Area of Hemisphere

A hemisphere is a shape in geometry that forms the foundation of domes, globes, bowls, and even stadium roofs. It is half of a sphere, but it comes with a unique set of properties and surface area formulas that are crucial for theory as well as real-world problem solving.

In this article, we will break down everything you need to know about the area of the hemisphere, along with the formula for surface area, step-by-step examples, and more.

1.0What is a Hemisphere?

The term “hemisphere” comes from the Greek roots: hemi (half) + sphaira (sphere), meaning half of a sphere. Imagine a ball or an orange, and you slice it right down the middle. What you have left are two hemispheres.

There are two parts in a hemisphere:

• Curved Surface - The dome-shaped, rounded outer part.
• Flat Base - The flat, circular face where the sphere was “cut.”

The radius (r) is the distance between the centre of the flat base and the point on the curved surface. The surface area of a hemisphere includes the following characteristics:

• The curved surface area (just the dome)
• Total surface area (dome plus the base)

2.0Surface Area of a Hemisphere

There are two things that you need to think about when you calculate the surface area of a hemisphere: whether you are calculating the surface area of the outer dome or the entire exterior, including the base. 

Curved Surface Area (CSA)

Curved surface area of a hemisphere refers to the area of the outer dome minus the base.

The formula for CSA is:

Curved Surface Area = 2πr²

Total Surface Area (TSA)

The total surface area of a hemisphere covers the curved surface and the flat base.

The formula for TSA is:

Total Surface Area = 3πr²

Since the hemisphere is half a sphere, both these formulas come from the formula of the surface area of a sphere, which is 4πr².

3.0How to Calculate the Surface Area of a Hemisphere

To calculate the surface area, you need to follow these simple steps:

• Find the radius

You will either be given the radius or the diameter. In the case of the latter, just divide it by 2.

• Choose the right formula

Use the curved surface area of the hemisphere if only the dome is involved. The total surface area of the hemisphere is curved, and the flat base is included.

• Put the values into the formula

Insert the value of r into the formula, simplify, and solve.

4.0Units of Surface Area

The surface area of the hemisphere is measured in square units. It can be in m², cm², mm², etc. It all depends on the unit of your radius.

5.0Solved Examples: Area of a Hemisphere

Problem 1:

A metal bowl has a radius of 5 cm. What is the total surface area of the bowl if it has a lid covering the flat base?

Solution:

The bowl has a curved surface as well as a flat base, so we will be calculating the total surface area:

TSA = 3πr²

= 3 x π x 5²

= 3 x π x 25

= 75 x 3.14 

= 235.5 cm²

Answer: The total surface area is 235.5 cm².

Problem 2:

A decorative dome has a curved surface area of 452.16 m². What is the radius of the dome?

Solution:

We have given the curved surface area.

CSA = 2πr²

452.16 = 2πr²

r² = $\frac{452.16}{2\pi }$

r² = 72

r = $\sqrt{72}$

r = 8.49 m

Answer: The radius of the decorative dome is 8.49 m.

Problem 3:

A toy manufacturer is painting the outer dome of a hemisphere-shaped toy with a radius of 4 cm. If 1 cm² of paint costs Rs. 2, what will be the cost to paint 50 such toys?

Solution:

To calculate the cost, we need the curved surface area for one toy:

CSA = 2πr²

= 2π x 4²

= 32π

CSA = 100.48 cm²

This brings the paint cost per toy to:

= 100.48 x 2

= Rs. 200.96

So, the cost for 50 toys will be:

= 200.96 x 50

= Rs. 10,048

Answer: The total cost of the painting will be Rs. 10,048.