Surface Area and Volume of Solids: Definition, Formulas, Examples

Surface Area and Volume of Solids: Surface area is explained as the sum of the areas of all the closed surfaces of a solid. There are three types of surface area: lateral surface area, curved surface area, and total surface area. Volume is a mathematical quantity that defines the capacity of a solid object.

We may see a variety of solid shapes all around us. We may calculate the surface area and volume of various solid shapes using specific formulas. A cube, cuboids, cylinders, cones, and other solid bodies can be used. We will go over how to calculate the surface area and volume for each of these shapes. Continue reading to learn more about Surface Area and Volume formulas, definitions, formulas, examples, etc.

What is Surface Area?

The amount of external space that covers a three-dimensional shape is called it’s surface area. The surface area of a solid shape is categorised into three types, namely lateral surface area, curved surface area, total surface area

Lateral Surface Area: The lateral surface area is known as the area of all the faces except the bottom and top faces or bases.

Curved Surface Area: The curved surface area is known as the area of all the curved regions of the solid.

Total Surface Area: The total surface area is the area of all the faces (including top and bottom faces) of the solid object.

Unit of Surface Area

The surface area of solids is measured in square units. For instance, if the dimensions are given in \(m,\) then the surface area will be in \({{\rm{m}}^2},\) which is the standard unit of surface area in the International System of Units (SI). Similarly, other units of surface area are \({\rm{c}}{{\rm{m}}^2},\,{\rm{m}}{{\rm{m}}^2},\,{{\rm{m}}^2},\) etc.

What is Volume?

The volume of a solid shape is defined as the amount of space it occupies. It is the space enclosed by a boundary or occupied by an object or the capacity to hold something.

Unit of Volume

The volume of solids is calculated using cubic units. For instance, if the dimensions are given in \(m,\) then the volume will be in \({{\rm{m}}^3}.\) Similarly, other units of volume can be \({\rm{c}}{{\rm{m}}^3},\,{\rm{m}}{{\rm{m}}^3},\,{\rm{inc}}{{\rm{h}}^3},\) etc. To measure the volume of liquids, we use litres.
We know,
\(1\) litre \( = 1000\,{\rm{c}}{{\rm{m}}^3}\)
\(1000\,{\rm{c}}{{\rm{m}}^3} = \frac{{1000}}{{100 \times 100 \times 100}}{{\rm{m}}^3} = 0.001\,{{\rm{m}}^3}\)
\( \Rightarrow 0.001\,{{\rm{m}}^3} = 1\) litre
Hence, \(1\,{{\rm{m}}^3} = 1000\) litre

Surface Area and Volume Formulas of Solids

It is easy to find the surface area and volume for cube and cuboid as they have flat surfaces. But for solids that contain curved regions such as a cone, cylinder, and sphere, the radius or the diameter of its curved region/surface plays a significant role in finding their volume and surface area.

Let us talk about the surface area and the volume of a few solid shapes like a
1. Cuboid
2. Cube
3. Cylinder
4. Cone
5. Frustum of the cone
6. Sphere
7. Hemisphere

Surface Area and Volume of a Cuboid

A cuboid is a solid three-dimensional figure that has six rectangular faces with eight vertices and twelve edges.

Lateral Surface Area of Cuboid

The figure clearly shows that it has four lateral flat faces, excluding the top and bottom faces. So, the sum of the faces, excluding the top and bottom faces, is known as the lateral surface area.
The formula of the lateral surface area of a cuboid \( = 2\left( {bh + lh} \right)\)

Total Surface Area of Cuboid

The total surface area is the sum of the lateral surface area and the top and bottom faces. \( = 2\left( {lb + hb + lh} \right)\)

Volume of a Cuboid

We know the volume of any polygonal three \( – \) dimensional figure \( = {\rm{area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{base}} \times {\rm{height}}\)
A cuboid has six rectangular surfaces.
Let us consider it as the base.
Since the area of the base \( = l \times b\)
Hence, the volume of the cuboid \( = l \times b \times h\)

Surface Area and Volume of a Cube

A cube is a solid three-dimensional figure that has six square faces with eight vertices and twelve edges. The concept of lateral surface area, total surface area, and volume of a cube is very similar to a cuboid.

Lateral Surface Area of Cube

Let us assume \(a\) is one of the edges of the cube.
Lateral surface area of the cube \( = 4{\left( a \right)^2}\)

Total Surface Area of Cube

Total Surface Area \( = 6{\left( a \right)^2}\)

Volume of a Cube

If we know the length of the cube’s edge, we can simply find out its volume.
The volume of a cube \( = {\rm{edge}} \times {\rm{edge}} \times {\rm{edge}} = {\left( {{\rm{edge}}} \right)^3} = {\left( a \right)^3}\,\left[ {\,{\rm{length}} = {\rm{breadth}} = {\rm{height}}} \right]\)

Surface Area and Volume of a Cylinder

A cylinder is a basic three-dimensional solid object with one curved surface and two circular surfaces at the ends. 

Curved Surface Area of a Cylinder

A cylinder has two flat surfaces at the bottom and top faces. The curved surface area does not involve two circular faces. If we open a cylinder, we will get one rectangle and two circles having the same radius.
Therefore, the area of the curved surface is \(2\pi r \times h = 2\pi rh\) (as the area of a rectangle is \({\rm{length}} \times {\rm{breadth}}\)).

Total Surface Area of a Cylinder

The total surface area of a cylinder means the sum of curved surface area and the area of two circular bases.
Therefore, the total surface area \( = 2\pi rh + \pi {r^2} + \pi {r^2} = 2\pi r\left( {h + r} \right)\) [As the area of the circle is \(\pi {r^2}\)]

Volume of a Cylinder

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

Volume of a cylinder \( = {\rm{area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{circular}}\,{\rm{base}} \times {\rm{height}}\)
Area of a circle \( = \pi {r^2}\)
The height of the right circular cylinder is \(h.\)
Volume of a cylinder \( = \pi {r^2}h\)

Surface Area and Volume of a Cone

A cone is a three-dimensional solid object with a circular base, one curved face, and a vertex and a circular edge.

In general, a cone is similar to a pyramid with one circular base and one curved surface. We can calculate the surface area and the volume of a cone if its radius and height are known. Here, \(r\) is the radius of the circular base and \(h\) is the height, and \(l\) is the slant height of the cone.
To know about the surface area of a right circular cone, cut the right circular cone and open it as shown as follows.

If we cut a right circular cone, we will get a sector with a radius equals to slant height \(l.\)

Curved Surface Area of a Cone

The curved surface area of the cone equals the area of the sector formed.
Therefore, the curved surface area \( = \pi rl\)

Total Surface Area of a Cone

The total surface area of a cone is the sum of the curved surface area and the area of its base.
Total surface area \( = {\rm{Area}}\,{\rm{of}}\,{\rm{base}} + {\rm{Curved}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{cone}} = \pi {r^2} + \pi rl\)

Volume of a Cone

The volume of a cone equals one-third of the product of the area of base and height of the right circular cone.
Volume \( = \frac{1}{3} \times {\rm{area}}\,{\rm{of}}\,{\rm{base}}\,\left( {{\rm{circle}}} \right) \times {\rm{height}}\)
Hence, the volume of a cone \( = \frac{1}{3} \times \pi {r^2} \times h = \frac{1}{3}\pi {r^2}h\)

Surface Area and Volume of Frustum of a Cone

If a plane parallel to a cone’s base cuts off a right circular cone, then the part of the cone between the base and the cutting plane of the cone is called a frustum of the cone.
A frustum of a cone has two unequal flat circular bases and a curved surface.
Let us now define some other terms related to the frustum of a cone, such as height, slant height, etc.

The curved surface area of the frustum of a cone \( = \pi \left( {{r_1} + {r_2}} \right)l\)
where, \(l = \sqrt {{h^2} + {{\left( {{r_1} – {r_2}} \right)}^2}} \)
The total surface area of the frustum of a cone \( = \pi l\left( {{r_1} + {r_2}} \right) + \pi r_1^2 + \pi r_2^2\)
where, \(l = \sqrt {{h^2} + {{\left( {{r_1} – {r_2}} \right)}^2}} \)

Volume of Frustum of a Cone

Let \(h\) be the height, \(l\) be the slant height and \({r_1}\) and \({r_2}\) be the radii of the bases \(\left( {{r_1} > {r_2}} \right)\) of the frustum of a cone. Then we can find the volume and is given by,
The volume of a frustum of the cone \( = \frac{1}{3}\pi h\left( {r_1^2 + r_2^2 + {r_1}{r_2}} \right)\)

Surface Area and Volume of a Sphere

A sphere is a three-dimensional solid figure which is round in shape.

The surface area of a sphere with radius \(r = 4\pi {r^2}.\)

Volume of a Sphere

The volume of a sphere with radius r can be calculated using the formula, \(V = \frac{4}{3}\pi {r^3}\,{\rm{cubic}}\,{\rm{units}}\)

Surface Area and Volume of a Hemisphere

A hemisphere is a three-dimensional solid shape that is accurately the half of a sphere.

Curved Surface Area of Solid hemisphere

The curved surface of the solid hemisphere is exactly half of the total surface area of a sphere as it does not include the circular base.
The total surface area of a sphere \( = 4\pi {r^2}\)
Therefore, the curved surface area of the hemisphere \( = \frac{1}{2} \times 4\pi {r^2} = 2\pi {r^2}.\)
Where the radius of the hemisphere is \(r.\)

Total Surface Area of a hemisphere

The total surface area consists of the circular base and the curved surface area of the solid hemisphere.
Hence, the total surface area of a hemisphere \( = \left( {2\pi {r^2} + \pi {r^2}} \right) = 3\pi {r^2}\)

Volume of Hemisphere

The volume of the hemisphere will be exactly half of the volume of a solid sphere.
Since the volume of a solid sphere \( = \frac{4}{3}\pi {r^3}\)
The volume of hemisphere \( = \frac{1}{2} \times \frac{4}{3}\pi {r^3} = \frac{2}{3}\pi {r^3}\)
Where \(r\) is the radius of the hemisphere.

Solved Examples About Surface Area and Volume of Solids

Let us look at some examples about Surface Area and Volume of Solids:

Q.1. What is the curved surface area of a hemisphere if the diameter is \(12\,{\rm{cm}}{\rm{.}}\)
Ans: The diameter is \(12\,{\rm{cm}}{\rm{.}}\)
The radius is \(\frac{{12}}{2}\,{\rm{cm = 6}}\,{\rm{cm}}\)
The curved surface area of the hemisphere \( = 2\pi {r^2} = 2 \times \frac{{22}}{7} \times 6 \times 6\,{\rm{c}}{{\rm{m}}^2}\)
\( = 226.28\,{\rm{c}}{{\rm{m}}^2}\)

Q.2. Find the length of the sides of the cube if its volume is \(216\,{\rm{c}}{{\rm{m}}^3}.\)
Ans: Given, the volume of the cube \( = 216\,{\rm{c}}{{\rm{m}}^3}\)
Let the length of the sides is \(a.\)
We know,
The volume of a cube \( = {\left( {{\rm{side}}} \right)^3}\)
Substituting the value we get,
\({a^3} = 216\)
\( \Rightarrow a = \sqrt[3]{{216}} \Rightarrow a = 6\,{\rm{cm}}\)
Hence, the side of the cube is \(6\,{\rm{cm}}{\rm{.}}\)

Q.3. If the diagonal is \(4\sqrt 3 \,{\rm{cm}}{\rm{.}}\) Find the volume of the cube.
Ans: We know, \(V = \frac{{{d^3}}}{{3\sqrt 3 }} \Rightarrow V = \frac{{{{\left( {4\sqrt 3 } \right)}^3}}}{{3\sqrt 3 }} = \frac{{64 \times 3\sqrt 3 }}{{3\sqrt 3 }} \Rightarrow V = 64\,{\rm{c}}{{\rm{m}}^3}\)
Hence, the volume of the cube is \(64\,{\rm{c}}{{\rm{m}}^3}.\)

Q.4. The slant height of a frustum of a cone is \(4\,{\rm{cm}},\) and the perimeters (circumference) of its circular ends are \(18\,{\rm{cm}}\) and \(6\,{\rm{cm}}{\rm{.}}\) Find the curved surface area of the frustum.
Ans: Given, \(l = 4\,{\rm{cm}}\)
Circumference of a circular end \( = 18\,{\rm{cm}}\)
\( \Rightarrow 2\pi {r_1} = 18\,{\rm{cm}}\)
\( \Rightarrow \pi {r_1} = 9\,{\rm{cm}}\)
Circumference of another circular end \( = 6\,{\rm{cm}}\)
\( \Rightarrow 2\pi {r_2} = 6\,{\rm{cm}}\)
\( \Rightarrow \pi {r_2} = 3\,{\rm{cm}}\)
We know that curved surface area \( = \pi l\left( {{r_1} + {r_2}} \right) = l\left( {\pi {r_1} + \pi {r_2}} \right)\)
\( = 4 \times \left( {9 + 3} \right){\rm{c}}{{\rm{m}}^2}\)
\( = 48\,{\rm{c}}{{\rm{m}}^2}\)
Hence, the curved surface area of the frustum is \(48\,{\rm{c}}{{\rm{m}}^2}.\)

Q.5. Madhu prepares a birthday cap with a piece of paper in the form of a right circular cone of radius \(3\) inches and height \(4\) inches. Find the slant height of the birthday cap made by her.
Ans: Given that, the birthday cap prepared by Madhu is in the shape of a right circular cone.
The radius of the cone \(\left( r \right) = 3\) inches and height of the cone \(\left( h \right) = 4\) inches.
It is known that the slant height of the cone is given by \(l = \sqrt {{r^2} + {h^2}} .\)
\( \Rightarrow l = \sqrt {{3^2} + {4^2}} \)
\( \Rightarrow l = \sqrt {9 + 16} \)
\( \Rightarrow l = \sqrt {25} = 5\) inches
Hence, the slant height of the birthday cap is \(5\) inches.

Summary On Surface Area and Volume of Solids

This article discussed the surface area and the volume of various solids. We learnt all formulas of surface area and volume. We also solved some examples of it.

Frequently Asked Questions on Surface Area and Volume of Solids

We have provided some frequently asked questions on Surface Area and Volume of Solids here:

Q.1. What do you mean by surface area and volume of solids?
Ans: The amount of external space that covers a three-dimensional shape is called the surface area. The volume of a solid shape is the amount of space it occupies. It is the space enclosed by a boundary or occupied by an object or the capacity to hold something.

Q.2. What is the surface area of a solid?
Ans: The amount of external space that covers a three-dimensional shape is called the surface area. The surface area of the solid shapes is categorised into three types. These are lateral surface area, curved surface area, total surface area.

Q.3. What is the surface area formula?
Ans: The formula of the surface area of the cuboid \( = 2\left( {lb + hb + lh} \right)\)
The formula of the surface area of a cube \( = 6{\left( a \right)^2}\)
The formula of the surface area of a cylinder \( = 2\pi r\left( {h + r} \right)\)
The formula of the surface area of a cone \( = \pi {r^2} + \pi rl\)
The formula of the surface area of a frustum of a cone \( = \pi l\left( {{r_1} + {r_2}} \right) + \pi r_1^2 + \pi r_2^2\)
The formula of the surface area of a sphere \( = 4\pi {r^2}\)
The formula of the surface area of a hemisphere \( = 3\pi {r^2}\)

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

Q.5. How do you find the surface area and volume of a solid?
Ans: If we know the dimensions of the solids, then we can easily find the surface area and volume of solids using the formulas.

We hope this detailed article on surface area and volume of solids helped you in your studies.