Volume formulas let you calculate how much space a 3D shape occupies. This quick-reference chart covers every common solid with the formula, variable definitions, and a worked example for each.
Complete Volume Formulas Table
| Shape | Formula | Variables |
|---|---|---|
| Cylinder | V = πr²h | r = radius, h = height |
| Sphere | V = (4/3)πr³ | r = radius |
| Cone | V = (1/3)πr²h | r = base radius, h = height |
| Cube | V = a³ | a = side length |
| Rectangular Prism | V = l × w × h | l = length, w = width, h = height |
| Pyramid (square) | V = (1/3)a²h | a = base side, h = height |
| Triangular Prism | V = (1/2)bh_t × L | b = base, h_t = triangle height, L = length |
Cylinder Volume: V = πr²h
What it means: The circular base area (πr²) extended through the height h.
Example: r = 5 cm, h = 12 cm → V = π × 25 × 12 = 942.48 cm³
Cylinders appear as cans, pipes, drums, silos, columns, and water tanks. The formula works for any right circular cylinder.
Sphere Volume: V = (4/3)πr³
What it means: The volume enclosed by a perfectly round surface at distance r from center.
Example: r = 6 cm → V = (4/3) × π × 216 = 904.78 cm³
Balls, globes, bubbles, and planets are spheres. Volume grows with the cube of radius, so small changes in r produce large volume changes.
Cone Volume: V = (1/3)πr²h
What it means: Exactly one-third of the cylinder that shares the same base and height.
Example: r = 3 cm, h = 7 cm → V = (1/3) × π × 9 × 7 = 65.97 cm³
Ice cream cones, funnels, traffic cones, and volcanic shapes use this formula. Always use perpendicular height, not slant height.
Cube Volume: V = a³
What it means: Side × side × side. All edges are equal.
Example: a = 4 cm → V = 64 cm³
Dice, sugar cubes, and Rubik's cubes are common examples. A cube is a special rectangular prism where l = w = h.
Rectangular Prism Volume: V = lwh
What it means: Three perpendicular dimensions multiplied together.
Example: 8 × 4 × 3 = 96 cm³
Most real-world boxes, rooms, pools, and containers are rectangular prisms. This is the most frequently used volume formula in practical applications.
Pyramid Volume: V = (1/3)Bh
What it means: One-third the volume of a prism with the same base and height.
Square base: B = a²
Triangular base: B = (√3/4)a²
Example (square): a = 5, h = 12 → V = (1/3) × 25 × 12 = 100 cm³
The 1/3 factor applies to all pyramids regardless of base shape. Egyptian pyramids, roof peaks, and crystal formations follow this formula.
Triangular Prism Volume: V = (1/2)bh_t × L
What it means: Triangle cross-section area × prism length.
Example: b = 5, hₜ = 4, L = 8 → V = (1/2) × 5 × 4 × 8 = 80 cm³
Toblerone boxes, roof sections, and wedge-shaped objects are triangular prisms. First find the triangle area, then multiply by length.
Key Relationships Between Volume Formulas
- Cone = (1/3) × Cylinder with same base and height
- Pyramid = (1/3) × Prism with same base and height
- Sphere = (2/3) × Cylinder that circumscribes it (Archimedes's result)
- Cube is a special case of rectangular prism (l = w = h)
Volume vs Surface Area
| Property | Volume | Surface Area |
|---|---|---|
| Measures | Space inside | Outside covering |
| Units | Cubic (m³, cm³, L) | Square (m², cm²) |
| Scales as | Length³ | Length² |
| Used for | Capacity, filling | Painting, wrapping |
Use the volume calculators: Cylinder, Sphere, Cone, Cube, Rectangular Prism, Pyramid, Triangular Prism.