The volume of the Hemisphere is an important topic and an important formula and keeps appearing in a lot of exams. To understand this properly, let us start with 3D shapes.

We know and see a lot of three-dimensional shapes in daily life. The 3D shapes have three measurements, i.e., length, breadth, and height.

We know that 3D shapes do not lie on an x-y axis. Most of the 3D objects are obtained from the rotation of the 2D objects.

The **sphere** is one of the best examples of a 3D body we can obtain by rotating a 2D shape, a circle.

**Hemisphere**

A sphere is a 3D solid figure, made up of all points in space, lying at a constant distance called the radius, from a fixed central point called the centre of the sphere.

Now, if a plane cuts the sphere into two halves passing through the centre, it forms two **hemispheres**.

**Equation of Hemisphere**

With radius ‘r’ as the centre of origin, it comes out to be,

**Equation of Hemisphere:** x^{2} + y^{2} + z^{2} = r^{2}

And the Cartesian equation is

\((x – x_0)^2 + (y – y_0)^2 + (z – z_0)^2 = r^2\)

**The Surface Area of a Hemisphere**

Let ‘r’ be the radius of the sphere, then

The **surface area of a sphere** is given by \(4\pi r^2\)

And since we know that hemisphere is exactly half of the sphere hence, its surface area will be

**Surface area(Hemisphere)** = Surface area of half of the sphere + Area of the circle on the bottom of hemisphere

= \(2 \pi r^2 + \pi r^2\)

= \(3 \pi r^2\)

**The Volume of a Hemisphere**

Let ‘r’ be the radius of the sphere, then

Volume of sphere = \(\frac{4}{3} \pi r^3\)

Volume of Hemisphere Formula = \(\frac{Volume of sphere}{2}\)

=\( \frac{\frac{4}{3} \pi r^3}{2}\)

So, **Volume of a hemisphere formula** = \(\frac{2}{3} \pi r^3\)

If one scoops out the inner of the hemisphere, it forms the hollow hemisphere.

The total Surface area(TSA) of the hollow hemisphere = Surface area of Hemisphere + surface area of inner surface – overlap

TSA (given **R** and **r** are outer and inner radii of the hollow hemisphere)

= \((2\pi R^2 + \pi R^2) + (2\pi r^2) – (\pi r^2)\)

= \(3\pi R^2 + \pi r^2\)

The **volume of the **hollow Hemisphere = Volume of the total hemisphere – Volume of the removed hemisphere

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

= \(\frac{2}{3} \pi (R^3 – r^3)\)

**Problems**

Q. **Find the volume of the Hemisphere with a radius 3cm.**

**Sol**. Volume = \(\frac{2}{3} \pi r^3\)

= \(\frac{2}{3} \pi (3)^3\)

= \(18\pi\)

Q. **If the volume of a hemisphere is \(6174\pi\), find its radius.**

Sol. Volume = \(\frac{2}{3} \pi r^3\)

=> \(6174\pi = \frac{2}{3} \pi r^3\)

Bringing 2/3 to the left

=> \(6174 \cdot \frac{3}{2} = r^3 \)

=> \(9261 = r^3 \)

We know that cube root of 9261 is 21.

Thus \( r = 21 cm \)

**FAQs**

**What is a Hemisphere?**When a plane cuts a sphere into two halves passing by its center, it forms two hemispheres.

**What is the volume of the Hemisphere?**Volume = \(\frac{2}{3} \pi r^3\)

**What is the volume of the hollow hemisphere?**Volume = \(\frac{2}{3} \pi (R^3 – r^3)\)

**What is the surface area of the hemisphere?**The surface area of the hemisphere = \(3\pi R^2\)

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