Volume of Hemisphere

The total space occupied by a hemisphere in a 3-dimensional region is called its volume. To recall, geometrically a hemisphere is an exact half of a sphere. Some of the common real-life examples of a hemisphere are northern/ southern half of the Earth, left/ right side of our brain, a bowl, an igloo, headphones, and domes in architecture.

Along with the area, the volume of a hemisphere is an important mathematical calculation that helps us solve many problems related to mensuration. Hence, we have provided the formula to calculate the Volume of Hemisphere as well as an online hemisphere volume calculator. Also, solve some practice questions on this topic to get confidence.

What is a Hemisphere? Definition & Equation

A sphere is defined as a set of points in three-dimension, and all the points lying on the surface is equidistant from the centre. When a plane cuts across the sphere at the centre or equal parts, it forms a hemisphere.

We can say, a hemisphere is exactly half of a sphere. In general, a sphere makes exactly two hemispheres. One such good example of the hemisphere is our earth. Our earth consists of two hemispheres, namely Southern Hemisphere and the Northern Hemisphere.

Hemisphere Equation:

When the radius “R” is centred at the origin, the equation for a 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 (x₀, y₀, z₀) is written as

$${\left( {x – {x_0}} \right)^2} + {\left( {y – {y_0}} \right)^2} + {\left( {z – {z_0}} \right)^2} = {R^2}$$

Therefore, the spherical coordinates of the hemisphere are given as follows

$$x = r\cos \theta \sin \emptyset $$$$y = r\sin \theta \cos \emptyset $$$$z = r\cos \emptyset $$

Volume of Hemisphere Calculator

Below, you can find the simple online calculator to find the volume of hemisphere. Just enter the value of the radius and click on “Calculate” button to get the corresponding value of the volume.

Volume of Hemisphere Formula

Wondering how the online volume calculator for hemisphere works? When you enter the value of the radius, the calculator calculates its volume based on a formula which we will discuss here.

$$Volume\, = \,{2 \over 3} \times \pi \times {r^3}$$

Formula Derivation:

Since a hemisphere is half of a full sphere, we can calculate the volume of a hemisphere just by halving the volume of the complete sphere. Let’s see how the formula for a sphere is derived.

Consider a sphere of radius r and divide it into pyramids. In this way, we see that the volume of the sphere is the same as the volume of all the pyramids of height, r and total base area equal to the surface area of the sphere as shown in the figure.

The total volume is calculated by adding the pyramids’ volumes.

Volume of the sphere = Sum of volumes of all pyramids

$$Volume\, = \,{1 \over 3}{A_1}r + {1 \over 3}{A_2}r + {1 \over 3}{A_3}r + \ldots {1 \over 3}{A_n}r = {1 \over 3}r\left( {Surface\,area\,of\,a\,sphere} \right)$$

$$ = {1 \over 3} \times 4\pi {r^2} \times r$$

Hence, volume of a sphere = $${4 \over 3}\pi {r^3}$$

Since volume of a hemisphere = Half of the volume of a sphere, we have:

$$Volume\,of\,hemisphere = {1 \over 2}\left( {{4 \over 3}\pi {r^3}} \right) = {2 \over 3}\pi {r^3}$$

Volume of Hemisphere Solved Examples

Question 1: Find the volume of a hemisphere whose radius is 8 cm.

Solution: Volume of a hemisphere = ⅔πr³, putting the value of r = 8 cm and π = 3.14, we get:

Volume = 1072.33 cm³

Question 2: A hemispherical bowl has a volume of 288 π cubic units. Find the diameter of the bowl.

Solution: Volume of a hemisphere = ⅔πr³, putting the value of V = 288 π, we get: r = 6 units.

Diameter = 2 times radius = 12 units

FAQs:

Here are some of the frequently asked questions on volume of hemisphere:

Q1: What is the formula for volume of hemisphere?
Ans: Volume of a hemisphere is given by the formula: ⅔πr3, where r is the radius.

Q2: What is the relation between volume of a sphere and a hemisphere?
Ans: The volume of a hemisphere is half of the volume of a sphere.

Q3: What is the cartesian equation for a hemisphere whose center is at the origin?
Ans: The cartesian equation for a hemispherical shape having the center at origin is: x² + y² + z² = R²

Q4. What is the definition of volume of hemisphere?
Ans: The volume of a hemisphere is also half of a sphere which is equal to two-thirds multiplied by pi multiplied by the radius to the power 3.

Q5. What is the SI unit of volume of hemisphere?
Ans: The volume of a hemisphere = (2/3)πr3 cubic units.

We hope this detailed article on the volume of hemisphere is helpful to you. If you have any queries, ping us through the comment box below and we will get back to you as soon as possible.