The volume of a solid is the measure of how many regions of space is occupied by an object. It is measured by the number of unit cubes it takes to fill up the solid. It is decided by counting the unit cubes in the solid.

For example, fill a jug with water up to its brim and keep it inside a bucket. Now slowly drop a cricket ball into that. Some quantity of water will be overflowed from the jug to the bucket. This quantity of water indicates the volume of the cricket ball dropped. In this article, we will provide you with detailed information about the volume of solids. Scroll down to read more!

Volume of Solids: Definition

Volume is the region occupied by the object/matter (solid, liquid, gas) inside a three-dimensional object. Alternatively, the volume of a three-dimensional object is generally defined as the capacity of the object, which can hold the matter.

For example, the volume of a cubical box indicates the amount of water or any substance that can be contained in it. If we consider the case of cup ice cream, the amount of ice cream that can be filled inside the cup without any gap is the volume of the ice cream cup.

Learn About Volume of Cube Here

Volume of Three-dimensional Shape

Generally, we know the volume of three-dimensional shape is a product of its area of base and height.
The formula can be written as:
\({\rm{Volume}} = {\rm{Area}}\,{\rm{of}}\,{\rm{base}} \times {\rm{Height}}\)

Units of Volume

The units of volume are cubic millietre \(\left( {{\rm{m}}{{\rm{m}}^{\rm{3}}}} \right){\rm{,}}\) cubic centimeter \(\left( {{\rm{c}}{{\rm{m}}^{\rm{3}}}} \right){\rm{,}}\) cubic meter \(\left( {{{\rm{m}}^{\rm{3}}}} \right)\) etc.

Volume of a Cube

A cube is a three-dimensional box-like figure represented in the three-dimensional plane. A cube has \(6\) square-shaped equal faces. Each face meets another face at \(90^\circ \) each.\(3\) sides of the cube intersect at the same vertex.
The volume of a cube is equal to the product of the edge length three times.

If each edge length is “\(a\)” then the,
The volume of a cube is \({a^3}.\)
\(V = {a^3}\,{\rm{cubic}}\,{\rm{units}}\)

Volume of a Cuboid

A cuboid is a three-dimensional box-like figure represented in the three-dimensional plane.
The volume of a cuboid is obtained by multiplying the length, breadth, and height.

The volume of a cuboid is \(V = {\rm{length}} \times {\rm{breadth}} \times {\rm{height}}\)
\(V = \left( {l \times b \times h} \right)\,{\rm{Cubic}}\,{\rm{units}}\)

Value of pi

A constant term \(\pi \) is used in the formula of the area and volume of many solid shapes. \(\pi \) is a constant term and is called Archimedes’ constant. Generally, we can define \(\pi \) is that it is the ratio of the circumference of a circle to its diameter. The value of it is \(3.14159.\) For the common use in practice, the value of \(\pi \) is approximately taken as \(3.14\) when used as a decimal number and is taken as \(\frac{{22}}{7}\) when used as a fraction to ease the calculation.

Volume of a Cylinder

The volume of a cylinder is the amount of space occupied by matter inside a cylinder or the measure of the capacity of a cylinder.

The volume of a cylinder is the same as the product of the area of the circular base and the height of the cylinder.

\({\rm{Volume}}\,{\rm{of}}\,{\rm{a}}\,{\rm{cylinder}} = {\rm{Area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{circle}} \times {\rm{Height}}\,{\rm{of}}\,{\rm{a}}\,{\rm{cylinder}}\)
Area of circle, \(A = \pi {r^2}\)
Height of the right circular cylinder is \(h\)
Volume of a cylinder \( = \pi {r^2} \times h = \pi {r^2}h\)
\(V = \pi {r^2}h\,{\rm{cubic}}\,{\rm{units}}\)

Volume of a Cylinder with Diameter

The volume of a cylinder with diameter \( = \pi {\left( {\frac{d}{2}} \right)^2}h\)

Volume of a Sphere Formula

The volume of a sphere is calculated by using one method by assuming the volume of the sphere is made up of numerous thin circular disks which are arranged one over the other. These circular disks will have continuously varying radius, but the centres of them will be collinear.

The volume of a sphere can be written as the product of the area of the circle and its thickness. Integrating the product of radius and thickness, we will get the volume of the sphere.
1. The volume of a Sphere Formula When the Radius is Given:
The formula to find the volume of a sphere is,
The volume of a sphere is \(V = \frac{4}{3}\pi {r^3}\,{\rm{cubic}}\,{\rm{units,}}\) where \(r\) is the radius of the sphere.
2. The Volume of a Sphere Formula When the Diameter is Given:
The volume of a sphere is \(V = \frac{4}{3}\pi {\left( {\frac{d}{2}} \right)^3}\,{\rm{cubic}}\,{\rm{units,}}\) where \(d\) is the diameter of the sphere.

Volume of a Hemisphere

The hemisphere is a three-dimensional shape/object and exactly the half of a sphere; the Volume of the hemisphere will also be exactly half of the Volume of a sphere.

The hemisphere is a three-dimensional shape/object and exactly the half of a sphere; the Volume of the hemisphere will also be exactly half of the Volume of a sphere.
The volume of a hemisphere is \(V = \frac{2}{3}\pi {r^3}\,{\rm{cubic}}\,{\rm{units,}}\) where \(r\) is the radius of the hemisphere.

Volume of a Cone

Commonly, we know a cone is like a pyramid with a circular base. We can find the volume of a cone if its height and radius are given.
Where, \(r \to \) radius of the circular base, \(h \to \) height, and \(l \to \) slant height of the cone.
Hence, the volume of a cone
\(V = \frac{1}{3}\pi {r^2}h\,{\rm{cubic}}\,{\rm{units}},\) where, \(r \to \) radius of the circular base, \(h \to \) height.

Solved Examples on Volume of Solids

Q.1. Find the volume of a cuboid of length \(20\,{\rm{cm,}}\) breadth \(12\,{\rm{cm,}}\) height \(10\,{\rm{cm}}{\rm{.}}\)
Ans:
Given: length \( = 20\,{\rm{cm,}}\) breadth \( = 12\,{\rm{cm,}}\) height \( = 10\,{\rm{cm}}{\rm{.}}\)
\({\rm{Volume}}\,{\rm{of}}\,{\rm{a}}\,{\rm{Cuboid}} = {\rm{length}} \times {\rm{breadth}} \times {\rm{height}}\)
\( \Rightarrow V = 20 \times 12 \times 10\)
\( \Rightarrow V = 2400\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\)
Hence, the obtained Volume of a Cuboid is \(2400\,{\rm{c}}{{\rm{m}}^{\rm{3}}}.\)

Q.2. Find the volume of the cube whose each edge is \(5\,{\rm{cm}}.\)
Ans: From the given edge \(a = 5\,{\rm{cm}}\)
The volume of a cube is \({a^3}.\)
\(V = {a^3}\,{\rm{cubic}}\,{\rm{units}}\)
\( \Rightarrow V = {5^3}\)
\( \Rightarrow V = 125\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\)
Hence, the Volume of a Cube is \(125\,{\rm{c}}{{\rm{m}}^{\rm{3}}}.\)

Q.3. Find the volume of the largest cone that can be carved out of a cube of side \(16.8\,{\rm{cm}}.\)
Ans: From the given,
Side of a cube \( = 16.8\,{\rm{cm}}\)
Height of the cone Side of a cube \( = 16.8\,{\rm{cm}}\)
Diameter of the cone \( = 16.8\,{\rm{cm}}\)
Then, the radius of the cone \(\left( r \right) = \frac{{16.8}}{2} = 8.4\,{\rm{cm}}\)

We know that,
The volume of a cone \(V = \frac{1}{3} \times \pi \times {r^2}h\)
\(V = \frac{1}{3} \times \frac{{22}}{7} \times {\left( {8.4} \right)^2} \times 16.8\)
\(V = 1241.86\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\)
Hence, \(1241.86\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\) is the volume of the largest cone that can be carved out of a cube of side.

Q.4. The volume of a cylindrical water tank is \(1000\,{{\rm{m}}^{\rm{3}}},\) and the radius of the base of the cylindrical tank is \(5\,{\rm{m}}.\) Calculate the height of the tank.
Ans:
Given: Radius \(\left( r \right)\) of base \( = 5\,{\rm{m}}\) and the volume of tank \( = 1000\,{{\rm{m}}^{\rm{3}}}\)
We know that the volume of a cylinder \( = \pi {r^2}h\)
\( \Rightarrow 1000 = \frac{{22}}{7} \times 5 \times 5 \times h\)
\( \Rightarrow h = 12.7\,{\rm{m}}\)
Hence, the height of the tank is \(12.7\,{\rm{m}}{\rm{.}}\)

Q.5. Find the volume of the wood used in pencil; if its outer radius is \({\rm{4}}\,{\rm{mm,}}\) its inner radius is \({\rm{3}}\,{\rm{mm,}}\) and its height is \(50\,{\rm{mm}}{\rm{.}}\)
Ans: From the given,
The outer radius of a pencil \( = 4\,{\rm{mm}}\)
The inner radius of a pencil \( = 3\,{\rm{mm}}\)
Height of a pencil \( = 50\,{\rm{mm}}\)

We know that the volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Now, the volume of the wood used in pencil \( = \frac{{22}}{7} \times \left( {{4^2} – {3^2}} \right) \times 50\,{\rm{m}}{{\rm{m}}^3}\)
\( = \frac{{22}}{7} \times 7 \times 50\,{\rm{m}}{{\rm{m}}^3} = 1100\,{\rm{m}}{{\rm{m}}^{\rm{3}}}\)
Hence, the volume of the wood used in pencil is \({\rm{1100}}\,{\rm{m}}{{\rm{m}}^{\rm{3}}}{\rm{.}}\)

Q.6. The Volume of a container is \(1440\,{m^3}.\) The length and breadth of the container are \({\rm{12}}\,{\rm{m}}\) and \({\rm{10}}\,{\rm{m,}}\) respectively. Find its height.
Ans: From the given, volume of the container \( = 1440\,{{\rm{m}}^{\rm{3}}}\)
Length \( = 12\,{\rm{m,}}\) Breadth \( = 10\,{\rm{m}}\)
The volume of the container, \(V = {\rm{length}} \times {\rm{breadth}} \times {\rm{height}}\)
\( \Rightarrow 1440 = 12 \times 10 \times {\rm{height}}\)
\( \Rightarrow {\rm{height = }}\frac{{1440}}{{120}} = 12\)
\( \Rightarrow {\rm{height}} = 12\,{\rm{m}}\)
Hence, the height of the container is \(12\,{\rm{m}}.\)

Summary

The volume of a solid is defined as the measure of how many regions of space is occupied by an object. It is calculated by the number of unit cubes it takes to cover the solid. In this article, we learned about the volume of solids like the cube, cuboid, cylinder, sphere, etc. and formulas to find their respective volumes. The volume of solids helps in solving different problems related to cubes, cuboids, spheres, etc. The volume of the three-dimensional shape is measured by calculating the product of its area of base and height.

FAQs on Volume of Solids

Q.1. Define volume.
Ans: Volume is the space occupied by the matter (solid, liquid, gas) inside the three-dimensional object, or the volume of a three-dimensional object is generally defined as the capacity of the object, which can hold the matter.

Q.2. How do we find the volume?
Ans: In general, the volume of a three-dimensional shape is a product of its area of base and height.
\({\rm{Volume}} = {\rm{area}}\,{\rm{of}}\,{\rm{base}} \times {\rm{height}}\)

Q.3. What is the volume of a solid cube?
Ans: The formula of the volume of a cube is given by,
\(V = {a^3}\,{\rm{cubic}}\,{\rm{units}}{\rm{.}}\)

Q.4. What are the formulas for volume?
Ans: The formula of volume of the cuboid:
\(V = \left( {l \times b \times h} \right)\,{\rm{cubic}}\,{\rm{units}}\)
The formula of volume of a cube, \(V = {a^3}\,{\rm{cubic}}\,{\rm{units}}\)
The formula of volume of a cylinder, \(V = \pi {r^2}h\,{\rm{cubic}}\,{\rm{units}}\) or
\(V = \pi {\left( {\frac{d}{2}} \right)^2}h\,{\rm{cubic}}\,{\rm{units}}\)
The formula of the volume of a cone, \(V = \frac{1}{3}\pi {r^2}h\,{\rm{cubic}}\,{\rm{units}}\)
The formula of the volume of a sphere, \(V = \frac{4}{3}\pi {r^2}\,{\rm{cubic}}\,{\rm{units}}\)

Q.5. What is the volume of a hemisphere?
Ans: As a hemisphere is a three-dimensional shape and exactly the half of a sphere, the volume of the hemisphere will also be exactly half of the volume of a sphere.
The volume of a hemisphere is \(V = \frac{2}{3}\pi {r^3}\,{\rm{cubic}}\,{\rm{units,}}\) where \(r\) is the radius of the hemisphere.

We hope this detailed article on the volume of solids proves helpful to you. If you have any doubts or queries regarding this topic, feel to ask us in the comment section and we will help you at the earliest.