Volume of Combined Solids with Composite Shape Examples
Volume of Combined Solids is found by dividing a composite 3D figure into familiar solids such as cuboids, cylinders, cones and spheres. Add the volumes of joined parts and subtract the volume of any hollow or removed part. This page covers formulas, methods, examples and common aptitude problems.
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What Is the Volume of Combined Solids?
First identify the component solids and their dimensions. Then use the appropriate volume formula for each part, keeping all measurements in the same unit. For joined parts, Volume = V₁ + V₂ + V₃ + ...; for a hollow or cut-out part, Volume = outer volume − removed volume. The final unit is cubic, such as cm³ or m³.
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Combined Solids: Volume and Surface Area
Learn how to calculate the volume and surface area of combined three-dimensional solids by identifying component shapes and applying the appropriate mensuration formulas.
Volume of Combined Solids Formula & Tricks
Important Formulas
Add the volumes of all non-overlapping solids forming the composite figure.
Subtract the volume of the cavity, hole or cut-out section from the outer solid.
Here l, b and h are the length, breadth and height.
Here r is the radius of the circular base and h is the height.
The radius is r and the perpendicular height is h.
A hemisphere has half the volume of a sphere with the same radius.
Quick Tricks
Draw an imaginary boundary wherever the shape changes. Calculate each simple solid separately instead of trying to use one formula for the entire figure.
When component solids have the same radius, take common factors such as πr² outside the expression. This reduces calculation time and errors.
A hole or hollow section occupies space that is absent from the object, so its volume must be subtracted from the outer volume.
All lengths must use the same unit. If dimensions are in centimetres, the answer is in cm³; if dimensions are in metres, the answer is in m³.
Volume of Combined Solids Concepts
Adding the Volumes of Joined Solids
Use V_total = V₁ + V₂ + ... . Identify each solid, write its dimensions, apply its volume formula and add the results. For a cylinder of radius 3 cm and height 10 cm joined to a hemisphere of radius 3 cm, V = π(3²)(10) + ⅔π(3³) = 90π + 18π = 108π cm³.
Subtracting a Hollow or Cut-Out Section
Use V_remaining = V_outer − V_removed. The outer and removed volumes must be expressed in the same cubic unit. If a cylindrical hole of radius 2 cm and height 10 cm is drilled through a cuboid measuring 10 cm × 8 cm × 10 cm, then V_remaining = 800 − 40π cm³.
Composite Solids with a Common Base
For a cylinder and cone with the same radius r, heights h₁ and h₂, V = πr²h₁ + ⅓πr²h₂ = πr²(h₁ + h₂/3). This form is useful when the cone and cylinder are placed one above the other.
Unit Conversion in Volume Problems
Since 1 m = 100 cm, 1 m³ = 100³ cm³ = 1,000,000 cm³. Similarly, 1 cm³ = 0.000001 m³. Convert every dimension to one unit before applying a volume formula.
Finding an Unknown Dimension from Combined Volume
Write the component-volume sum or difference, substitute the known values and solve for the unknown. For a cylinder plus hemisphere with common radius r, V = πr²h + ⅔πr³; therefore h = (V − ⅔πr³)/(πr²).
Volume of Combined Solids Visual Guide & Data
Use the table and chart below to understand the data better.
Volume Formulas for Common Component Solids
These formulas are used to calculate the parts of a composite solid.
| Solid | Volume formula | Required dimensions |
|---|---|---|
| Cuboid | l × b × h | Length, breadth and height |
| Cube | a³ | Side a |
| Cylinder | πr²h | Radius r and height h |
| Cone | ⅓πr²h | Radius r and perpendicular height h |
| Sphere | ⁴⁄₃πr³ | Radius r |
| Hemisphere | ⅔πr³ | Radius r |
Note: Use the same value of π throughout a calculation, usually 22/7 or 3.14 when the question permits.
Cylinder Joined to a Hemisphere
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Volume of Combined Solids: Revision Points
Remember the component formulas and the addition or subtraction rule for composite figures.
- Split the composite figure into familiar, non-overlapping solids.
- Add component volumes for joined parts: V_total = V₁ + V₂ + ... .
- Subtract the volume of a cavity or removed section.
- Use perpendicular height for cylinders and cones.
- Convert all dimensions to the same unit before calculating.
- Volume is expressed in cubic units.
- For a common radius, factor out πr² where possible.
- A hemisphere has volume ⅔πr³, which is half the volume of a sphere.
Volume of Combined Solids FAQs
What is the general volume of combined solids formula?
For joined solids, V_total = V₁ + V₂ + V₃ + ... . If a section is removed, use V_remaining = V_outer − V_removed.
How do you find the volume of a cylinder topped by a hemisphere?
Add their volumes: V = πr²h + ⅔πr³, where r is the common radius and h is the cylinder height.
What is the volume of a cylinder of radius 3 cm and height 10 cm joined to a hemisphere of radius 3 cm?
V = 90π + 18π = 108π cm³, approximately 339.43 cm³.
Should the volume of a hole be added or subtracted?
It should be subtracted because the hole represents the part removed from the outer solid. Thus, remaining volume = outer volume − hole volume.
What is the difference between the volume of a sphere and a hemisphere?
A sphere has volume ⁴⁄₃πr³, while a hemisphere has volume ⅔πr³. The hemisphere has exactly half the volume of the sphere with the same radius.
How many cubic centimetres are in 1 cubic metre?
1 m³ = 100³ cm³ = 1,000,000 cm³. The linear conversion factor is cubed for volume.