Volume of Hemisphere

Hemisphere Definition:

Were you aware that the prefix “Hemi” means half and sphere stands for a three-dimensional shape, like basketball or marble, perfectly round? What does the term "hemisphere" mean, then? Ok, the hemisphere is a 3D shape, half a region with a flat, circular side, also known as a face.


A sphere is defined as a set of three-dimensional points and the centre is at equidistant with all the points on the surface. Just for say, when an aircraft takes a round of the earth from earth’s midpoint, it covers two hemispheres of the earth. It can be said that the globe is made of two hemispheres. A globe generally produces two hemispheres exactly. Our earth is composed of two hemispheres, the southern and the northern hemispheres.


In the above figure, the earth’s hemispheres are shown. These hemispheres are equally divided with their respective counterparts.


Real-World Examples:

You’ll find hemispheres around you when you look around. Let’s talk about the supermarket. There are many regions that can be halved for two hemispheres. A grape-fruit cut half gives rise to two hemispheres; cutting cherry into two halves would give rise to two hemispheres.


There are many other non-food representations of hemispheres in the real world, including the planet itself, before we go too far. Every spherical planet in the universe can be conceptually divided into the Northern and Southern hemispheres just like Earth, and the Central, and the Western and Prime Meridian hemisphere.


Hemispheres in Math:

As a student, you read different parts of mathematics like algebra, computation, and geometry to name only a few. You will learn more about hemisphere in geometry.


Generally, the surface area of the hemisphere is sometimes considered, the sum of the surfaces of all the structures covering the surface of the hemisphere. For instance, you wanted to wrap a wrapping paper in the hemisphere type object. You will need to learn the surface area to find out how much wrapping paper you need.


Sometimes you have to know the volume of the globe, which is a quantity of space within a hemisphere, in maths lessons like algebra or geometry. Compare it to the human body is a way of understanding the difference between surface and size. The skin required to cover the body is the plain, whereas the insides are the body’s weight, such as organ, bones, and blood.


All around us are hemispheres. The geometric description of the hemisphere is a three-dimensional structure half a sphere with a smooth, circular face. On the other side, a sphere is a 3-dimensional and almost perfectly round shaped. You can also learn how to measure a hemisphere’s surface area and volume and many other ways when learning geometry.


We studied various kinds of three-dimensional forms in geometry. The solids have three different dimensions, such as length, width, and height, in the three-dimensional shapes. The 3D types don’t lie on a journal. Typically, most 3D objects are derived from the two-dimensional object rotation. The sphere formed from the 2D rotation that is called a circle is one of the best examples of the 3D form. Our earth is one of the other examples of a sphere.


Hemisphere Volume:

Derive the formula for sphere and then divide it by two, as, a sphere represents two hemispheres equally.


In the following figure, a cylindrical differential element is shown with radius x and altitude dy.


Then, the volume of the cylindrical element is -

\[dV = \pi x^{2} dy\]


From o to r, the sum of the cylindrical elements is forming a hemisphere and twice the hemispheres will five the volume of the sphere.

Therefore,

\[V = 2\pi \int_{0}^{r} x^{2} dy\]

Now, from circle’s equation -

\[x^{2} + y^{2} = r^{2}\],

and, \[x^{2} = r^{2} - y^{2}\]

\[V = 2\pi \int_{0}^{r} x^{2} dy\]

\[V = 2\pi [r^{2}y - \frac{y^{3}}{3} ]_{0}^{r}\]

\[V = [(r^{3} - \frac{r^{3}}{3}) - (0 - \frac{0^{3}}{3})]\]

\[V = 2\pi [\frac{2r^{3}}{3}]\]

\[V = \frac{4\pi r^{3}}{3}\]


Above, we have derived the formula of volume of a sphere now divide it by 2 and the volume of the hemisphere will be defined.

\[V = \frac{4\pi r^{3}}{2\times 3}\]

\[V = \frac{2\pi r^{3}}{3}\]


The volume is measured in cubic units.

Volume of Hemisphere = \[\frac{2}{3} \pi \times radius^{3}\] cubic units.

Total Surface Area = \[3\pi \times radius^{2}\] sq. units.

Archimedes determined and derived the volume of hemisphere.


Hemisphere Equation:

When the radius R is been centred at the point of origin, 

Then the hemisphere equation is given by:

\[x^{2} + y^{2} + z^{2} = R^{2}\]

Hemisphere’s spherical coordinates are given below-

\[X = r cos\theta sin\phi\]

\[Y = r sin\theta cos\phi\]

\[Z = rcos\phi\]


Solved Numerical:

A sample numerical on hemisphere for the better understanding of the hemisphere equation use is been given below:

Question:

Determine the volume of the hemisphere having a radius of 6 cm?

Solution:

Given:

The radius of the hemisphere = 6cm

The volume of a hemisphere can be expressed as:

\[Volume = \frac{2}{3} \pi \times radius^{3}\] cubic units.

= \[\frac{2}{3} \times 3.14 \times 6^{3}\]

= \[\frac{2}{3} \times 3.14 \times 6 \times 6 \times 6\]

= 452.16 cubic cms.

Therefore, the volume of the hemisphere is 452.16 cubic cms.


Question:

The volume of a hemisphere is 2500 cm^3. Find the radius of the Hemisphere.

Solution:

Given:

Volume = 2500 cm^3

\[\text{The volume of hemisphere} = \frac{2}{3} \times radius^{3}\]

⇒ 2500 = \[\frac{2}{3} \pi \times r^{3}\]

⇒ 2500 x 3 = \[2 \pi r^{3}\]

⇒ \[r^{3} = \frac{7500}{2\times \pi}\]

⇒ \[r^{3} = \frac{7500}{2\times 3.14}\]

⇒ \[r^{3} = \frac{7500}{6.28}\]

⇒ \[r^{3} = 1194.267\]

⇒ \[r = \sqrt[3]{1194.267}\]

⇒ r = 10.6096358

Therefore, the radius of the hemisphere is 10.6096358.