Surface Area Calculator
Surface Area Formulas
Quick Formulas
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Surface Area Formulas
What Is Surface Area?
Surface area is the total area of all surfaces of a three-dimensional object. It measures how much material would be needed to cover or wrap the object. Surface area is measured in square units (square inches, square centimeters, etc.).
Surface Area Formulas
| Shape | Formula | Variables |
|---|---|---|
| Cube | SA = 6s² | s = side length |
| Rectangular Box | SA = 2(lw + lh + wh) | l = length, w = width, h = height |
| Sphere | SA = 4πr² | r = radius |
| Cylinder | SA = 2πrh + 2πr² | r = radius, h = height |
| Cone | SA = πrs + πr² | r = radius, s = slant height |
| Square Pyramid | SA = b² + 2bs | b = base length, s = slant height |
Lateral vs Total Surface Area
Lateral Surface Area
The area of just the sides, excluding the top and bottom (bases). Think of it as wrapping paper around the sides only.
Total Surface Area
The area of all surfaces including the bases. This is what you need to paint or cover the entire object.
Step-by-Step Examples
Example 1: Cube with side = 5 cm
Example 2: Sphere with radius = 4 in
Example 3: Cylinder (r = 3, h = 7)
Real-World Applications
Paint Coverage
Calculate how much paint needed to cover a room, tank, or structure.
Wrapping Material
Determine paper or fabric needed to wrap gifts or cover objects.
Manufacturing
Calculate material costs for products based on surface coverage.
Remember
- Surface area uses square units (cm², m², in², ft²)
- For cones and pyramids, slant height ≠ vertical height
- A sphere has no lateral area—it's all one curved surface
- Double-check units: all measurements must use the same unit
Frequently Asked Questions
What is the difference between lateral and total surface area?
Lateral surface area includes only the sides of a 3D shape, excluding the top and bottom (bases). Total surface area includes everything - sides plus bases. For a cylinder: lateral area is just the curved part (2 pi r h), while total area adds both circular ends (2 pi r h + 2 pi r squared).
How much paint do I need for a room?
Calculate the wall surface area (perimeter times height), subtract windows and doors, then divide by paint coverage (typically 350-400 sq ft per gallon). For a 12x14 ft room with 8 ft ceilings: (2 x 12 + 2 x 14) x 8 = 416 sq ft. Minus a 20 sq ft door and two 15 sq ft windows = 366 sq ft, or about 1 gallon.
Why does a sphere have the minimum surface area for a given volume?
A sphere encloses the maximum volume with the minimum surface area because it has no edges or corners that 'waste' surface. This is why bubbles are spherical - nature minimizes surface tension. For a given volume, a sphere uses about 52% less surface area than a cube.
How do I find the slant height for a cone or pyramid?
Use the Pythagorean theorem. For a cone: slant height equals the square root of (radius squared + height squared). For a pyramid: slant height equals the square root of (half-base squared + height squared). The slant height is needed for lateral surface area calculations.