Volume
In mathematics, ‘Volume’ is a mathematical quantity that shows the amount of three-dimensional space occupied by an object or a closed surface. The unit of volume is in cubic units such as m^{3}, cm^{3}, in^{3} etc.
Volume is also termed as capacity, sometimes. For example, the amount of water a cylindrical jar can occupy is measured by its volume. Check volume of cylinder here.
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Definition
A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder or a sphere.
Different shapes have different volumes. In 3d geometry, we have studied the various shapes and solids such as cube, cuboid, cylinder, cone, etc., that are defined in three dimensions. For all these shapes, we are going to learn to find the volume.
Unit of Volume
Volume of a solid is measured in cubic units. For example, if dimensions are given in meters, then the volume will be in cubic meters. This is the standard unit of volume in the International System of Units (SI). Similarly, other units of volume are cubic centimeters, cubic foot, cubic inches, etc.
Example
If a cuboid has dimensions of 10cm x 3cm x 5cm, then find its volume.
Sol: Volume of cuboid = length x width x height
V = 10cm x 3cm x 5cm
V = 150 cubic centimeters
Hence, we can see here the unit of volume of cuboid is measured in cubic centimeters.
Volume of liquid
Basically, the volume of a liquid is measured in liters, where 1 liter is equal to 1000 cubic centimeters.
1 liter = 1000 cubic centimeters = 0.001 cubic meters
Hence,
1 cubic meters = 1000 liters
Also, to measure the volume of a small amount of liquid, we use milliliters.
1 milliliters = 0.001 liter = 1 cubic centimeters
Volume of liquid is also commonly measured in gallons.
1 liter = 0.264172 US liquid gallon.
Volume Formulas
The formula to calculate the volume of a solid in a three-dimensional space, is to find the product of dimensions. Basically, the volume is equal to the product of area and height of shape.
Volume = Base Area x Height
For shapes having flat surfaces such as cube and cuboid, it is easy to find the volume. But for curved shapes like cone, cylinder and sphere, we have to consider the dimensions of its curved surface as well, such as radius or diameter.
Let us check the formulas for all the shapes.
Volume of Shapes
Name of geometrical shape | Volume formula |
Cube | V = a^{3}, where a is the edge-length of cube |
Cuboid | V = length x width x height |
Cone | V = ⅓ πr²h
Where r is the radius and h is the height of cone |
Cylinder | V = πr²h,
Where r is the radius and h is the height of cylinder |
Sphere | V = 4/3 πr^{3},
Where r is the radius of sphere |
Volume of other shapes
Volume of Frustum | πh/3 (R^{2}+r^{2}+Rr)
Where ‘R’ and ‘r’ are radius of base and top of frustum |
Volume of Prism | Base Area x Height |
Volume of Pyramid | ⅓ (Area of base) (Height) |
Volume of Hemisphere | ⅔ (πr^{3})
Where r is the radius |
Volume Related Facts
- The cube has all its sides equal, therefore, the volume will be equal to the cube of its side length.
- If the radius and height of a cone and a cylinder are the same, then volume of cone is equal to one-third of volume of cylinder.
- The formula of volume of cuboid and rectangular prism is the same.
- The volume of the prism depends on the shape of its base. For example, if the base is square, then the volume will be (side^{2} x height).
Volume Related Articles
Solved Examples on Volume
Q.1: Find the volume of a cube if its side length is equal to 4 cm.
Solution: Given, length of cube = 4cm
As we know,
Volume of cube = Side^{3}
Therefore,
Volume of cube with 4cm length = 4^{3} (cm)^{3}
Volume = 64 cm^{3}
Q.2: What is the volume of the cone if the radius is 2cm and height is 5cm.
Solution: Given, radius of circular base of cone = 2cm
Height of cone = 5cm
As we know,
Volume of cone = ⅓ πr^{2}h
Volume = ⅓ π (2)^{2} (5)
Volume = ⅓ x 22/7 x 4 x 5
Volume = 20.93 cu.cm.
Q.3: The volume of a cube is 512 cm^{3}, Its surface area is?
Solution: Since, a^{3} = 512 = 8 x 8 x 8
⇒ a = 8 cm
∵ Surface area of cube = 6a^{2}
=[6 x (8)2] cm^{2}
=384 cm^{2}
Q.4: A hemisphere has 3 cm radius. Calculate its volume?
Solution: Volume of the hemisphere = (2/3)πr^{3}
= (2/3) x π x 3^{3}
= (2/3) x π x 27
= 18π cm^{3}
Q.5. Calculate the volume of a triangular prism, whose triangular base has height = 6 cm, length of base = 8cm. Height of prism = 10 cm.
Solution: Area of triangular base of prism = ½ x 8 x 6 = 24 cm^{2}
Volume of prism = 24 x 10 = 240 cm^{2}
Practice Questions
- Find the volume of spherical shape with a diameter equal to 12cm.
- A gas cylinder is flattened on top and bottom, to get two parallel circular faces on either side of the curved surface. Find the volume if the radius of circular bases is 10 cm and the height of the cylinder is 50 cm.
Worksheets
1. Calculate the volume of cuboid with following dimensions:
- 2cm x 3cm x 4cm
- 4cm x 18cm x 5cm
- 10cm x 10cm x 10cm
- 6m x 9cm x 11 cm
2. Find the volume of cube having side-length equal to:
- 3cm
- 9 inches
- 7.5 cm
- 4.4 cm
- 0.5 cm
Frequently Asked Questions on volume
What is volume?
Volume is a three-dimensional quantity that is used to measure the capacity of a solid shape. It means, the amount of three-dimensional space that a closed figure can occupy is measured by its volume.
What is the formula for volume?
Unlike area, where two dimensions are multiplied to find the region covered by a shape, volume is measured by multiplying the area of the shape with its third dimension. For example, if the area of the rectangle is length x breadth, then the volume of cuboid formed by elongating the dimensions of the rectangle in space, is equal to length x breadth x height.
What is the unit of volume?
In Maths, volume is measured in cubic units, such as cubic meters, cubic centimeters, cubic millimeters, etc.
The volume of liquid is usually measured in liters and gallons.
Is the volume of cube and cuboid the same?
A cube has all its faces in square shape and a cuboid has rectangular faces. Therefore, the edge lengths of cube are equal to each other whereas for cuboid they are different. Hence, a cube and a cuboid cannot have the same volume.
What is the volume of a pyramid?
The volume of a pyramid is equal to the one-third of the product of area of its base and its height.