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# Surface Area of Spheres

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Surface Area of Spheres

Have you ever made a pinata? Have you ever smashed one? Take a look at this dilemma.

The Olympics at Montgomery Middle School was a huge success. The students and guests had a terrific time and all decided that this event would be an annual event from now on.

For the final celebration, the students hosted a huge party in the gymnasium. Joe took on creating a piñata that students could enjoy. In fact, he made four of them for the party. When finished with the basic shape, Joe began decorating the sphere shaped piñata with green construction paper.

Each sphere has a radius of 2.4 feet and Joe made four piñatas. How much construction paper will he need to cover each piñata completely?

Figuring out the surface area of a sphere can be a tricky thing to figure out. For this problem, you will need to know how to solve for the surface area of a sphere. Pay close attention throughout this Concept and you will know how to solve the problem by the end of it.

### Guidance

A sphere is a solid figure that exists in three-dimensional space. Spheres consist of all the points that are equidistant from a center point. Every point on the sphere is the distance of the radius from the center. Spheres are perfectly round.

Did you know you can calculate the surface area of a sphere? Do you know how to do this?

Surface area is the total of the areas of each face in a solid figure.

Spheres do not have any faces because they are round. Still, we can think of its surface as a flat plane that we can unroll. Imagine you could wrap a sphere in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure exactly represents its surface area.

If we could “unroll” the sphere and show it as a rectangle, the rectangle would have a width that is equivalent to the diameter of the sphere. Its length would be the same as the circumference of the sphere (recall that circumference is the distance around a circle). Now this gives us something we can work with, because we can use the area formula for rectangles to find the area of the “unrolled” sphere.

The formula for the area of rectangles is \begin{align*}A = lw\end{align*}. In other words, we multiply the length and the width. Now let’s think about this in terms of the sphere. We have said that the length is the same as the circumference and that the width is the same as the diameter. This gives us \begin{align*}A = Cd\end{align*}. Now let’s substitute the formula for circumference in for \begin{align*}C\end{align*}, which is \begin{align*}2 \pi r\end{align*}, where \begin{align*}r\end{align*} is the radius. Now we have \begin{align*}A = 2 \pi rd\end{align*}. There’s one more substitution we can make to make the calculations easier. Remember, the diameter of a circle or sphere is exactly twice the radius. Let’s us \begin{align*}2r\end{align*} instead of \begin{align*}d\end{align*}. Now we have \begin{align*}A = 2 \pi r \times 2r\end{align*}. Finally, we can simplify this by multiplying to get \begin{align*}4 \pi r^2\end{align*}. Let’s sum all of this up:

The formula for finding the surface area of a sphere is \begin{align*}4 \pi r^2\end{align*}.

Write this formula down in your notebook.

All we need to do is substitute the measure of the radius for \begin{align*}r\end{align*} in the formula and solve for \begin{align*}SA\end{align*}, the surface area.

Let’s give it a try.

What is the surface area of the sphere below?

We can see that the radius of the sphere is 8, so we put this into the formula and solve.

Calculating numbers with pi is a bit complicated, because pi is actually a decimal number that goes on forever.

The most exact measure of the sphere’s surface area is to leave it as \begin{align*}256 \pi\end{align*}. However, we often round the decimal to 3.14 to represent pi. Then we would multiply 256 by 3.14 to get a surface area of 803.84 square centimeters for this sphere. Remember, we always use square units to measure area, because we measure area in two dimensions.

Find the surface area of each sphere.

#### Example A

A sphere with a radius of 5 inches.

Solution: \begin{align*}314 \ in^2\end{align*}

#### Example B

A sphere with a radius of 8 meters.

Solution: \begin{align*}803.84 \ m^2\end{align*}

#### Example C

A sphere with a radius of 12 feet.

Solution: \begin{align*}1808.64 \ ft^2\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

This problem is asking us about covering the surface of a piñata, so we will need to calculate its surface area. The problem tells us that the radius of the piñata is 2.4 feet, so we can put this into the surface area formula and solve.

The piñata has a surface area of 72.35 square feet when we approximate pi as 3.14, so this is how much paper Joe will need to cover one.

But Joe made four piñatas, so we need to multiply this answer by 4.

\begin{align*}72.35(4) = 289.4 \ sq. feet\end{align*}

### Vocabulary

Surface Area
the measurement of the outer covering of a solid figure.

### Guided Practice

Here is one for you to try on your own.

This sphere has a radius of 8.5 inches. What is the surface area of the sphere?

Solution

To figure this out, we can use the formula for finding the surface area of a sphere and substitute in the given value for radius.

### Practice

Directions: Find the surface area of each sphere. Use 3.14 to approximate pi.

1. A sphere with a radius of 4 in.
2. A sphere with a radius of 2 in.
3. A sphere with a radius of 3.5 ft.
4. A sphere with a radius of 6.7 in.
5. A sphere with a radius of 12 cm.
6. A sphere with a radius of 1.6 ft.
7. A sphere with a radius of 9 m.
8. A sphere with a diameter of 9 m.
9. A sphere with a diameter of 18 in.
10. A sphere with a diameter of 10 cm.
11. A sphere with a diameter of 12 m.
12. A sphere with a diameter of 13 ft.
13. A sphere with a diameter of 15 m.
1. What is the surface area of a sphere whose diameter is 22 centimeters?
2. Bruce is making a sculpture in his art class that is made of 3 spheres. Each sphere has a radius of 2.3 feet. He will paint them all with blue poster paint. If each bottle of paint covers 20 square feet, how many bottles will Bruce need to buy?

### Vocabulary Language: English

Surface Area

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.