How to find the volume of a sphere

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Geometry › How to find the volume of a sphere

Questions 1 - 10

1

What is the volume of a sphere with a diameter of 6 in?

$36\pi\ in^{3}$

$216\pi\ in^{3}$

$108\pi\ in^{3}$

$72\pi\ in^{3}$

$288\pi\ in^{3}$

Explanation

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

2

A sphere is cut in half and is placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 7^2 \times 15=735\pi$

Next, find the volume of the half sphere.

$\text{Volume of Half sphere}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius}^3)=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 7^3=\frac{686}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=735\pi + \frac{686}{3}\pi=3027.45$

Remember to round to places after the decimal.

3

True or false: A sphere with radius 1 has volume $\frac{4}{3} \pi$ .

True

False

Explanation

Given radius , the volume of a sphere can be calculated according to the formula

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

Set :

$V = \frac{4}{3} \pi \cdot 1 ^{3}= \frac{4}{3} \pi \cdot 1= \frac{4}{3} \pi$

The statement is true.

4

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(3)^3=18\pi=56.55$

Make sure to round to places after the decimal.

5

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(4)^3=\frac{128}{3}\pi=134.04$

Make sure to round to places after the decimal.

6

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\frac{1}{2}$ , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(\frac{1}{2})^3=\frac{1}{12}\pi=0.26$

Make sure to round to places after the decimal.

7

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(5)^3=\frac{250}{3}\pi=261.80$

Make sure to round to places after the decimal.

8

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(6)^3=144\pi=452.39$

Make sure to round to places after the decimal.

9

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is , what is the volume of the figure?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by .

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(7)^3=\frac{686}{3}\pi=718.38$

Make sure to round to places after the decimal.

10

A sphere with a radius of $\frac{8}{3}$ is cut out of a cube that has a side length of . What is the volume of the resulting figure?

Explanation

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{8}{3})^3=\frac{2048}{81}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=6^3=216$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=216-\frac{2048}{81}\pi=136.57$

Make sure to round to places after the decimal.

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