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Volume of Hemisphere – Formula, Derivation & Examples

volume of hemisphere

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A hemisphere is a 3D geometric shape that is half of a sphere with one side flat and the other side as a circular bowl. Circular shapes take the shape of a hemisphere when observed as three-dimensional structures, e.g., a bowl or a mushroom head. The space occupied by a hemisphere is called the volume or capacity of hemisphere. The volume of hemisphere depends on its radius. The volume of hemisphere is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a hemisphere and its formula.

Volume of Hemisphere Formula

The volume of hemisphere is the number of unit cubes that can fit into it. The unit of volume is cubic units, such as $m^{3}$, $cm^{3}$, $ft^{3}$, $in^{3}$, and so on.

The volume of a hemisphere is two-thirds of a product of $\pi$ and the cube of its radius, i.e., $\text{Volume of hemisphere} = \frac {2}{3} \pi r^{3}$, where $r$ is the radius of the hemisphere.

Derivation of Volume of Hemisphere Formula

Let us see how the formula for the volume of a hemisphere is derived. Since a hemisphere is half of a sphere, we can divide the volume of a sphere by $2$ to get the volume of its hemisphere. 

Now considering that the radius of a sphere is $r$.

Then the volume of the sphere is given by the formula $V = \frac {4}{3} \pi r^{3}$. So, the formula for the volume of a hemisphere will become $\frac {1}{2}$ of $\frac {4}{3} \pi r^{3} = \frac {1}{2} \times \frac {4}{3} \pi r^{3} = \frac {2}{3} \pi r^{3}$.

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

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Maths can be really interesting for kids

The volume of a Hollow Hemisphere

Consider a hollow hemisphere such that the radius of the outer sphere is $R$ and the radius of the inner sphere is $r$, such that the thickness of the layer of a sphere is $R – r$.

volume of hemisphere

$\text{Volume of the layer of a hemisphere} = \text{Volume of Outer Hemisphere} – \text{Volume of Inner Hemisphere}$

$ = \frac {2}{3} \pi R^{3} – \frac {2}{3} \pi r^{3} = \frac {2}{3} \pi (R^{3} – r^{3})$.

Maths in Real Life

Examples

Ex 1: Find the volume of a hemispherical bowl of radius $7 cm$.

Radius of a hemispherical bowl $r = 7 cm$

Volume of a hemispherical bowk = $\frac {2}{3} \pi r^{3} = \frac {2}{3} \times \frac {22}{7} \times 7^{3} = 718.67 cm^{3}$.

Ex 2: The volume of a hemispherical vessel is $19404 c.c.$. Find the radius of the vessel.

Volume of the hemispherical vessel = $19404 cm^{3}$.

Let the radius of the hemispherical vessel = $r$.

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

Therefore, $\frac {2}{3} \pi r^{3} = 19404 => \frac {2}{3} \times \frac{22}{7} \times r^{3} = 19404 => \frac{44}{21} \times r^{3} = 19404$

$ => r^{3} = 19404 \times \frac{21}{44} => r^{3} = 9261 => r^{3} = \sqrt[3]{9261} =>r = 21 cm$.

Ex 3: The circumference of a hemispherical bowl’s edge is $132 cm$. Determine the capacity of the bowl. 

Let the radius of the hemispherical bowl = $r$.

Circumference of a hemispherical bowl = $2 \pi r$.

Therefore, $2 \pi r = 132 => 2 \times \frac{22}{7} \times r = 132 => \frac{44}{7} \times r = 132$

$ => r = 132 \times \frac{7}{44} =>r = 21 cm$

Volume of a hemispherical bowl = $2 \pi r^{3} = 2 \times \frac{22}{7} \times 7^{3} = 2156 c.c.$

Ex 4: What is the volume of a hemisphere having a diameter of $10 cm$?

Diameter of a hemispherical bowl $d = 10 cm$.

Volume of a hemisphere is given by $V = \frac {1}{12} \pi d^{3}$.

Therefore, $V = \frac {1}{12} \times \frac{22}{7} \times 10^{3} =  261.91 cm^{3}$.

Hemisphere – A 3D Solid Shape

The word hemisphere comes from the two words hemi(Greek hemisus meaning half) and sphere. So hemisphere is a 3D geometric shape that is half of a sphere with one side flat and the other side as a circular bowl. A hemisphere is formed when a sphere is cut at the exact centre along its diameter leaving behind two equal parts. The flat side of the hemisphere is known as the base or the face of the hemisphere.

volume of hemisphere

The important elements of a hemisphere are as follows:

  • Radius: The length of the line segment drawn between the centre of the hemisphere to any point on its surface. If $O$ is the centre of the hemisphere and $A$ is any point on its surface, then the distance $OA$ is its radius. 
  • Diameter: The length of the line segment from one point on the surface of the hemisphere to the other point which is exactly opposite to it, passing through the centre is called the diameter of the hemisphere. The length of the diameter is exactly double the length of the radius.
  • Circumference: The length of the great semicircle of the hemisphere is called its circumference.
volume of hemisphere

Properties of a Hemisphere

Since a hemisphere is the exact half of a sphere, a hemisphere and a sphere have quite similar properties. The properties of the hemisphere are

  • A hemisphere has a curved surface area.
  • Just like a sphere, there are no edges and no vertices in a hemisphere.
  • It is not a polyhedron since polyhedrons are made up of polygons, but a hemisphere has one circular base and one curved surface.
  • The diameter of a hemisphere is a line segment that passes through the centre and touches the two opposite points on the base of the hemisphere.
  • The radius of a hemisphere is a line segment from the centre to a point on the curved surface of the hemisphere.

Practice Problems

  1. Find the volume of a hemisphere of radius $6 cm$.
  2. What is the volume of a hemisphere with a diameter of $40 mm$?
  3. If the radius of a hemisphere is doubled, how much its volume increases?
  4. The volume of a hemisphere is $261.5$ cubic metres. Find the radius of the hemisphere.
  5. A hemispherical bowl of internal radius $5 cm$ is made up of a metal sheet of thickness $0.25 cm$. Find the amount of metal used in the bowl.

FAQs

What are the volume of a hemisphere and its formula?

The space occupied by a hemisphere is called the volume of a sphere. The formula to find its volume is $\frac {2}{3} \pi r^{3}$, where $r$ is the radius of the hemisphere and $\pi$ is a constant with value $\frac{22}{7}$ or $3.14$.

How do you find the volume of a sphere and hemisphere?

The formula used to find the volume of
sphere is $\frac {4}{3} \pi r^{3}$
hemisphere is $\frac {2}{3} \pi r^{3}$
Where $r$ is the radius of the sphere or hemisphere
$\pi$ is a constant with value $\frac{22}{7}$ or $3.14$

What is the TSA of a hemisphere?

The total surface area of a hemisphere = flat surface area + curved surface area. The curved surface area of a hemisphere $=2 \pi r^2$ and flat surface area $= \pi r^2$. 
Therefore, the total surface area of a hemisphere $=2 \pi r^2 + \pi r^2 = 3 ]pi r^2$

What are CSA and TSA of the hemisphere?

The formula of Curved Surface Area (CSA) does not take into account the circular base. The Total Surface Area (TSA), on the other hand, is a combination of the curved area along with the area of the base 
a) The formula for CSA = $2 \pi r^2$
b) The formula for TSA = $2 \pi r^2$

Conclusion

The volume of hemisphere is the number of cubic units, occupied by the hemisphere completely and is calculated by using its radius. If $r$ is the radius of a hemisphere, then the formula for computing its volume is $\frac {2}{3} \pi r^{3}$.

Recommended Reading

FAQs

What is the volume of a hemisphere?

The volume of a hemisphere is the amount of space in it. The volume of a hemisphere of radius $r$ is $\frac {2}{3}\pi r^{3}$.

How do you find the volume of a hemisphere?

The formula used to find the volume of a hemisphere of $r$ is $\frac {2}{3}\pi r^{3}$.

How the amount of material used in a hemisphere is calculated?

The formula for the amount of material used in a sphere is $\frac {2}{3} \pi \left(R^{3} – r^{3} \right)$
where $R$ is the outer radius
$r$ is the inner radius
Thickness of material = $R – r$

What is the volume of a hemisphere using diameter?

Let us consider a sphere of radius $r$ and diameter $d$. The volume of a sphere of radius $r$ is $V = \frac {2}{3} \pi r^{3}$. 
We also know that $r = \frac {d}{2}$. 
By substituting this in the above formula we get $V = \frac {1}{12} \pi d^{3}$.

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