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ISEE Upper Level Math : Spheres

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Example Questions

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

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

Possible Answers:

$108\text{in}^3$ $\displaystyle 108\text{in}^3$

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

$324\text{in}^3$ $\displaystyle 324\text{in}^3$

$594\text{in}^3$ $\displaystyle 594\text{in}^3$

$972\text{in}^3$ $\displaystyle 972\text{in}^3$

Correct answer:

$972\text{in}^3$ $\displaystyle 972\text{in}^3$

Explanation:

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

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

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

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

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

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

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

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

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

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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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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

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

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

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

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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 $\frac{1}{4}$ $\displaystyle \frac{1}{4}$.

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Write the formula to find the volume of a sphere.

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

Substitute the radius into the formula.

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

Evaluate the equation.

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

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

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Example Question #11 : Spheres

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

36 inches = $36 \div 12 = 3$ $\displaystyle 36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ $\displaystyle \frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ $\displaystyle r = \frac{3}{2}$, substitute in the surface area formula, and solve for $\displaystyle A$:

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

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

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

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

$A =9 \pi$ $\displaystyle A =9 \pi$

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

Give the surface area of a sphere with diameter $\displaystyle 8$.

Possible Answers:

$64 \pi$ $\displaystyle 64 \pi$

$32 \pi$ $\displaystyle 32 \pi$

$128 \pi$ $\displaystyle 128 \pi$

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

$256 \pi$ $\displaystyle 256 \pi$

Correct answer:

$64 \pi$ $\displaystyle 64 \pi$

Explanation:

A sphere with diameter $\displaystyle 8$ has radius half that, or $\displaystyle 4$, so substitute r = 4 $\displaystyle r = 4$ into the formula for the surface area of a sphere:

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

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Example Question #401 : Geometry

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

Possible Answers:

50 m^2 $\displaystyle 50 m^2$

$100\pi m^2$ $\displaystyle 100\pi m^2$

100 m^2 $\displaystyle 100 m^2$

$50\pi m^2$ $\displaystyle 50\pi m^2$

Correct answer:

$100\pi m^2$ $\displaystyle 100\pi m^2$

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:

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

Plug in our radius and solve!

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

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Example Question #402 : Geometry

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:

$144 \pi in^2$ $\displaystyle 144 \pi in^2$

$192\pi in^2$ $\displaystyle 192\pi in^2$

$576\pi in^2$ $\displaystyle 576\pi in^2$

Not enough information provided

Correct answer:

$576\pi in^2$ $\displaystyle 576\pi in^2$

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:

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

Now, just plug in our known radius and simplify:

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

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Example Question #11 : Spheres

Find the surface area of a sphere with a diameter of 10in.

Possible Answers:

$125\pi \text{ cm}^2$ $\displaystyle 125\pi \text{ cm}^2$

$100\pi \text{ cm}^2$ $\displaystyle 100\pi \text{ cm}^2$

$36\pi \text{ cm}^2$ $\displaystyle 36\pi \text{ cm}^2$

$50\pi \text{ cm}^2$ $\displaystyle 50\pi \text{ cm}^2$

$75\pi \text{ cm}^2$ $\displaystyle 75\pi \text{ cm}^2$

Correct answer:

$100\pi \text{ cm}^2$ $\displaystyle 100\pi \text{ cm}^2$

Explanation:

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

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 10cm. We also know the diameter is two times the radius. Therefore, the radius is 5cm.

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

$SA = 4 \cdot \pi \cdot (5\text{cm})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (5\text{cm})^2$

$SA = 4 \cdot \pi \cdot 25\text{cm}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 25\text{cm}^2$

$SA = 100\text{cm}^2 \cdot \pi$ $\displaystyle SA = 100\text{cm}^2 \cdot \pi$

$SA = 100\pi \text{ cm}^2$ $\displaystyle SA = 100\pi \text{ cm}^2$

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

Find the surface area of a sphere with a radius of 10in.

Possible Answers:

$125\pi \text{ in}^2$ $\displaystyle 125\pi \text{ in}^2$

$100\pi \text{ in}^2$ $\displaystyle 100\pi \text{ in}^2$

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

$375\pi \text{ in}^2$ $\displaystyle 375\pi \text{ in}^2$

$250\pi \text{ in}^2$ $\displaystyle 250\pi \text{ in}^2$

Correct answer:

$400\pi \text{ in}^2$ $\displaystyle 400\pi \text{ in}^2$

Explanation:

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

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 10in.

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

$SA = 4 \cdot \pi \cdot (10\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400\pi \text{ in}^2$ $\displaystyle SA = 400\pi \text{ in}^2$

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

Find the surface area of a sphere with a diameter of 18in.

Possible Answers:

$138\pi \text{ in}^2$ $\displaystyle 138\pi \text{ in}^2$

$421\pi \text{ in}^2$ $\displaystyle 421\pi \text{ in}^2$

$212\pi \text{ in}^2$ $\displaystyle 212\pi \text{ in}^2$

$256\pi \text{ in}^2$ $\displaystyle 256\pi \text{ in}^2$

$324\pi \text{ in}^2$ $\displaystyle 324\pi \text{ in}^2$

Correct answer:

$324\pi \text{ in}^2$ $\displaystyle 324\pi \text{ in}^2$

Explanation:

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

$SA = 4 \cdot \pi \cdot r^2$ $\displaystyle SA = 4 \cdot \pi \cdot r^2$

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

$SA = 4 \cdot \pi \cdot (9\text{in})^2$ $\displaystyle SA = 4 \cdot \pi \cdot (9\text{in})^2$

$SA = 4 \cdot \pi \cdot 81\text{in}^2$ $\displaystyle SA = 4 \cdot \pi \cdot 81\text{in}^2$

$SA = 324 \pi \text{ in}^2$ $\displaystyle SA = 324 \pi \text{ in}^2$

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ISEE Upper Level Math : Spheres

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

All ISEE Upper Level Math Resources

Example Questions

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 #11 : Spheres

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

Example Question #401 : Geometry

Example Question #402 : Geometry

Example Question #11 : Spheres

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

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

All ISEE Upper Level Math Resources

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