Volume Ratio of Solids with Similar Shape Examples
Volume Ratio of Solids compares the space occupied by two or more three-dimensional figures. The ratio is obtained by dividing their volumes and simplifying the result. For similar solids, the volume ratio equals the cube of the ratio of corresponding lengths. This page covers formulas and examples for cubes, cuboids, cylinders, cones and spheres.
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What Is the Volume Ratio of Solids?
For solids with volumes V₁ and V₂, volume ratio = V₁/V₂ or V₁:V₂. If two solids are similar and their corresponding linear dimensions are in the ratio a:b, their volumes are in the ratio a³:b³. For example, cubes with edges 2 cm and 4 cm have volume ratio 2³:4³ = 8:64 = 1:8.
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Inscribed Shapes and Volume Ratios
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Volume Ratio of Solids Formula & Tricks
Important Formulas
Calculate both volumes in the same unit, form the ratio and divide by their common factor.
If corresponding lengths of two similar solids are in the ratio a:b, their volumes are in the cube ratio.
The volume of a cuboid is length × breadth × height.
The volume of a cylinder depends on the square of its radius and its height.
The volume of a cone is one-third of the volume of a cylinder with the same radius and height.
The volume of a sphere depends on the cube of its radius.
Quick Tricks
When all corresponding dimensions have the same scale factor, cube that factor instead of calculating both complete volumes.
For cylinders or spheres, common factors such as π and the same numerical constants cancel directly in the ratio.
Convert all dimensions to one unit first. Since volume has cubic units, a length conversion factor must be cubed.
Volume Ratio of Solids Concepts
Volume Ratio of Similar Solids
If the corresponding edges, radii or heights are in the ratio a:b, then V₁:V₂ = a³:b³. This applies to cubes, spheres and other solids whose corresponding dimensions are scaled by the same factor.
Ratios of Cubes and Cuboids
For cubes with edges a and b, V₁:V₂ = a³:b³. For cuboids, V₁:V₂ = l₁b₁h₁:l₂b₂h₂. If two cuboids have dimensions 2 × 3 × 4 cm and 3 × 4 × 5 cm, their volume ratio is 24:60 = 2:5.
Ratios of Cylinders and Cones
For cylinders, V₁:V₂ = r₁²h₁:r₂²h₂. For cones, V₁:V₂ = r₁²h₁:r₂²h₂. A cylinder and a cone cannot be compared using this cancellation unless their different constants are included.
Ratios of Spheres
For spheres, V₁:V₂ = r₁³:r₂³. Since radius is half the diameter for both spheres, the same rule gives V₁:V₂ = d₁³:d₂³. For diameters 4 cm and 10 cm, the ratio is 4³:10³ = 64:1000 = 8:125.
Comparing Different Types of Solids
A sphere of radius 3 cm has volume (4/3)π(3³) = 36π cm³. A cylinder with radius 3 cm and height 4 cm has volume π(3²)(4) = 36π cm³. Therefore, their volume ratio is 1:1.
Volume Ratio of Solids Visual Guide & Data
Use the table and chart below to understand the data better.
Volume formulas used in solid ratios
Common solids and the expressions used to calculate their volumes.
| Solid | Volume formula | Ratio expression |
|---|---|---|
| Cube | a³ | a₁³:a₂³ |
| Cuboid | lbh | l₁b₁h₁:l₂b₂h₂ |
| Cylinder | πr²h | r₁²h₁:r₂²h₂ |
| Cone | (1/3)πr²h | r₁²h₁:r₂²h₂ |
| Sphere | (4/3)πr³ | r₁³:r₂³ |
Note: The simplified ratio expressions for cylinders and cones assume both solids are of the same type, so their common constants cancel.
Dimensions used in a cylinder volume ratio
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Volume Ratio of Solids: Revision Points
Use the solid-specific volume formula, keep units consistent and simplify the resulting ratio.
- Volume ratio = V₁:V₂.
- For similar solids, volume ratio = cube of the corresponding length ratio.
- Cube ratio for edges a and b is a³:b³.
- Cylinder ratio for dimensions r₁, h₁ and r₂, h₂ is r₁²h₁:r₂²h₂.
- Cone ratio has the same expression when both figures are cones.
- Sphere ratio is r₁³:r₂³ or d₁³:d₂³.
- Convert dimensions to the same unit before calculating volume.
- A conversion factor for length becomes its cube when applied to volume.
Volume Ratio of Solids FAQs
What is the volume ratio formula for two solids?
The formula is V₁:V₂ = volume of solid 1 : volume of solid 2. Find both volumes in the same unit and simplify the ratio.
What is the volume ratio of cubes with edges 2 cm and 5 cm?
The ratio is 2³:5³ = 8:125.
What is the volume ratio of two cylinders with equal heights and radii 4 cm and 6 cm?
Since the heights and π are common, the ratio is 4²:6² = 16:36 = 4:9.
What is the volume ratio of spheres with diameters 6 cm and 9 cm?
The ratio is 6³:9³ = 216:729 = 8:27.
Can the cube rule be used for any two solids?
No. It applies directly to similar solids with corresponding dimensions in one common scale ratio. For different shapes, calculate each volume using its own formula.
What is the volume ratio of a cone and a cylinder with the same radius and height?
The cone-to-cylinder volume ratio is (1/3)πr²h : πr²h = 1:3.