Surface Area of a Hemisphere – Definition, Formula, Examples

What Is the Surface Area of a Hemisphere?

The surface area of a hemisphere is defined as the area covered by the outer surface of a hemisphere. The surface area of a hemisphere is measured in square units.

The surface of a hemisphere consists of a flat circular base and curved surface. 

A hemisphere is a three-dimensional shape that is half of a sphere. The hemisphere can either be hollow or solid.

A sphere is a three-dimensional shape. It has no edges. If you cut a sphere into exactly two halves by a plane, each half is called a hemisphere. Simply put, the hemisphere is half of a sphere.

You can see that a hemisphere has one curved surface and 1 flat surface. 

Surface Area of a Hemisphere: Formulas

Description	Formula
Base Area of Hemisphere	B $= \pi r^{2}$
Curved Surface Area of Hemisphere	CSA $= 2\pi r^{2}$
Total Surface Area of Hemisphere	TSA $= 3\pi r^{2}$

Curved Surface Area of Hemisphere

To calculate the total surface area of the sphere, we use the formula:

Total Surface Area of the sphere Formula = 4πr² 

The curved region of the hemisphere is exactly half of the curved region of the sphere. 

Surface area of half a sphere = Curved Surface Area of a Hemisphere

So, we simply need to divide the total surface area of the sphere by 2 to get the curved surface area of the hemisphere.

Curved Surface Area of Hemisphere Formula = 2πr²

where ‘r’ is the radius of the hemisphere

Base Area of Hemisphere

The base of the hemisphere is a flat circular shape. 

Area of circle = πr²

Area of the circular base of a hemisphere = πr²

Total Surface Area of Hemisphere

The total surface area of the hemisphere is defined as the sum of the space occupied by the curved area and the flat surface area (or the base area) of the hemisphere.

Thus, the total surface area of the hemisphere can be calculated using the following formula:

Total surface area of a hemisphere = Curved Surface Area + Base Area 

TSA = 2πr² + r² 

TSA = 3πr²

How to Find the Surface Area of a Hemisphere

Depending on whether we are calculating the curved surface area or the total surface area of a hemisphere, we have to choose the correct formulas.

Lets see steps of how to find curved surface area and total surface area of hemisphere.

Curved Surface Area of Hemisphere

Curved Surface Area of the hemisphere = 2πr² 

Example: Find the curved surface area of a hemisphere with radius 2 units.

Note down the radius (r) of the hemisphere.

r= 2 units

Substitute the value in the formula CSA = 2πr²

CSA = 2πr²

CSA $= 2 \times \frac{22}{7} 2^{2}$

CSA = 25.14 square units

Total Surface Area of Hemisphere

The formula to calculate the total surface area (TSA) of a hemisphere with radius ‘r’ is given as, TSA of hemisphere = 3πr² 

Example: Find the curved surface area of a hemisphere with radius 7 units.

r = 7 units

TSA of hemisphere = 3πr²

TSA of hemisphere $= 3 \times \frac{22}{7} 7^{2}$

TSA of hemisphere = 462 square units 

Surface Area of Hollow Hemisphere

A hollow hemisphere is a three-dimensional shape whose base looks like a flat ring. It is hollow on the inside.

A hollow hemisphere’s inner and outer circular bases can each have a different radius.
Because of this, a hollow hemisphere has two curved surfaces, such as an inner and outer curve. A hemisphere’s thickness is determined by the difference between its inner and outer radii.

Surface Area of a Hollow Hemisphere Formula and Derivation

If ‘r’ is the inner radius (radius of the inside circular base), and ‘R’ is the outer radius (radius of the outside circular base), we can derive the following formulas.

• The curved surface area of the outer hemisphere = 2πR²

• The curved surface area of the inner hemisphere = 2πr²

Observing the figure given below, we can see that a ring is formed at the base of a hollow hemisphere. 

Area of the ring = Area of outer hemisphere – area of inner hemisphere = (R² – r²)

So, the Total surface area of the hollow hemisphere can be calculated as

TSA of a hollow hemisphere = 2πR² + 2πr² + π(R² – r²)

TSA of hollow hemisphere = 2π(R² + r²) + (R² – r²)

TSA of hollow hemisphere = 3πR² + r²

Facts about Surface Area of Hemisphere

Conclusion

In this article, we learnt about the surface area of the hemisphere and hollow hemisphere. We also learnt the formulas for the same. Let’s use this knowledge to solve some examples and practice problems.

Solved Examples on Surface Area of a Hemisphere

Example 1: Find the surface area of the hemisphere if its radius is 14 units?

Solution: 

The radius of the hemisphere is 14 units.

We know that the total surface area of the hemisphere = 3πr²

 Thus, TSA $=  3\times \frac{22}{7} \times 14 \times 14$

                   $= 3 \times 22 \times 2 \times 14$

                   $= 1848$ square units.

Therefore, the total surface area of the hemisphere is 1848 square units.

Example 2: Find the surface area of a hollow hemisphere whose outer radius measures 20 units and inner radius is 10 units. (Use π = 3.14)

Solution:

The outer radius (R) of the hemisphere = 20 units

The inner radius (r) = 10 units

Substituting the value of R and r in the formula, we get

TSA of hollow hemisphere = 3πR² + r²

                                           = [3π × (20)²] + [π × (10)²]

                                           = (3 × 3.14 × 400) + (3.14 × 100)

                                           = 3768 + 314

                                           = 4082 square units

Therefore, the surface area of the hollow hemisphere = 4082 square units.

Example 3: Calculate the radius of a hemisphere if its curved surface area is 500 sq. in.

Solution:

A = 500

Curved surface area of hemisphere A = 2πr²

A =  2πr²

$r^{2} = \frac{A}{2\pi}$

$r^{2} = \frac{500}{2 \times 3.14}$

$r^{2} = \frac{250}{3.14}$

r² = 79.617

r = $\sqrt{79.617}$

r = 8.922

Thus, the radius of the hemisphere = 8.922 in.

Practice problems on Surface Area of a Hemisphere

Frequently asked questions about Surface Area of a Hemisphere