Hemisphere: Definition, Formulas and Solved Examples

In geometry, the hemisphere is a three-dimensional solid figure that is exactly half of the sphere. It can be said that if a sphere is divided into two equal parts, we will get two hemispheres. This three-dimensional figure can be compared with a half watermelon, half lemon, etc. The hemisphere has a curved surface and a flat surface. The flat surface is circular, and it can be said as the circular base. This article will discuss various aspects of the hemisphere like the types of hemispheres, properties, surface areas such as curved surface area, total surface area, and volume of hemispheres.

Definition of a Hemisphere

A hemisphere is a \(3D\) geometric figure which is exactly half of a sphere with flat and curved surfaces. So, a hemisphere is an accurate half of a sphere. Thus, two identical hemispheres make a sphere.
One such decent example of the hemisphere is the planet earth. The earth is made of two hemispheres, such as the Southern hemisphere and the Northern hemisphere.

Properties of a Hemisphere

1. A hemisphere consists of only one curved surface area.
2. It has a circular base that is known as the flat face.
3. A hemisphere has no vertices as a sphere.
4. A hemisphere has one curved edge (where the curved face and the flat face meet). 
5. It cannot be considered a polyhedron since polyhedrons are formed by polygons.

Types of Hemisphere

There are two types of hemispheres:

1. Solid hemisphere
2. Hollow hemisphere

1. Solid Hemisphere
A solid hemisphere is a three-dimensional object in the form of the exact half of a sphere and filled with the material it is made up of.
For example, half of a lemon.

2. Hollow Hemisphere
A hollow hemisphere is a three-dimensional object with only the outer circular bowl boundary, and nothing is filled inside.
Such good examples of a hollow hemisphere is an igloo, the exact half of a coconut shell, etc.

Surface Area of Hemisphere

A hemisphere is an exact half of a perfectly round three-dimensional geometrical shape. The surface area of the hemisphere is determined as the number of square units needed to cover the surface.
The surface area of a hemisphere includes the area of its curved surface and flat circular face.
We have discussed earlier the solid and the hollow hemisphere. Now, let us know about their surface areas, namely curved surface area and lateral surface area.

Surface Area of Solid Hemisphere

The surface area of a solid hemisphere is classified into two types, namely;

1. Curved Surface Area (CSA)
2. Total Surface Area (TSA)

1. Curved Surface Area of a Solid Hemisphere
The curved surface area is the area of all the curved regions of the solid. The curved surface of the solid hemisphere will also be accurately half of the total surface area of a solid sphere, and it does not consist of a circular base.
We know that the total surface area of a sphere is \(A = 4\pi {r^2}\)

Therefore, the curved surface area of a hemisphere is \(A = \frac{1}{2} \times 4\pi {r^2} = 2\pi {r^2}\).

Where \(r\) is the radius of the hemisphere.

2. Total Surface Area of a Solid Hemisphere

The total surface area is the area of all the faces of the solid object. The total surface area includes the circular base and the curved surface area of the solid hemisphere.
\(A = \left( {2\pi {r^2} + \pi {r^2}} \right) = 3\pi {r^2}\)

Surface Area of Hollow Hemisphere

A hollow hemisphere can have two different radii for its inner circular base and its outer circular base. So, a hollow hemisphere has two curved surfaces, such as inner curved surface and outer curved surface. The difference between the outer radius and the inner radius of the hemisphere is considered the thickness. So, we can get the flat surface of the hollow hemisphere if we subtract the circular base with the inner radius from the circular base with the outer radius.

The surface area of a hollow hemisphere is classified into two types, namely:

1. Curved Surface Area (CSA)
2. Total Surface Area (TSA)

Curved Surface Area
A hollow hemisphere can have two curved surfaces, such as inner curved surface and outer curved surface with different radii.
Let us say, \({r_1}\) is the inner radius of the hemisphere and  \({r_2}\) is the radius of the outer hemisphere.
So, the curved surface area of the inner hemisphere is \(2\pi r_1^2\)
the curved surface area of the outer hemisphere is \(2\pi r_2^2\)
Thus, the curved surface of the hemisphere\(=\)curved surface of the inner hemisphere\(+\)curved surface of the outer hemisphere

Therefore, the curved surface of the hemisphere\( = 2\pi r_1^2 + 2\pi r_2^2 \Rightarrow 2\pi \left( {r_1^2 + r_2^2} \right)\)

Total Surface Area
The total surface area of a hollow hemisphere includes the circular ring base and the curved surface area of the outer hemisphere and the inner hemisphere.
We know, \({r_1}\) is the inner radius of the hemisphere and  \({r_2}\) is the radius of the outer hemisphere.
Area of a circular ring\(=\)Area of an outer circular base\(-\)Area of an inner circular base.Area of a circular ring \( = \pi r_1^2 – \pi r_2^2 = \pi \left( {r_1^2 – r_2^2} \right)\)

The total surface of the hemisphere=curved surface of the inner hemisphere+ curved surface of the outer hemisphere+area of the circular ring
Therefore, the total surface area of the hemisphere is \(2\pi \left( {r_1^2 + r_2^2} \right) + \pi \left( {r_1^2 – r_2^2} \right)\)

Volume of Hemisphere

The volume of a hemisphere of the same radius can be found by calculating half of the volume of a sphere of the same radius.
The unit of volume is in \({\rm{cubic}}\,{\rm{units}}\) such as \({\rm{cubic}}\,{\rm{metre,}}\,{\rm{cubic}}\,{\rm{centimetre}}\) etc.

We know the solid hemisphere is a three-dimensional shape that is accurately the half of a solid sphere. So, the volume of the solid hemisphere will be accurately half of the volume of a solid sphere.
Since the formula of the volume of a solid sphere is \(V = \frac{4}{3}\pi {r^3}\).

So, the volume of a solid hemisphere is \(V = \frac{1}{2} \times \frac{4}{3}\pi {r^3} = \frac{2}{3}\pi {r^3}\)

Where \(r\) is the radius of the hemisphere.

Solved Examples – Hemisphere

Q.1. What is the total surface area of a solid hemispherical object of radius \({\rm{7}}\,{\rm{cm}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans: We know that the total surface area of a solid hemisphere is calculated as
\(A = 3\pi {r^2}\)
Given, the radius of the object \( = 7\;{\rm{cm}}\)
So, the total surface area of the object \(9 = 3 \times \frac{{22}}{7} \times 7 \times 7\;{\rm{c}}{{\rm{m}}^2}\)
\( = \frac{{3 \times 22 \times 7 \times 7}}{7}\;{\rm{c}}{{\rm{m}}^2}\)
\( = 462\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the total surface area of the solid hemispherical object is \(462\;{\rm{c}}{{\rm{m}}^2}.\)

Q.2. A hemispherical bowl has an inner radius of \({\rm{14}}\,{\rm{cm}}\). How much amount of water it can contain? considering \(\pi = \frac{{22}}{7}.\)
Ans: We know that the volume of a hemisphere is calculated as
\(V = \frac{2}{3}\pi {r^3}\)
Given, the radius of the ball \( = 14\;{\rm{cm}}\)
So, the volume of the ball \( = \frac{2}{3} \times \frac{{22}}{7} \times 14 \times 14 \times 14\;{\rm{c}}{{\rm{m}}^3}\)
\( = 5749.34\;{\rm{c}}{{\rm{m}}^3}\left( {{\rm{approx}}} \right)\)
Hence, the amount of water held by the hemispherical bowl of radius \({\rm{14}}\,{\rm{cm}}\) is \(5749.34\;{\rm{c}}{{\rm{m}}^3}\left( {{\rm{approx}}} \right).\)

Q.4. Find the radius of the hemisphere when its total surface area is \(462\,{\rm{square}}\,{\rm{units,}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans: Given, the total surface area of a hemisphere \(462\,{\rm{square}}\,{\rm{units}}{\rm{.}}\)
We know that the total surface area of a solid hemisphere is calculated as
\(A = 3\pi {r^2}\)
So, \(3\pi {r^2} = 462\)
\( \Rightarrow {r^2} = \frac{{462}}{{3\pi }} \Rightarrow {r^2} = \frac{{462 \times 7}}{{3 \times 22}} = 49\)
\( \Rightarrow r = 7\,{\rm{units}}\)
Hence, the radius of the hemisphere is \(7\,{\rm{units}}{\rm{.}}\)

Q.5. Find the volume of the hemisphere when its area of the circular base is \({\rm{154}}\,{\rm{square}}\,{\rm{units}}{\rm{.}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans: Given, the area of the circular base is \({\rm{154}}\,{\rm{square}}\,{\rm{units}}{\rm{.}}\)
The formula of the area of the circular base is \(\pi {r^2}\).
So, \(\pi {r^2} = 154 \Rightarrow {r^2} = \frac{{154}}{\pi } \Rightarrow {r^2} = \frac{{154 \times 7}}{{22}} = 49\)
\( \Rightarrow r = 7\,{\rm{units}}\)
Now, we know that the volume of a hemisphere is calculated as
\(V = \frac{2}{3}\pi {r^3}\)
the radius of the hemisphere \( = 7\,{\rm{units}}\)
So, the volume of the hemisphere \( = \frac{2}{3} \times \frac{{22}}{7} \times 7 \times 7 \times 7\,{\rm{uni}}{{\rm{t}}^{\rm{3}}}\)
\( = 718.67\,{\rm{uni}}{{\rm{t}}^{\rm{3}}}{\rm{(approx)}}\)
Hence, the volume of the hemisphere is \(718.67\,{\rm{uni}}{{\rm{t}}^{\rm{3}}}{\rm{(approx)}}{\rm{.}}\)

Summary

In this article, we learned about the hemisphere. We also learned about the types of the hemisphere, i.e., solid hemisphere and hollow hemisphere. We also learned the formula to find the volume, the formula to calculate the surface area of a hemisphere. In this article, we also studied the properties of hemisphere, and their real-life examples.

Frequently Asked Questions (FAQs) – Hemisphere

Q.1. What is the unit of volume of a hemisphere?
Ans: The unit of volume of a hemisphere is in cubic units like \({{\rm{m}}^3},\,{\rm{c}}{{\rm{m}}^3}\) etc.

Q.2. What is the difference between the equator and hemisphere?
Ans: The earth is made of two hemispheres, such as the Southern hemisphere and the Northern hemisphere, and the equator is an imaginary largest circle around the earth. Therefore, if we divide the earth into imaginary two hemispheres, we can compare the equator with the circumference of the circular base of a hemisphere.

Q.3.What is the formula for calculating the volume of a hemisphere?
Ans: The formula for finding the volume of a hemisphere is \(V = \frac{2}{3}\pi {r^3},\) where \(r\) is the radius of the hemisphere.

Q.4. What is the formula of the total surface area of a solid sphere?
Ans: The total surface area includes the circular base and the curved surface area of the solid hemisphere.
\(A = \left( {2\pi {r^2} + \pi {r^2}} \right) = 3\pi {r^2}\)

Q.5.What does hemisphere literally mean?
Ans: A hemisphere is a \(3D\) geometric figure which is exactly half of a sphere with flat and curved surfaces. So, a hemisphere is an accurate half of a sphere. Thus, two identical hemispheres make a sphere.

We hope this detailed Maths article on the Hemisphere has been informative to you. If you have any queries on this article, ping us through the comment box below and we will get back to you as soon as possible.