# Volume of Hemisphere

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.

Index

## 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: x2 + y2 + z2 = r2

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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