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ISEE Upper Level Math : Solid Geometry

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Example Questions

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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:

$576 \pi in^3$

$1728\pi in^3$

$2304\pi in^3$

2304 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 : How To Find The Volume Of A Sphere

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

Possible Answers:

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

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

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

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

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

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 #5 : How To Find The Volume Of A Sphere

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

Possible Answers:

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

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

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

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

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

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 #6 : How To Find The Volume Of A Sphere

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

Possible Answers:

$972\text{in}^3$

$324\text{in}^3$

$594\text{in}^3$

$108\text{in}^3$

$288\text{in}^3$

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 #7 : How To Find The Volume Of A Sphere

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

Possible Answers:

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

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

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

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

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

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 #8 : How To Find The Volume Of A Sphere

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

Possible Answers:

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

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

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

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

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

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

In terms of $\pi$ , give the surface area, in square feet, of a spherical tank with diameter 36 inches.

Possible Answers:

$36\pi \textrm{ ft}^{2}$

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

$18 \pi \textrm{ ft}^{2}$

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

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

Correct answer:

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

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 surface area formula, and solve for :

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

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

$A =\frac{ 4}{1} \cdot \frac{3}{2}\cdot \frac{3}{2} \cdot \pi$

$A =\frac{ 1}{1} \cdot \frac{3}{1}\cdot \frac{3}{1} \cdot \pi$

$A =9 \pi$

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

Give the surface area of a sphere with diameter .

Possible Answers:

$64 \pi$

$32 \pi$

$256 \pi$

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

$128 \pi$

Correct answer:

$64 \pi$

Explanation:

A sphere with diameter has radius half that, or , so substitute r = 4 into the formula for the surface area of a sphere:

$A = 4 \pi r^{2} = 4 \pi \cdot 4^{2}= 64 \pi$

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

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

Possible Answers:

50 m^2

$50\pi m^2$

$100\pi m^2$

100 m^2

Correct answer:

$100\pi m^2$

Explanation:

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

To find the surface area of a sphere, use the following:

$SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$SA_{sphere}=4\pi (5m)^2=100\pi m^2$

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

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

Possible Answers:

$192\pi in^2$

Not enough information provided

$144 \pi in^2$

$576\pi in^2$

Correct answer:

$576\pi in^2$

Explanation:

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

First, recall the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, just plug in our known radius and simplify:

$SA_{sphere}=4 \pi (12in)^2=576\pi in^2$

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ISEE Upper Level Math : Solid Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #3 : Spheres

Example Question #4 : How To Find The Volume Of A Sphere

Example Question #5 : How To Find The Volume Of A Sphere

Example Question #6 : How To Find The Volume Of A Sphere

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

Example Question #8 : How To Find The Volume Of A Sphere

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

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

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

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

All ISEE Upper Level Math Resources

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