Volume & surface area formulas
Volume is the space inside a solid, in cubic units; surface area is the total area of its outer surface, in square units. For a cylinder, V = πr²h and S = 2πr² + 2πrh. For a sphere, V = ⁴⁄₃πr³ and S = 4πr².
What each letter means
- r, h
- Radius and perpendicular height
- w, h, d
- Width, height, and depth (or l × w × h)
- ℓ
- Slant height of a cone or pyramid face
- V, S
- Volume (cubic units) and surface area (square units)
Volume & surface area formulas calculator
What fits inside, and what covers the outside. Change a value and the diagram, result, and working update together.
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Use the same unit for every dimension. Changing the unit relabels the measurements; it does not convert them.
From dimensions to an answer
The idea
Volume measures three-dimensional space in cubic units. Surface area measures the total outside covering in square units. A rectangular prism has volume whd and surface area 2(wh + wd + hd). A sphere has volume 4πr³/3 and surface area 4πr².
Volume and surface area formulas chart
| Solid | Volume | Total surface area |
|---|---|---|
| Cube (edge s) | V = s³ | S = 6s² |
| Rectangular prism | V = whd | S = 2(wh + wd + hd) |
| Cylinder | V = πr²h | S = 2πr² + 2πrh |
| Cone | V = ⅓πr²h | S = πr² + πrℓ, ℓ = √(r² + h²) |
| Sphere | V = ⁴⁄₃πr³ | S = 4πr² |
| Hemisphere | V = ⅔πr³ | S = 3πr² (with base) |
| Square pyramid (base b) | V = ⅓b²h | S = b² + 2bℓ |
| Any prism | V = base area × length | S = 2 × base area + base perimeter × length |
Worked examples
1Cube
- Edge 3 cm: V = 3³ = 27 cm³ and S = 6 × 3² = 54 cm².
- Double the edge to 6 cm: V = 216 cm³ (8 times larger) and S = 216 cm² (4 times larger).
Answer: Volume grows with the cube of the scale factor; surface area with its square.
2Cylinder
- r = 3 cm and h = 10 cm.
- V = π × 3² × 10 = 90π ≈ 282.74 cm³.
- S = 2π × 3² + 2π × 3 × 10 = 18π + 60π = 78π ≈ 245.04 cm².
Answer: About 282.74 cm³ and 245.04 cm².
3Cone
- r = 3 and vertical height h = 4, so ℓ = √(3² + 4²) = 5.
- V = ⅓ × π × 9 × 4 = 12π ≈ 37.70.
- S = π × 3 × (3 + 5) = 24π ≈ 75.40.
Answer: About 37.70 cubic units and 75.40 square units.
4Sphere
- r = 6 cm.
- V = ⁴⁄₃ × π × 6³ = 288π ≈ 904.78 cm³.
- S = 4π × 6² = 144π ≈ 452.39 cm².
Answer: About 904.78 cm³ (just over 0.9 liters) and 452.39 cm².
Common mistakes
- Mixing up square and cubic units.
- Using slant height for volume, or vertical height for a cone’s curved surface.
- Forgetting a base, or including a missing top, in surface area.
- Converting capacity incorrectly: 1 cm³ = 1 mL and 1,000 cm³ = 1 L.
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Keep exploring
Volume & surface area formulas: common questions
Wondering about the why? Start here.
What is the difference between volume and surface area?
Volume measures how much space is inside a solid, in cubic units such as cm³. Surface area measures how much material covers the outside, in square units such as cm². Filling a box is about volume; wrapping it is about surface area.
Open the volume calculatorCan you convert volume to surface area?
Not directly. Different shapes with the same volume can have different surface areas. You need the dimensions: for a cube, find the edge s = ∛V, then S = 6s². A 64 cm³ cube has edge 4 cm and surface area 96 cm².
How do cubic centimeters relate to liters?
One cubic centimeter holds one milliliter, so 1,000 cm³ = 1 liter. One cubic meter holds 1,000 liters.