How to find the volume of a sphere - Intermediate Geometry

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Question

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

Answer

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.

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Question

A sphere has a circumference of $8\pi$ , what is its volume?

Answer

The circumference is given by $2\pi r$ which yields a radius of 4. The volume is given by $\frac{4}{3}\pi r^{3}$

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Question

Given a sphere of radius 7cm, what is the volume of the sphere?

Answer

$\frac{4}{3}\cdot \pi \cdot r^3\rightarrow \frac{4}{3}\cdot \pi \cdot (7)^3$

Simplified, it is $\frac{1372}{3}; \pi ; cm^3$

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Question

What is the volume of a sphere with surface area 1,000 square centimeters?

Answer

Use the surface area formula to find the radius, then use the volume formula to find the volume.

$A=4\pi r^{2}$

$4\pi r^{2} =1,000$

$r^{2} =\frac{1,000}{4\pi }=\frac{250}{\pi }$

$r =\sqrt{\frac{250}{\pi }}$

$V=\frac{4}{3} \pi r^{3} = \frac{4}{3} \pi \left (\sqrt{\frac{250}{\pi }} \right )^{3} \approx 2,974$

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Question

The circumference of a sphere is $27 \pi :cm$ . What is the sphere's volume?

Answer

The formula to find the volume of a sphere is: $V=\frac{4}{3} \cdot \pi \cdot r^3$

Finding the volume is simple. All that we need is the radius!

The only information the problem provides is the circumference. In order to find the radius, we have to think how circumference relates to radius.

Since the equation for circumference is $C=\pi \cdot d$ , where d stands for diameter, and radius is half of the diameter, the two have diameter in common.

The first step in solving this problem is to determine the diameter from the circumference:

$C= \pi \cdot d$

$27 \cdot \pi = \pi \cdot d$

$\frac{27 \cdot \pi}{ \pi} = \frac{\pi \cdot d}{\pi}$

Because the diameter is , the radius must be .

Now we are ready to solve for the volume after substituting in our value.

$V= \frac{4}{3} \cdot \pi \cdot (13.5)^3$

$V= \frac{4}{3} \cdot \pi \cdot (2,460.375)$

$V = 3280.5 \cdot \pi$

$V =10,305.99 \approx10,306:cm^3$

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Question

A cube with a side length of 20 inches has a sphere inscribed within. What is the volume of the sphere?

Answer

The cube has a side length of 20 inches. Since the sphere is inscribed within the cube its diameter measures 20 inches; the radius will be 10 inches.

The volume of a sphere is given by

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

$V=\frac{4}{3}\pi (10)^3=\mathbf{4,189\hspace{1mm}in^3}$

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Question

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

If the radius of the sphere is , find the volume of the figure.

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(2)^3=\frac{16}{3}\pi=16.76$

Make sure to round to places after the decimal.

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Question

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?

Answer

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.

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Question

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?

Answer

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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Question

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?

Answer

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.

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Question

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?

Answer

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.

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Question

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?

Answer

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.

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Question

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?

Answer

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.

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Question

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?

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(8)^3=\frac{1024}{3}\pi=1072.33$

Make sure to round to places after the decimal.

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Question

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

If the radius of the sphere is $\frac{1}{3}$ , what is the volume of the figure?

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(\frac{1}{3})^3=\frac{2}{81}\pi=0.08$

Make sure to round to places after the decimal.

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Question

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?

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(9)^3=486\pi=1526.81$

Make sure to round to places after the decimal.

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Question

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?

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(10)^3=\frac{2000}{3}\pi=2094.40$

Make sure to round to places after the decimal.

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Question

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?

Answer

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.

$\text{Volume of Figure}=\frac{2}{3}\pi(11)^3=\frac{2662}{3}\pi=2787.64$

Make sure to round to places after the decimal.

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Question

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

Find the volume of the figure.

Answer

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:

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

$\text{Volume of Cylinder}=\pi\times 6^2 \times 12=432 \pi$

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.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 6^3=144\pi$

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

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

$\text{Volume of Figure}=432\pi + 144\pi=576\pi=1809.56$

Remember to round to places after the decimal.

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Question

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.

Answer

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.

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