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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Spheres

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

Possible Answers:

$\frac{196\pi}{3}$

$\frac{64\pi}{3}$

$48\pi$

$\frac{2048\pi}{3}$

$\frac{256\pi}{3}$

Correct answer:

$\frac{256\pi}{3}$

Explanation:

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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Example Question #1 : Spheres

Possible Answers:

$\frac{1256}{3}\; \pi \; cm^3$

$98\pi\; cm^3$

$\frac{1372}{3}\; \pi \; cm^3$

$\frac{218}{3}\pi\; cm^3$

$\frac{196}{3}\pi\; cm^3$

Correct answer:

$\frac{1372}{3}\; \pi \; cm^3$

Explanation:

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

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

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Example Question #134 : Solid Geometry

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

Possible Answers:

$108\pi\ in^{3}$

$288\pi\ in^{3}$

$216\pi\ in^{3}$

$36\pi\ in^{3}$

$72\pi\ in^{3}$

Correct answer:

$36\pi\ in^{3}$

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.

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Example Question #1 : How To Find The Volume Of A Sphere

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

Possible Answers:

$10,403\:cm^3$

$82,488\:cm^3$

$10,308\:cm^3$

$9,983\:cm^3$

$10,306\:cm^3$

Correct answer:

$10,306\:cm^3$

Explanation:

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:

$C= \pi \cdot d$

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

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

27 = d

Because the diameter is $27\:cm$ , the radius must be $13.5\:cm$ .

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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Example Question #5 : Spheres

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

Possible Answers:

$V=3,330\hspace{1mm}in^3$

$V=2,500\hspace{1mm}in^3$

$V=4,189\hspace{1mm}in^3$

$V=1,509\hspace{1mm}in^3$

$V=5,239\hspace{1mm}in^3$

Correct answer:

$V=4,189\hspace{1mm}in^3$

Explanation:

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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Example Question #2 : Spheres

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.

Possible Answers:

18.55

16.30

16.76

15.09

Correct answer:

16.76

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.

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

Make sure to round to places after the decimal.

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Example Question #3 : Spheres

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?

Possible Answers:

55.08

56.55

51.18

58.63

Correct answer:

56.55

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.

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Example Question #242 : Solid Geometry

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?

Possible Answers:

134.04

151.03

144.07

125.09

Correct answer:

134.04

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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Example Question #9 : How To Find The Volume Of A Sphere

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?

Possible Answers:

0.26

0.36

0.16

0.46

Correct answer:

0.26

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.

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Example Question #4 : Spheres

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?

Possible Answers:

288.09

230.36

212.20

261.80

Correct answer:

261.80

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.

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : Spheres

Example Question #1 : Spheres

Example Question #134 : Solid Geometry

Example Question #1 : How To Find The Volume Of A Sphere

Example Question #5 : Spheres

Example Question #2 : Spheres

Example Question #3 : Spheres

Example Question #242 : Solid Geometry

Example Question #9 : How To Find The Volume Of A Sphere

Example Question #4 : Spheres

All Intermediate Geometry Resources

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