Surface Area of Spheres Examples

A big cantaloupe has a diameter of about 26 inches. How much surface area does its peel cover? 

If we know the diameter of the cantaloupe, we can find its radius faster than a Hufflepuff. It's 13 inches. Take that, Cedric Diggory. Now we can use our surface area formula.

SA = 4πr²
SA = 4π(13 in)²
SA ≈ 2124 in²

That's all it takes.

You ate half the cantaloupe and now you're full. How much saran wrap will you need to cover the rest of the cantaloupe and save it for later? 

The radius is 13 inches, but that's old news. The surface area of the entire cantaloupe is half of the outer peel and the midsection where you cut it (the great circle). That means our formula looks like this:

SA = 2πr² + πr² = 3πr²

We know the radius. What are we waiting for?

SA = 3π(13 in)²
SA ≈ 1593 in²

That is, of course, assuming we want to cover the peel of the cantaloupe, too.

Find the surface area of the solid.

This wondrous object is a cylinder glued to a hemisphere. If we imagine having to wrap it up, we need to cover the outside of the hemisphere, the lateral surface of the cylinder, and the base of the cylinder.

SA = 2πr² + 2πrh + πr² = 3πr² + 2πrh

We don't have to worry about different radii because the radius of the cylinder's base is the same as the radius of the hemisphere. The height of the cylinder is 7 inches, and the radius is 3 inches.

SA = 3π(3 in)² + 2π(3 in)(7 in)
SA = 27π + 42π = 69π
SA ≈ 216.8 in²

Done and done.

As part of your pottery class, you decide to make a bowl in the shape of a hemisphere. It's a perfect hemisphere, so clearly you rock at pottery. If you want to paint the entire thing blue, how much surface area will you cover?

The bowl is actually made up of two hemispheres: an inner one and an outer one. The surface area of the whole bowl is the surface area of the inner and outer hemispheres plus the rim of the bowl.

You're probably thinking, 'Hold the iPhone! We never said anything about this rim business! What's all that about?' Stay with us and you'll be holding your own iPhone in no time.

All we need to do is subtract the area of the little circle from the big circle. We end up with a surface area equation that looks something like this, if r_o is the radius of the outer hemisphere and r_i is the radius of the inner hemisphere.

SA = 2πr_o² + 2πr_i² + (πr_o² – πr_i²)

Make sense? Good. Now we can plug in our radii.

SA = 2π(10 in)² + 2π(8 in)² + π(10 in)² – π(8 in)²

Solve.

SA = 200π + 128π + 100π – 64π = 364π
SA ≈ 1144 in²

And that's for one layer of paint.