Volume Calculator
Calculate volume and surface area for spheres, cylinders, cones, cubes, rectangular prisms, and pyramids. Step-by-step formulas included.
About Volume Calculator
Calculate the volume and surface area of six common 3D shapes: sphere, cylinder, cone, cube, rectangular prism, and pyramid. Select a shape, enter the dimensions in your chosen unit, and see the results with formulas shown step by step.
Volume and Surface Area Formulas
| Shape | Volume | Surface Area |
|---|---|---|
| Sphere | (4/3)πr³ | 4πr² |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | (1/3)πr²h | πr² + πr√(r² + h²) |
| Cube | s³ | 6s² |
| Rectangular prism | lwh | 2(lw + lh + wh) |
| Pyramid | (1/3)lwh | lw + l√((w/2)² + h²) + w√((l/2)² + h²) |
Worked Examples
Sphere (r = 8 cm):
- Volume = (4/3) x π x 8³ = (4/3) x π x 512 = 2,144.66 cm³
- Surface area = 4 x π x 64 = 804.25 cm²
Cylinder (r = 5 cm, h = 12 cm):
- Volume = π x 25 x 12 = 942.48 cm³
- Surface area = 2π(25) + 2π(5)(12) = 157.08 + 376.99 = 534.07 cm²
Cone (r = 4 cm, h = 9 cm):
- Volume = (1/3) x π x 16 x 9 = 150.80 cm³
- Slant height = √(16 + 81) = √97 = 9.849
- Surface area = π(16) + π(4)(9.849) = 50.27 + 123.72 = 173.98 cm²
Volume Relationships Between Shapes
There are elegant relationships between the volumes of shapes that share the same base and height:
| Relationship | Factor | Example (r=5, h=10) |
|---|---|---|
| Cylinder | 1x (baseline) | πr²h = 785.40 cm³ |
| Cone (same base and height) | 1/3 of cylinder | (1/3)πr²h = 261.80 cm³ |
| Sphere (diameter = h = 2r) | 2/3 of cylinder | (4/3)πr³ = 523.60 cm³ |
A cone is exactly 1/3 the volume of a cylinder with the same base and height. A sphere fitting exactly inside a cylinder (diameter = height = 2r) takes up exactly 2/3 of the cylinder's volume. This relationship was proved by Archimedes and he considered it his greatest mathematical achievement.
Everyday Volume Reference
| Object | Approximate Shape | Dimensions | Volume |
|---|---|---|---|
| Tennis ball | Sphere | r = 3.35 cm | 157 cm³ |
| Soda can | Cylinder | r = 3.3 cm, h = 12.2 cm | 418 cm³ (355 mL usable) |
| Basketball | Sphere | r = 12.1 cm | 7,415 cm³ |
| Swimming pool | Rectangular prism | 12m x 6m x 1.5m | 108 m³ (108,000 L) |
| Earth | Sphere | r = 6,371 km | 1.083 x 10¹² km³ |
Unit Conversions for Volume
| From | To | Multiply by |
|---|---|---|
| cm³ | mL | 1 (they are equal) |
| cm³ | litres | 0.001 |
| m³ | litres | 1,000 |
| in³ | cm³ | 16.387 |
| ft³ | litres | 28.317 |
| US gallon | litres | 3.785 |
Practical Applications
- Shipping: Package volume (l x w x h) determines shipping cost and container fit
- Aquariums: A 60 x 30 x 40 cm tank holds 72,000 cm³ = 72 litres of water
- Concrete: A cylindrical pier footing (r=15cm, h=60cm) needs πr²h = 42,412 cm³ = 0.042 m³ of concrete
- Storage: A 10 x 10 x 8 ft storage unit has 800 ft³ of space (about 22.7 m³)
For 2D shape areas, the area calculator covers rectangles, triangles, circles, and more. For surface area specifically, the surface area calculator provides additional shape options.
All calculations run entirely in your browser. No data is sent anywhere.
Frequently Asked Questions
What shapes does this volume calculator support?
It supports six common 3D shapes: sphere, cylinder, cone, cube, rectangular prism (box), and pyramid. Each shape has its own input fields and formulas.
How do I calculate the volume of a sphere?
Use the formula V = (4/3) times pi times r cubed. Just enter the radius and the tool does the rest. For a sphere with radius 5 cm, the volume is about 523.6 cubic cm.
Does it also calculate surface area?
Yes. Every shape shows both the volume and the surface area alongside the formulas used. The surface area uses the appropriate formula for each shape.
Can I change the measurement unit?
Yes. Select your unit (cm, m, inches, feet, mm, or yards) from the dropdown. The results display in the matching cubic and squared units. The math is the same regardless of unit.
What is the difference between a cone and a pyramid?
A cone has a circular base and tapers to a point, using V = (1/3) times pi times r squared times h. A pyramid has a rectangular base and uses V = (1/3) times length times width times height.
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