How to find the volume of a sphere

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Math › How to find the volume of a sphere

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is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

In terms of $\pi$ , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

$\frac{9}{2} \pi \textrm{ ft}^{3}$

$9 \pi \textrm{ ft}^{3}$

$18 \pi \textrm{ ft}^{3}$

$36 \pi \textrm{ ft}^{3}$

$3 \pi \textrm{ ft}^{3}$

Explanation

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ , substitute in the volume formula, and solve for :

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

$V = \frac{4}{3} \pi\left \cdot \left( \frac{3}{2} \right ) ^{3}$

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

Find the volume of the following sphere.

Sphere

$288 \pi m^3$

$270 \pi m^3$

$300 \pi m^3$

$312 \pi m^3$

$360 \pi m^3$

Explanation

The formula for the volume of a sphere is:

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

where is the radius of the sphere.

Plugging in our values, we get:

$V=\frac{4}{3} \pi (6m)^3$

$V=\frac{4}{3} \pi 216m^3 = 288 \pi m^3$

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.

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.

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.

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.

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.

Find the volume of the following sphere.

Sphere

$972 \pi m^3$

$982 \pi m^3$

$992 \pi m^3$

$962 \pi m^3$

$952 \pi m^3$

Explanation

The formula for the volume of a sphere is:

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

Where is the radius of the sphere

Plugging in our values, we get:

$V = \frac{4 \pi (9m)^3}{3}$

$V = 972 \pi m^3$

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.

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