Geometry

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# Surface Area of a Hemisphere

## Hemisphere Surface Area Formula

The formula for total surface area of a hemisphere is given as:**SA = 3πr ^{2}**

Where

*SA*is the surface area and

*r*is the radius.

This formula calculates the

*total*surface area of a hemisphere, which includes the top dome

*and*the bottom circle. To only calculate the top dome surface area, use:

SA = 2πr

^{2}.

### Derivation of the Formula

To derive the formula for surface area of a hemisphere, we simply divide the sphere surface area formula by 2 and then add in the area of the circle at its bottom. The full formula for surface area is derived by the following steps:

1.) Surface area of a sphere: SA = 4πr^{2}

2.) Area of a circle: πr^{2}

3.) Surface area of a hemisphere: SA = 2πr^{2} + πr^{2}

4.) Simplifying this gives us the following:

5.) SA = 3πr^{2}

Note how the surface area of a sphere is divided by two before being added to the circle area. This process allows us to account for all exterior surfaces of the hemisphere. If we ever must solve for surface area of a hemisphere but cannot remember the exact hemisphere formula, we can follow the method shown above to get the correct solution.

### What is the use of Finding Surface Area of a Hemisphere?

Other than being tasked directly with finding the surface area of a hemisphere (also called a semisphere or a half sphere), there are many real-world situations where our formula will be useful.

An example is when a painting company is hired to paint the inside of a dome building. They need to know what the interior wall and floor surface area are so they can purchase the correct amount of paint. By determining the radius of the inside hemisphere shape of the dome building, they can quickly determine surface area and therefore how much paint to buy.

### Example Problem

Find the total surface area of a hemisphere with a radius of 5.

Solution:

1.) Let’s plug the radius value 5 into the formula SA = 3πr^{2}.

2.) SA = 3π(5)^{2} = 3π(25) = 75π

3.) **The surface area of the hemisphere is 75π.**