Volume Formulas for All 3D Shapes
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Volume Formulas for All 3D Shapes
TL;DR
Volume measures the space a 3D shape occupies, expressed in cubic units (cm³, m³, in³). Each shape has its own formula - cube uses V = a³, cuboid uses l × b × h, cylinder uses πr²h, sphere uses ⁴⁄₃ πr³, and cones and pyramids hold exactly one-third of their corresponding cylinder or prism.
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Bhanzu Team Last updated on April 25, 202612 min read
A volume formula tells you how much three-dimensional space a shape occupies. Volume is always measured in cubic units - cm³, m³, in³, or ft³ - because every 3D shape has length, breadth, and height. The table below lists the volume formulas for every common 3D shape, followed by individual sections, worked examples, and the most common mistakes students make.
Volume Formulas: Quick Reference Table
The single table below covers every shape in this article. Use it as a lookup; the rest of the page works through each formula in detail.
| Shape | Formula | Variables | Quick Example |
|---|---|---|---|
| Cube | V = a³ | a = side | a = 4 → V = 64 cubic units |
| Cuboid | V = l × b × h | l, b, h = length, breadth, height | 6 × 4 × 3 = 72 |
| Cylinder | V = πr²h | r = radius, h = height | r = 3, h = 8 → 226.08 |
| Hollow Cylinder | V = πh(R² − r²) | R = outer radius, r = inner radius | R = 5, r = 3, h = 10 → 502.4 |
| Cone | V = ⅓ πr²h | r = radius, h = height | r = 4, h = 9 → 150.72 |
| Frustum | V = ⅓ πh(R² + r² + Rr) | R, r = larger, smaller radii | R = 6, r = 3, h = 7 → 461.58 |
| Sphere | V = ⁴⁄₃ πr³ | r = radius | r = 6 → 904.32 |
| Hemisphere | V = ⅔ πr³ | r = radius | r = 6 → 452.16 |
| Prism (general) | V = Base Area × Height | base shape varies | base 12, h = 10 → 120 |
| Pyramid (general) | V = ⅓ × Base Area × Height | base shape varies | base 81, h = 10 → 270 |
| Ellipsoid | V = ⁴⁄₃ πabc | a, b, c = three semi-axes | a = 3, b = 4, c = 5 → 251.2 |
Caption: Master volume formula reference table for all common 3D shapes.
What is Volume?
Volume is the amount of three-dimensional space a solid object occupies. It's measured in cubic units - cm³, m³, in³, ft³ - because volume extends in three perpendicular directions. Capacity is the closely related idea: how much a hollow shape can hold inside it, usually a liquid. For most everyday shapes, capacity equals volume.
The Three Relationships That Unlock Half the Formulas
Several formulas in the table above come from a single source formula. Knowing these three relationships means there are really only three formulas to memorise, not six.
| Derived Formula | Source Formula | Relationship |
|---|---|---|
| Cone | Cylinder | Cone = ⅓ × Cylinder (same base, same height) |
| Pyramid | Prism | Pyramid = ⅓ × Prism (same base, same height) |
| Hemisphere | Sphere | Hemisphere = ½ × Sphere |
If you know the volume of a cylinder, the cone with the same base and height is one-third of it. The same logic links pyramids to prisms. And a hemisphere is exactly half of the sphere it came from. (This isn't an approximation - it's geometric fact.)
Volume Formulas for Each 3D Shape
Volume of a Cube
A cube is a 3D shape with six identical square faces. All sides are equal, so the volume formula uses just one variable.
Volume of a Cube:
V = a³
| Variable | Meaning |
|---|---|
| a | Length of one side |
Use this formula when the side length of the cube is known.
Worked Example
Find the volume of a cube with side 4 cm.
V = 4³ = 4 × 4 × 4 = 64 cm³
If only the diagonal is given, use the alternative form:
V = (√3 × d³) / 9, where d is the length of the cube's space diagonal.
Volume of a Cuboid (Rectangular Prism)
A cuboid has six rectangular faces. Length, breadth, and height can all be different - a matchbox, a brick, and a room are all cuboids.
Volume of a Cuboid:
V = l × b × h
| Variable | Meaning |
|---|---|
| l | Length |
| b | Breadth (width) |
| h | Height |
Worked Example
A cuboid has length 6 cm, breadth 4 cm, height 3 cm.
V = 6 × 4 × 3 = 72 cm³
The cube is a special case of the cuboid where l = b = h.
Curriculum reference: CCSS 5.MD.C.5b; NCERT Class 8 Chapter 11.
Volume of a Cylinder
A cylinder has two identical circular bases joined by a curved surface. Tin cans, water tanks, and pipes are everyday cylinders.
Volume of a Cylinder:
V = πr²h
| Variable | Meaning |
|---|---|
| r | Radius of the base |
| h | Height (perpendicular distance between the two circular faces) |
| π | ≈ 3.14 |
Worked Example
A cylinder has radius 3 units and height 8 units.
V = π × 3² × 8 = π × 9 × 8 = 72π ≈ 226.08 cubic units
Volume of a Hollow Cylinder
A hollow cylinder is a cylinder with a smaller cylinder removed from its centre. Pipes and tubes are hollow cylinders.
Volume of a Hollow Cylinder:
V = πh(R² − r²)
| Variable | Meaning |
|---|---|
| R | Outer radius |
| r | Inner radius |
| h | Height (length of the pipe) |
Worked Example
A pipe has outer radius 5 cm, inner radius 3 cm, and length 10 cm.
V = π × 10 × (5² − 3²) = π × 10 × (25 − 9) = π × 10 × 16 = 160π ≈ 502.4 cm³
The formula is just the volume of the outer cylinder minus the volume of the empty cylinder inside it.
Volume of a Cone
A cone has a circular base that tapers to a single point (the apex). Ice-cream cones and party hats are familiar cones.
Volume of a Cone:
V = ⅓ πr²h
| Variable | Meaning |
|---|---|
| r | Radius of the circular base |
| h | Perpendicular height — from base to apex (not the slant) |
A cone holds exactly one-third the volume of a cylinder with the same base and the same height. That's where the ⅓ comes from.
Worked Example
A cone has radius 4 units and height 9 units.
V = ⅓ × π × 4² × 9 = ⅓ × π × 16 × 9 = 48π ≈ 150.72 cubic units
The slant height (the diagonal distance from apex to the edge of the base) is a different measurement - used in surface area, not volume. Always use perpendicular height in the volume formula.
Volume of a Frustum (Cone with Top Sliced Off)
A frustum is what's left when the top of a cone is cut off parallel to the base — the shape of a bucket or a lampshade.
Volume of a Frustum:
V = ⅓ πh(R² + r² + Rr)
| Variable | Meaning |
|---|---|
| R | Larger radius (bottom circle) |
| r | Smaller radius (top circle) |
| h | Perpendicular height between the two circles |
Worked Example
A frustum has bottom radius 6 cm, top radius 3 cm, and height 7 cm.
V = ⅓ × π × 7 × (6² + 3² + 6 × 3) V = ⅓ × π × 7 × (36 + 9 + 18) V = ⅓ × π × 7 × 63 = 147π ≈ 461.58 cm³
Volume of a Sphere
A sphere is a perfectly round 3D shape — every point on the surface is the same distance from the centre. Footballs and planets are spheres.
Volume of a Sphere:
V = ⁴⁄₃ πr³
| Variable | Meaning |
|---|---|
| r | Radius (centre to surface) |
Worked Example
Find the volume of a sphere with radius 6 cm.
V = ⁴⁄₃ × π × 6³ = ⁴⁄₃ × π × 216 = 288π ≈ 904.32 cm³
If the diameter is given instead, halve it to get the radius before substituting.
Curriculum reference: CCSS 8.G.C.9; NCERT Class 9 Chapter 13.
Volume of a Hemisphere
A hemisphere is exactly half of a sphere — a bowl or a dome.
Volume of a Hemisphere:
V = ⅔ πr³
| Variable | Meaning |
|---|---|
| r | Radius |
The formula is half the volume of the sphere with the same radius. ½ × ⁴⁄₃ πr³ = ⅔ πr³.
Worked Example
A hemisphere has radius 6 cm.
V = ⅔ × π × 6³ = ⅔ × π × 216 = 144π ≈ 452.16 cm³
Volume of a Prism
A prism has two identical, parallel bases connected by rectangular faces. The base can be any polygon - a triangle, square, pentagon, hexagon. The cube and cuboid are both prisms with rectangular bases.
Volume of a Prism:
V = Base Area × Height
| Variable | Meaning |
|---|---|
| Base Area | Area of the polygon base |
| Height | Perpendicular distance between the two bases |
Worked Example
A triangular prism has a right-triangle base with legs 4 cm and 6 cm. The prism is 10 cm long.
Base Area = ½ × 4 × 6 = 12 cm²
V = 12 × 10 = 120 cm³
Volume of a Pyramid
A pyramid has a polygon base and triangular faces meeting at a single apex. The base can be a triangle, square, rectangle, or any polygon.
Volume of a Pyramid:
V = ⅓ × Base Area × Height
| Variable | Meaning |
|---|---|
| Base Area | Area of the polygon base |
| Height | Perpendicular distance from base to apex |
A pyramid holds exactly one-third of a prism with the same base and the same height. That's where the ⅓ comes from - the same geometric reason a cone holds one-third of its cylinder.
Worked Example
A square pyramid has base side 9 cm and height 10 cm.
Base Area = 9 × 9 = 81 cm²
V = ⅓ × 81 × 10 = 270 cm³
Volume of an Ellipsoid
An ellipsoid is a 3D oval - a stretched sphere. A rugby ball and a chicken egg are close to ellipsoids.
Volume of an Ellipsoid:
V = ⁴⁄₃ πabc
| Variable | Meaning |
|---|---|
| a | Semi-axis along the x-direction |
| b | Semi-axis along the y-direction |
| c | Semi-axis along the z-direction |
A semi-axis is half the length of the ellipsoid measured along that direction (similar to how the radius is half the diameter of a sphere).
Worked Example
An ellipsoid has semi-axes a = 3, b = 4, c = 5.
V = ⁴⁄₃ × π × 3 × 4 × 5 = ⁴⁄₃ × π × 60 = 80π ≈ 251.2 cubic units
When all three semi-axes are equal (a = b = c = r), the formula collapses to ⁴⁄₃ πr³ - the volume of a sphere. The sphere is just an ellipsoid with three equal radii.
Which Volume Formula Do You Need? A Quick Decision Guide
If you know what measurements you have but aren't sure which formula applies, this table maps inputs to formulas.
| If you know... | The shape is likely... | Use this formula |
|---|---|---|
| A single side length | Cube | V = a³ |
| Length, breadth, height | Cuboid | V = l × b × h |
| Radius and height; flat top and bottom | Cylinder | V = πr²h |
| Outer radius, inner radius, length | Hollow cylinder | V = πh(R² − r²) |
| Radius and height; tapers to a point | Cone | V = ⅓ πr²h |
| Two radii and a height; a sliced-off cone | Frustum | V = ⅓ πh(R² + r² + Rr) |
| Radius only; round all over | Sphere | V = ⁴⁄₃ πr³ |
| Radius only; one flat circular face | Hemisphere | V = ⅔ πr³ |
| Base area and height; uniform cross-section | Prism | V = Base Area × Height |
| Base area and height; meets at a point | Pyramid | V = ⅓ × Base Area × Height |
| Three semi-axes (a, b, c) | Ellipsoid | V = ⁴⁄₃ πabc |
Common Mistakes When Using Volume Formulas
Using diameter instead of radius. Every formula above that contains an r uses radius. If a problem gives the diameter, divide by 2 first.
Forgetting to cube the units. Volume is always in cubic units. An answer written as "64 cm" instead of "64 cm³" is incomplete.
Mixing units in the same formula. If one dimension is in metres and another in centimetres, convert everything to the same unit before substituting.
Confusing slant height with perpendicular height. Cones and pyramids both have a slant edge and a perpendicular (vertical) height. The volume formula uses the perpendicular height. The slant is for surface area.
Forgetting the ⅓ on cones and pyramids. Cones and pyramids have a ⅓ in front. Cylinders and prisms don't. Mixing up which formula has the ⅓ is one of the most common slips in mensuration.
Volume Formulas in the Curriculum
Volume formulas appear from Grade 5 onwards in most curricula and become a standalone chapter by Grade 8 or 9.
| Curriculum | Where Volume Appears |
|---|---|
| CCSS Grade 5 | 5.MD.C.5b - rectangular prisms with whole-number edges |
| CCSS Grade 6 | 6.G.A.2 - fractional edge lengths |
| CCSS Grade 7 | 7.G.B.6 - composite shapes |
| CCSS Grade 8 | 8.G.C.9 - cones, cylinders, spheres |
| NCERT Class 8 | Chapter 11 - Mensuration (cube, cuboid, cylinder) |
| NCERT Class 9 | Chapter 13 - Surface Areas and Volumes (cone, sphere, hemisphere) |
| NCERT Class 10 | Chapter 13 - combinations of solids and frustum |
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Frequently Asked Questions
What is the volume formula in math?
A volume formula tells you how much 3D space a shape occupies. The formula depends on the shape - V = a³ for a cube, V = πr²h for a cylinder, V = ⁴⁄₃ πr³ for a sphere, and so on
What is the unit of volume?
Volume is measured in cubic units. Common ones are cm³, m³, in³, and ft³. The unit is always cubed because volume extends in three directions.
What is the difference between volume and capacity?
Volume is the space a 3D shape occupies. Capacity is how much that shape can hold - usually a liquid. For most everyday shapes the two values are equal, though capacity is more often expressed in litres or millilitres.
Which shapes have a ⅓ in their volume formula, and why?
Cones and pyramids both have a ⅓ in their volume formula. A cone holds exactly one-third of the cylinder with the same base and the same height. A pyramid holds exactly one-third of the prism with the same base and the same height. The ⅓ is geometric fact, not a coincidence.
Is the formula for volume the same as the formula for area?
No. Area measures 2D space and is given in square units (cm², m²). Volume measures 3D space and is given in cubic units (cm³, m³). Area applies to flat shapes like circles and rectangles; volume applies to solid shapes like cylinders and cuboids.
✍️ Written By
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