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ISEE Upper Level Math : How to find the volume of a sphere

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Example Questions

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

In terms of $\pi$ , give the volume, in cubic inches, of a spherical water tank with a diameter of 20 feet.

Possible Answers:

$230,400 \pi\textrm{ in}^{3}$

$57,600 \pi\textrm{ in}^{3}$

$307,200\pi\textrm{ in}^{3}$

$18,432,000\pi\textrm{ in}^{3}$

$2,304,000 \pi \textrm{ in}^{3}$

Correct answer:

$2,304,000 \pi \textrm{ in}^{3}$

Explanation:

20 feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set r = 120 , substitute in the volume formula, and solve for :

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

$V = \frac{4}{3} \pi \cdot 120^{3}$

$V = \frac{4}{3} \pi \cdot 1,728,000$

$V = 2,304,000 \pi$ cubic inches

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Example Question #2 : How To Find The Volume Of A Sphere

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

Possible Answers:

$27,000,000\pi \textrm{ cm}^{3}$

$1,125,000\pi \textrm{ cm}^{3}$

$3,375,000\pi \textrm{ cm}^{3}$

$4,500,000\pi \textrm{ cm}^{3}$

$36,000,000\pi \textrm{ cm}^{3}$

Correct answer:

$4,500,000\pi \textrm{ cm}^{3}$

Explanation:

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute r = 150 in the volume formula:

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

$V= \frac{4}{3} \cdot \pi \cdot 150^{3}$

$V= \frac{4}{3} \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

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Example Question #1 : Spheres

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

Possible Answers:

$166\frac{2}{3}\pi m^3$

$16\frac{2}{3}\pi m^3$

$165 \pi m^3$

$166\frac{2}{3} m^3$

Correct answer:

$166\frac{2}{3}\pi m^3$

Explanation:

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

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

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

All we have to do is plug in 5 meters and simplify:

$V_{sphere}=\frac{4}{3}\pi (5m)^3=\frac{4}{3}\pi*125m^3=166\frac{2}{3}\pi m^3$

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Example Question #2 : Spheres

You have a ball with a radius of 12 cm, what is its volume?

Possible Answers:

$192 \pi cm^3$

$404 \pi cm^3$

$2304 \pi cm^3$

$144 \pi cm^3$

Correct answer:

$2304 \pi cm^3$

Explanation:

You have a ball with a radius of 12 cm, what is its volume?

The volume of a sphere can be found via the following formula:

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

We know our radius, so all we need to do is plug in and simplify:

$V_{sphere}=\frac{4}{3}\pi (12cm)^3=2304 \pi cm^3$

So we have our answer:

$2304 \pi cm^3$

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Example Question #3 : Spheres

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

Possible Answers:

$2304\pi in^3$

$1728\pi in^3$

2304 in^3

$576 \pi in^3$

Correct answer:

$2304\pi in^3$

Explanation:

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

Alright, let's begin with the volume of a sphere formula:

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

Now, plug in 12 and simplify:

$V_{sphere}=\frac{4}{3}\pi(12in)^3=\frac{4}{3}\pi*1728in^3=2304\pi in^3$

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Example Question #4 : Spheres

Find the volume of a sphere with a diameter of 6cm.

Possible Answers:

$42\pi \text{ cm}^3$

$36\pi \text{ cm}^3$

$108\pi \text{ cm}^3$

$24\pi \text{ cm}^3$

$216\pi \text{ cm}^3$

Correct answer:

$36\pi \text{ cm}^3$

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.

Now, we know the diameter of the sphere is 6cm. We also know that the diameter is two times the radius. Therefore, the radius is 3cm.

Knowing this, we can substitute into the formula. We get

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

$V = \frac{4}{3} \cdot \pi \cdot 3\text{cm} \cdot 3\text{cm} \cdot 3\text{cm}$

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

Now, before we continue, we can simplify. The 3 and the 27 can both be divided by 3. We get

$V = \frac{4}{1} \cdot \pi \cdot 9\text{cm}^3$

$V = 4 \cdot \pi \cdot 9\text{cm}^3$

$V = 36\pi \text{ cm}^3$

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Example Question #7 : How To Find The Volume Of A Sphere

Find the volume of a sphere with a diameter of 12in.

Possible Answers:

$96\pi \text{ in}^3$

$288\pi \text{ in}^3$

$144\pi \text{ in}^3$

$200\pi \text{ in}^3$

$256\pi \text{ in}^3$

Correct answer:

$288\pi \text{ in}^3$

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.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in.

Knowing this, we can substitute into the formula. We get

$V = \frac{4}{3} \cdot \pi \cdot (6\text{in})^3$

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

Now, we can simplify before we multiply to make things easier. The 3 and a 6 can both be divided by 3. So, we get

$V = \frac{4}{1} \cdot \pi \cdot 2\text{in} \cdot 6\text{in} \cdot 6\text{in}$

$V = 4 \cdot \pi \cdot 72\text{in}^3$

$V = 288\pi \text{ in}^3$

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

Find the volume of a sphere with a diameter of 18in.

Possible Answers:

$288\text{in}^3$

$972\text{in}^3$

$594\text{in}^3$

$324\text{in}^3$

$108\text{in}^3$

Correct answer:

$972\text{in}^3$

Explanation:

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

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 18in. We also know the diameter is two times the radius. Therefore, the radius is 9in.

Knowing this, we can substitute into the formula. We get

$V = \frac{4}{3} \cdot \pi \cdot (9\text{in})^3$

$V = \frac{4}{3} \cdot \pi \cdot 9\text{in} \cdot 9\text{in} \cdot 9\text{in}$

Now, we can simplify before we multiply to make things easier. The 3 and a 9 can both be divided by 3. So, we get

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

$V = 4 \cdot \pi \cdot 243\text{in}^3$

$V = 972\pi \text{ in}^3$

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Example Question #393 : Geometry

Find the volume of a sphere with a diameter of 6cm.

Possible Answers:

$72\pi \text{ cm}^3$

$63\pi \text{ cm}^3$

$36\pi \text{ cm}^3$

$54\pi \text{ cm}^3$

$108\pi \text{ cm}^3$

Correct answer:

$36\pi \text{ cm}^3$

Explanation:

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

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 6cm. We also know the diameter is two times the radius. Therefore, the radius is 3cm.

So, we get

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

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

$V = \frac{4}{3} \cdot \frac{27\text{cm}^3}{1} \cdot \pi$

$V = \frac{4}{1} \cdot \frac{9\text{cm}^3}{1} \cdot \pi$

$V = \frac{4 \cdot 9\text{cm}^3}{1 \cdot 1} \cdot \pi$

$V = \frac{36\text{cm}^3}{1} \cdot \pi$

$V = 36\pi \text{ cm}^3$

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

Find the volume of a sphere with the radius of $\frac{1}{4}$ .

Possible Answers:

$\frac{1}{9}\pi$

$\frac{1}{72}\pi$

$\frac{1}{48}\pi$

$\frac{1}{24}\pi$

$\frac{1}{96}\pi$

Correct answer:

$\frac{1}{48}\pi$

Explanation:

Write the formula to find the volume of a sphere.

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

Substitute the radius into the formula.

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

Evaluate the equation.

$V=\frac{4}{3}\pi (\frac{1}{64}) = \frac{1}{3}\pi (\frac{1}{16}) = \frac{1}{48}\pi$

The answer is: $\frac{1}{48}\pi$

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Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

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ISEE Upper Level Math : How to find the volume of a sphere

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

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

Example Question #2 : How To Find The Volume Of A Sphere

Example Question #1 : Spheres

Example Question #2 : Spheres

Example Question #3 : Spheres

Example Question #4 : Spheres

Example Question #7 : How To Find The Volume Of A Sphere

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

Example Question #393 : Geometry

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

All ISEE Upper Level Math Resources

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