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GED Math : 3-Dimensional Geometry

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Example Questions

Example Question #1 : Volume Of A Sphere

A water tank takes the shape of a sphere whose exterior has radius 18 feet; the tank is three inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

Use 3.14 for $\pi$ .

Possible Answers:

$3,800 \textrm{ ft}^{3}$

$22,400 \textrm{ ft}^{3}$

$23,400 \textrm{ ft}^{3}$

$4,000 \textrm{ ft}^{3}$

Correct answer:

$23,400 \textrm{ ft}^{3}$

Explanation:

Three inches is equal to 0.25 feet, so the radius of the interior of the tank is

18 - 0.25 = 17.75 feet.

The amount of water the tank holds is the volume of the interior of the tank, which is

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

$V \approx \frac{4}{3} \cdot 3.14 \cdot 17.75^{3} \approx 23,413$ ,

which rounds to 23,400 cubic feet.

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Example Question #1 : 3 Dimensional Geometry

The contents of a full spherical glass 3 inches in radius are poured into an empty cylindrical glass 6 inches in radius and 6 inches high. What percent of the cylindrical glass is taken up by the contents?

Possible Answers:

$33\frac{1}{3} \%$

$8\frac{1}{3} \%$

$12\frac{1}{2} \%$

$16 \frac{2}{3} \%$

Correct answer:

$16 \frac{2}{3} \%$

Explanation:

The volume of the spherical glass is

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

where r = 4 :

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

$V = \frac{4}{3} \pi \cdot 27$

$V = 36 \pi$

The volume of the cylindrical glass is

$V = \pi r^{2}h$ ,

where r = 6, h = 6 :

$V = \pi \cdot 6^{2} \cdot 6$

$V = \pi \cdot 36 \cdot 6$

$V = 216 \pi$

The contents of the spherical glass will take up

$\frac{36 \pi } {216 \pi} \times 100 \% = 16 \frac{1}{6} \%$

of the capacity of the cylindrical glass.

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Example Question #2 : 3 Dimensional Geometry

The contents of a full cylindrical glass 4 inches in radius and 8 inches high are poured into an empty spherical glass 6 inches in radius. What percent of the spherical glass is taken up by the contents?

Possible Answers:

$44 \frac{4}{9} \%$

$66 \frac{2}{3} \%$

$100 \%$

$88 \frac{8}{9} \%$

Correct answer:

$44 \frac{4}{9} \%$

Explanation:

The volume of the cylindrical glass is

$V = \pi r^{2}h$ ,

where r = 4, h = 8 :

$V = \pi \cdot 4^{2} \cdot 8$

$V = \pi \cdot 16 \cdot 8$

$V = 128 \pi$

The volume of the spherical glass is

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

where r = 6 :

$V = \frac{4}{3} \pi \cdot 6^{3}$

$V = 216 \cdot \frac{4}{3} \pi$

$V = 288 \pi$

The contents of the cylindrical glass will take up

$\frac{128 \pi} {288 \pi} \times 100 \% = 44 \frac{4}{9} \%$

of the capacity of the spherical glass.

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Example Question #1 : 3 Dimensional Geometry

Find the volume of the sphere with a diameter of 5.

Possible Answers:

$200\pi$

$\frac{125}{6}\pi$

$\frac{125}{3}\pi$

$\frac{200}{3}\pi$

$75\pi$

Correct answer:

$\frac{125}{6}\pi$

Explanation:

Write the formula for the volume of a sphere.

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

Substitute the radius, which is half the diameter.

$r=\frac{5}{2}$

$V=\frac{4}{3}\pi (\frac{5}{2})^3 = \frac{4}{3}\pi (\frac{5}{2})(\frac{5}{2})(\frac{5}{2})= \frac{1}{3}\pi (\frac{5}{1})(\frac{5}{1})(\frac{5}{2})$

Simplify the terms.

The answer is: $\frac{125}{6}\pi$

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Example Question #1 : 3 Dimensional Geometry

Determine the volume of a sphere with a radius of 5.

Possible Answers:

$\frac{100}{3}\pi$

$\frac{250}{3}\pi$

$100\pi$

$\frac{1000}{3}\pi$

$\frac{500}{3}\pi$

Correct answer:

$\frac{500}{3}\pi$

Explanation:

Write the formula for the volume of a sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}\pi (5)^3 = \frac{500}{3}\pi$

The answer is: $\frac{500}{3}\pi$

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Example Question #4 : Volume Of A Sphere

Determine the volume of a sphere if the diameter is $\sqrt[3]{5}$ .

Possible Answers:

$\frac{20}{3}\pi$

$20\pi$

$\frac{200}{3}\pi$

$200\pi$

$\textup{The answer is not given.}$

Correct answer:

$\frac{20}{3}\pi$

Explanation:

Write the formula for the volume of the sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}\pi (\sqrt[3]{5})^3$

Any value cubed rooted that is raised to the power of three will leave only the integer behind.

$V=\frac{4}{3}\pi (5) = \frac{20}{3}\pi$

The answer is: $\frac{20}{3}\pi$

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Example Question #4 : 3 Dimensional Geometry

Determine the volume of a sphere with a diameter of 3.

Possible Answers:

$3\pi$

$\frac{9}{8}\pi$

$9\pi$

$\frac{3}{2}\pi$

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

Correct answer:

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

Explanation:

Given the diameter is 3, the radius is half the diameter, or $\frac{3}{2}$ .

Write the formula for the volume of a circle.

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

Substitute the radius and solve.

$V= \frac{4}{3} \pi (\frac{3}{2})^3=\frac{4}{3} \pi (\frac{27}{8})$

Reduce the fractions.

The answer is: $\frac{9}{2}\pi$

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Example Question #5 : Volume Of A Sphere

Determine the volume of a sphere if the diameter is $2\pi^2$ .

Possible Answers:

$\frac{4}{3}\pi^9$

$\frac{4}{3}\pi^{10}$

$\frac{4}{3}\pi^6$

$\frac{4}{3}\pi^8$

$\frac{4}{3}\pi^7$

Correct answer:

$\frac{4}{3}\pi^7$

Explanation:

Write the formula for the volume of a sphere.

$V_{sphere} = \frac{4}{3}\pi r^3$

The radius is half the diameter.

$r= \frac{d}{2} =\frac{ 2\pi^2}{2}=\pi^2$

Substitute the radius.

$V_{sphere} = \frac{4}{3}\pi(\pi^2)^3=\frac{4}{3}\pi(\pi^2)(\pi^2)(\pi^2)$

The answer is: $\frac{4}{3}\pi^7$

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Example Question #1 : Volume Of A Sphere

Find the volume of a sphere if the radius of the sphere is $3\pi^3$ .

Possible Answers:

$36\pi^{28}$

$36\pi^{7}$

$36\pi^{10}$

$4\pi^{10}$

$36\pi^{27}$

Correct answer:

$36\pi^{10}$

Explanation:

Write the volume formula for the sphere.

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

Substitute the radius into the formula.

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

Simplify the terms.

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

The answer is: $36\pi^{10}$

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Example Question #8 : Volume Of A Sphere

Let $\pi = 3.14$

A sphere has a radius of 3in. Find the volume.

Possible Answers:

$25.12\text{in}^3$

$28.26\text{in}^3$

$113.04\text{in}^3$

$84.78\text{in}^3$

$339.12\text{in}^3$

Correct answer:

$113.04\text{in}^3$

Explanation:

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

We know $\pi = 3.14$ .

We know the radius of the sphere is 3in.

Now, we can substitute. We get

$V = \frac{4}{3} \cdot 3.14 \cdot (3\text{in})^3$

$V = \frac{4}{3} \cdot 3.14 \cdot 27\text{in}^3$

Now, we can simplify to make things easier. The 3 and the 27 can both be divided by 3. So, we get

$V = \frac{4}{1} \cdot 3.14 \cdot 9\text{in}^3$

$V = 4 \cdot 3.14 \cdot 9\text{in}^3$

$V = 4 \cdot 9\text{in}^3 \cdot 3.14$

$V = 36\text{in}^3 \cdot 3.14$

$V = 113.04\text{in}^3$

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GED Math : 3-Dimensional Geometry

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #1 : Volume Of A Sphere

Example Question #1 : 3 Dimensional Geometry

Example Question #2 : 3 Dimensional Geometry

Example Question #1 : 3 Dimensional Geometry

Example Question #1 : 3 Dimensional Geometry

Example Question #4 : Volume Of A Sphere

Example Question #4 : 3 Dimensional Geometry

Example Question #5 : Volume Of A Sphere

Example Question #1 : Volume Of A Sphere

Example Question #8 : Volume Of A Sphere

All GED Math Resources

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