Volume of a Combination of Solids
Volume is a physical quantity that measures the space occupied by a solid. It is measured in cubic units. The volume of a combination of solids given by adding up the volumes of each individual solid in the combination of solids. For example, in a figure made up of a hemisphere and a cone (like a top), the sum of the volumes of each solid gives us the volume of the combination of solids.
Formulas for Volume of Important Solids
Since we can arrive at the volume of a solid only after summing up the volumes of individual solids, it is good to know the formulas of certain important solids before we start solving problems.
Sometimes the surface area is also used to solve certain trick questions, so it is best to know them also. The surface area is basically the amount of area that is exposed by the solid to the outer world. Think of it as the area of the net of the solid.
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In the formulas given in the above image, the quantities defined by the letters are stated below:
v stands for volume
SA, here, stands for the total surface area.
l stands for length
b and w stand for breadth and width respectively (they are the same)
h stands for height
s stands for slant height
r stands for radius
Solved Examples
Example 1: A cuboidal room is of dimensions 10 m x 10 m x 10 m. Find the length of the longest pole that can be placed in this room.
Solution:
The longest pole that can be placed in the room is basically the longest diagonal of the room. The longest diagonal can be determined using the following formula:
\[\sqrt{(l^{2} + b^{2} + h^{2})}\]
= \[\sqrt{(10^{2} + 10^{2} + 10^{2})}\]
= \[\sqrt{300}\]
= 17.32 m
The length of the longest diagonal of the room is 17.32 m. Hence, the longest pole that can be placed in this room has a length of 17.32 m.
Example 2: A cube is of side 2 cm. Eight of them are joined together to form a bigger cube. Calculate the volume of the bigger cube.
Solution:
The volume of a cube can be determined through the following formula:
s x s x s = 2 x 2 x 2
= 8 cm3
The volume of a combination of solids is the sum of the volumes of each individual solid used in making it. The large cube is actually a combination of 8 smaller cubes. So, the volume of the large cube can be calculated as:
Volume of the small cube x 8
= 8 x 8
= 64 cm3
Therefore, the volume of the large cube is 64 cm3.
Example 3: An ice-cream cone is of radius 7 cm. It has a total length of 21 cm. Find the volume of the ice-cream held in this cone.
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The volume of a hemisphere is half the volume of a sphere i.e. volume of a hemisphere can be calculated using the following formula:
(2/3) π r3
= \[(\frac{2}{3})\] x \[(\frac{22}{7})\] x (7)3
= \[(\frac{2}{3})\] x (22) x (7)2
= \[(\frac{2156}{3})\]
= 718.67 cm3
We know that the total length of the solid is 21 cm. The radius is 7 cm. The radius is the same even if we take it from top to bottom. So, the height of the cone will be calculated as:
Height of Cone = Total Height — Radius of Hemisphere
= 21 — 7
= 14 cm
The height of the cone is 14 cm. Now the volume of the cone can be calculated using the following formula:
\[(\frac{1}{3})\] π r2 h
= \[(\frac{1}{3})\] \[(\frac{22}{7})\] x (7)2 x 14
= \[(\frac{1}{3})\] x (22) x 7 x 14
= \[(\frac{2156}{3})\]
= 718.67 cm3
The volume of a combination of solids is the sum of the volumes of each individual solid used in making it.
Volume of Ice-Cream Cone = Volume of Hemisphere + Volume of Cone
= 718.67 cm3 + 718.67 cm3
= 1437.34 cm3
Therefore, the volume of the ice-cream cone is 1437.34 cm3. This is the maximum amount of ice-cream it can hold at any point of time.
FAQs on Volume of Combination of Solids
1. What is the volume of a combination of solids?
The volume of a combination of solids refers to the total three-dimensional space occupied by a composite object. It is calculated by identifying the basic geometric shapes that make up the object (like cylinders, cones, or hemispheres) and then summing up their individual volumes. This principle assumes that the solids are joined together without any overlap.
2. How do you calculate the volume of a combined solid?
To calculate the volume of a combined solid, follow these simple steps:
- Step 1: Deconstruct the composite shape into its individual, basic solid components (e.g., a tent might be a cylinder and a cone).
- Step 2: Calculate the volume of each individual solid using its standard formula (e.g., Volume of cone = ⅓πr²h).
- Step 3: Add the volumes of all the individual solids together to get the total volume of the combined object.
3. What are some real-world examples of combinations of solids?
Combinations of solids are very common in everyday objects. Some typical examples include:
- An ice cream cone, which is a combination of a cone and a hemisphere.
- A pharmaceutical capsule, formed by a cylinder with two hemispheres at its ends.
- A circus tent or a silo, which is often a combination of a cylinder and a cone.
- A spinning top (lattu), which combines a cone and a hemisphere.
4. What are the key solid shapes I need to know for calculating the volume of combined solids as per the CBSE Class 10 syllabus?
As per the CBSE Class 10 syllabus for the 2025-26 session, you must be proficient with the volume formulas for the following basic solids to solve problems on combined shapes:
- Cube
- Cuboid
- Sphere
- Hemisphere
- Right Circular Cylinder
- Right Circular Cone
5. Is there a single formula for the volume of any combined solid?
No, there is no single, universal formula for the volume of a combined solid. The 'formula' is actually a method or principle: you find the total volume by adding the volumes of the individual standard solids that form the composite shape. The key is to correctly identify the constituent shapes and apply their respective, well-known volume formulas.
6. How is calculating the volume of a combined solid different from calculating its surface area?
This is a crucial distinction. For volume, you simply add the volumes of the individual solids. However, for surface area, you must only add the areas of the exposed surfaces. The faces where the solids are joined together are not part of the total surface area and must be excluded from the calculation. This is the most common conceptual trap students face.
7. What if one solid is carved out from another? Do we still add the volumes?
No. In cases where a shape is hollowed out or carved from another (for example, a conical cavity scooped out of a solid cylinder), you must subtract the volumes. The final volume would be the volume of the original solid minus the volume of the part that was removed. This is a common type of problem in exams that tests your understanding of volume as the space an object occupies.
