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Volume of Sphere

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Volume of Spheres

Have you ever calculated volume? Take a look at this dilemma.

Maria has a paperweight that is a glass sphere. The sphere is full of red, sparkly liquid. If the diameter of the paperweight is 6 inches, how much red liquid does it contain?

You will know how to solve this dilemma by the end of the Concept.

Guidance

Volume is the measure of three-dimensional space a figure takes up. We can also think of it as how much space the figure “holds.”

For spheres, finding volume is a bit complicated because it doesn’t have any flat surfaces.

To find the volume of spheres, we can use pyramids. Imagine a pyramid with its base on the surface of the sphere and its point as the center of the sphere. The radius of the sphere would be the height of the pyramid.

The pyramid makes up a portion of the sphere’s volume. If we can fill the whole sphere with pyramids like this, we would know the volume of the sphere. It would be equal to the volumes of all the pyramids put together. How many pyramids would it take to fill a sphere? That depends on the surface area of the sphere. We can combine the surface area of a sphere with the volume formula for a pyramid to calculate the volume of all the pyramids contained within the sphere.

Let’s look at how this information can give us the formula for finding the volume of a sphere.

$V = \frac{1}{3} Bh$

Volume formula for a pyramid, where $B$ represents the area of its base

$V = \frac{1}{3} \times \text{surface area of sphere} \times r$

The surface area of the sphere is equal to the area of the bases of all the pyramids. The height of the pyramid is equal to the radius of the sphere, so we substitute $r$ for $h$ .

$V = \frac{1}{3} \times 4 \pi r^2 \times r$

We can simplify the formula by combining like terms.

$V = \frac{4}{3} \pi r^3$

The formula for finding the volume of a sphere is $V = \frac{4}{3} \pi r^3$ .

Write this formula down in your notebook.

Again, all we need to know is the radius of the sphere. We put the value in for $r$ in the formula and solve for $V$ , the volume.

Let’s try it.

Find the volume of the sphere below.

We know that the radius of the sphere is 6 meters, so we put this value in for $r$ and solve.

$V & = \frac{4}{3} \pi r^3\\V & = \frac{4}{3} \pi (6^3)\\V & = \frac{4}{3}(216) \pi\\V & = 288 \pi$

We can leave the volume as $288 \pi$ , or we can use 3.14 to approximate an answer. This gives us $288 \times 3.14 = 904.32 \ cubic \ meters$ . Remember, we measure volume in three dimensions, so we use cubic units.

Find the volume of each sphere.You may round to the nearest hundredth when necessary.

Example A

A sphere with a radius of 4 inches.

Solution: $267.95 \ in^3$

Example B

A sphere with a radius of 5 ft.

Solution: $523.33 \ ft^3$

Example C

A sphere with a radius of 3.5 inches.

Solution: $179.50 \ in^3$

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

First of all, what is the problem asking us to find? We need to find how much liquid the paperweight contains. This amount will be the volume of the paperweight, so we’ll need to use the volume formula. Now let’s see if we know the radius of the paperweight. We know that the diameter is 6 inches. Therefore the radius is $6 \div 2 = 3 \ inches$ . Now we can put this into the volume formula and solve.

$V & = \frac{4}{3} \pi r^3\\V & = \frac{4}{3} \pi (3^3)\\V & = \frac{4}{3} (27) \pi\\V & = 36 \pi$

The sphere has a volume of $36 \pi$ . We can approximate a numeric value if we use 3.14 for pi. This gives us a volume of $36 \times 3.14 = 113.04$ cubic inches.

Vocabulary

Sphere
a perfectly round solid figure where all of the points of it are equidistant from a center point.
Volume
the amount of capacity contained inside a solid figure.

Guided Practice

Here is one for you to try on your own.

Find the volume of the following sphere.

Solution

Let's use the formula for finding the volume of a sphere with the given value for the radius.

$V & = \frac{4}{3} \pi r^3\\V & = \frac{4}{3} \pi (8^3)\\V & = \frac{4}{3} (512) \pi\\V & = 682.67 \pi$

Practice

Directions: Find the volume of each sphere. You may round to the nearest hundredth when necessary.

1. A sphere with a radius of 3 m.
2. A sphere with a radius of 2.5 m.
3. A sphere with a radius of 5 in.
4. A sphere with a radius of 6 in.
5. A sphere with a radius of 7 ft.
6. A sphere with a radius of 4.5 cm.
7. A sphere with a radius of 5.5 m.
8. A sphere with a radius of 13 mm.
9. A sphere with a diameter of 8 in.
10. A sphere with a diameter of 10 ft.
11. A sphere with a diameter of 3 m.
12. A sphere with a diameter of 13 m.
13. A sphere with a diameter of 22 ft.

Directions: Use what you have learned to solve each problem.

1. A sphere has a diameter of 12 feet. What is its volume?
2. Kelly has a perfume bottle in the shape of a sphere. The diameter of the bottle is 6 inches. How much perfume does Kelly have left if the bottle is only half full?