Volume of Combination of Solids

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.