Volume of Hemisphere: Definition, Formula, Examples

What is a Hemisphere?

The prefix “hemi” has a Greek origin and it means “half.” Thus, a hemisphere simply refers to the half of the sphere. If a sphere is divided into two equal parts, we will get two hemispheres.

You may have come across the word hemisphere in the real life context of planet Earth. For example, if you cut the Earth right on its equator, you’d have two halves: the northern and southern hemispheres. 

Hemisphere: Definition

In geometry, the hemisphere is a 3D solid figure that is exactly half of the sphere. When a sphere is cut into two equal parts at the center then the hemisphere is formed. It has 1 flat circular base and 1 curved surface. 

Hemisphere Volume: Formula

The volume of a hemisphere is the total capacity of the hemisphere. The volume of hemisphere is measured in terms of cubic units, such as $in^3,\; ft^3$, etc.

The volume of sphere is calculated using the formula $V = \frac{4}{3} \pi r^3$.

So, what is the volume of a hemisphere? 

We know the hemisphere is simply half of the sphere. The volume of the hemisphere is also half the volume of the sphere.

Thus, the formula for volume of a hemisphere $= \frac{2}{3} \pi r^3$

How to Find the Volume of a Hemisphere

The volume is the amount of space that a 3D shape takes up.

Let us recall the formula of the volume of the hemisphere first.

Volume of Hemisphere $= \frac{2}{3} \pi r^3$

Here, the symbol stands for pi. We use the value of as 3.14 or $\frac{22}{7}$. 

The r stands for radius, and it is the distance from the center point of the sphere. 

We simply have to put the values in the above formula and find the volume of the hemisphere.

Facts about Volume of Hemisphere

• A hemisphere consists of only one curved surface area.

• The volume of the hemisphere is given as $\frac{2}{3} \pi r^3$ cubic units, where “r” is the radius of the hemisphere and   is equal to $\frac{22}{7}$ or 3.14 approximately.

• A hemisphere has two surface areas, i.e., curved surface area and the total surface area. The curved surface area of a hemisphere is only the area covered by the curved surface, while the surface area of a hemisphere is the sum of the curved surface area and the base area of the hemisphere.

• As volume is a 3D concept including length, width, and height, volume of the hemisphere is measured in cubic units. 

• The spherical coordinates of the hemisphere are $x = r \; cos \theta sin\; \Phi ,\; y = r sin \theta cos\; \Phi$, and $z = r\; cos\; \Phi$.

• Hemisphere Equation

When the radius “R” is centered at the origin, the equation of hemisphere is given by

$x^2 + y^2 + z^2 = R^2$

The Cartesian form or equation of a hemisphere with the radius “R” at the point $(x_0,\; y_0,\; z_0)$ is written as follows:

$(x\;-\;x_0)^2 + (y\;-\;y_0)^2 + (z\;-\;z_0)^2 = R^2$

The spherical coordinates of the hemisphere are given as follows:

$x = r\; cos\; \theta sin\; \Phi$

$y = r\; sin\; \theta cos\; \Phi$

$z = r\; cos\; \Phi$

Conclusion

The volume of a hemisphere is the total capacity of the hemisphere. In this article, we learnt in detail about the volume of the hemisphere formula, definition, and facts. Let’s solve some examples and practice problems to understand the concept better.

Solved Examples on the Volume of a Hemisphere

1. The diameter of a hemisphere is 6 ft. Calculate the volume.

Solution: 

Given data,

Diameter of hemisphere $= 6$ ft.

Radius of hemisphere $= 3$ ft.

We know that, volume of hemisphere $= \frac{2}{3} \pi r^3$

                                                            $= \frac{2}{3} \times 3.14 \times 3^3$

                                                            $= \frac{2}{3} \times 3.14 \times 27$

                                                            $= 2 \times 3.14 \times 9$

                                                            $= 56.52\; ft^3$

The volume of the hemisphere is $56.52\; ft^3$.

2. A hemispherical bowl has an inner radius of 4 inches. How much water can it contain?

Solution:

Radius of hemispherical bowl $= 4$ in.

We know that, volume of hemisphere $= \frac{2}{3} \pi r^3$

                                                            $= \frac{2}{3} \times 3.14 \times 4^3$

                                                            $= \frac{2}{3} \times 3.14 \times 64$

                                                            $= \frac{401.92}{3}$

                                                            $= 133.97\; in^3$

Thus, the hemispherical bowl can contain $133.97\; in^3$ of water.

3. Find the radius of the hemispherical bowl if the area of its base is 154 unit square. $( \pi = \frac{22}{7})$

Solution: 

Area of the base $= 154$ unit square.

We know that the area of the base of the hemisphere $= \pi r^2$

So, substituting value in the above formula the equation becomes: 

 $154 = \frac{22}{7} \times r^2$

 $r^2  = 154 \times \frac{7}{22}$

 $r^2  = \frac{1078}{22}$

 $r^2  =  49$

 $r   = \sqrt{49}$

 $r   = 7$ units.

The radius of the hemispherical bowl is 7 units.

4. Find the volume of the hemisphere when its area of the circular base is 616 sq. units.

Solution: 

Given data: 

Area of circular base $= 616$ sq.units.

We know that:

The formula for the area of the circular base $= \pi r^2$

So, substituting value in the above formula the equation becomes,

 $616 = \frac{22}{7} \times r^2$

   $r^2  =  616 \times \frac{7}{22}$

   $r^2  = \frac{4312}{22}$

   $r^2  = 196$

   $r   = \sqrt{196}$

   $r   = 14$ units.

Thus, the volume of the hemisphere equation can be written as

Volume of hemisphere $= \frac{2}{3} \pi r^3$

                                     $= \frac{2}{3} \times 3.14 \times 14^3$

                                    $ = \frac{2}{3} \times 3.14 \times 2744$

                                     $= \frac{17232.32}{3}$

                                     $= 5744.10\; unit^3$

The volume of the hemisphere is $5744.10\; unit^3$.

5. A sphere with a radius of 5 inches is divided into two equal halves. Calculate the volume of each produced hemisphere.

Solution: 

Given data, 

Radius of sphere $= 5$ inch.

We know that the sphere and hemisphere have the same radius.

Thus, radius of hemisphere $= 5$ inch

Volume of each hemisphere $= V = \frac{2}{3} \pi r^3$

                                                     $=  \frac{2}{3} \times 3.14 \times 5^3$

                                                     $= \frac{2}{3} \times 3.14 \times 125$

                                                     $= \frac{785}{3}$

                                                     $= 261.66\; in^3$

The volume of each hemisphere is $261.66\; in^3$

Practice Problems on the Volume of a Hemisphere

Frequently Asked Questions on the Volume of a Hemisphere