(8) = 3.14159 × 24 = 75.40 cm²
4. Total: 28.27 + 75.40 = 103.67 cm²

Note: If you only have height (not slant height) for a cone, calculate slant height as √(r² + h²) first.

Common mistakes to avoid

Mixing up radius and diameter: Radius is half the diameter. If your circle measures 10 cm across, the radius is 5 cm, not 10 cm. Using the wrong value doubles your result incorrectly.

Forgetting both bases: Cylinders and cones each have a circular base. Don't include only the curved surface—add the base area too (unless the shape is open-ended, which should be specified in your problem).

Unit inconsistency: If one dimension is in meters and another in centimeters, convert everything first. Mixing units produces meaningless numbers.

Confusing slant height with vertical height on cones: The slant height runs along the cone's slope from apex to edge. The vertical height goes straight down. For surface area, you need the slant height. If given only vertical height, use the Pythagorean theorem: slant height = √(radius² + height²).