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High School Math : Solid Geometry

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #21 : Spheres

What is the surface area of a sphere with a radius of ?

Possible Answers:

576

$144\pi$

$288\pi$

$24\pi$

$576\pi$

Correct answer:

$576\pi$

Explanation:

To solve for the surface area of a sphere you must remember the formula:

First, plug the radius into the equation for :

$SA=(4)(12^{2})(\pi)$

Since $(12^{2})=144$ , the surface area is $(4)(144)(\pi)=576\pi$ .

The answer is therefore $576\pi$ .

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

What is the diameter of a sphere with a volume of $36\pi$ ?

Possible Answers:

Correct answer:

Explanation:

To find the diameter of a sphere we must use the equation for the volume of a sphere to find the radius which is half of the diameter.

The equation is

First we enter the volume into the equation yielding $36\pi=(\frac{4}{3})(\pi)(r^{3})$

We then divide each side by to get $36=(\frac{4}{3})(r^{3})$

We then multiply each side by to get $27=r^{3}$

We then take the cubic root of each side to solve for the radius $\sqrt[3]{27}=\sqrt[3]{r^{3}}$

The radius is r=3

We then multiply the radius by 2 to find the diameter 3*2=6

The answer for the diameter is .

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

The volume of a sphere is $2 8 8 \pi$ . What is the diameter?

Possible Answers:

14.7

29.4

Correct answer:

Explanation:

To find the diameter of the sphere, we need to find the radius.

The volume of a sphere is $V=\frac{4}{3}\pi r^3$ .

Plug in our given values.

$2 88\pi=\frac{4}{3}\pi r^3$

Notice that the $\pi$ 's cancel out.

$288=\frac{4}{3}r^3$

$\frac{3}{4}*288=r^3$

216=r^3

$\sqrt[3]{216}=\sqrt[3]{r^3}$

6= r

The diameter is twice the radius, so d=2r .

d=2(6)

d=12

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Example Question #3 : How To Find The Diameter Of A Sphere

If the surface area of a sphere is $64\pi$ , find the diameter of this sphere.

Possible Answers:

$4\pi$

$4\sqrt{\pi }$

$\sqrt{8\pi }$

Correct answer:

Explanation:

The standard equation to find the surface area of a sphere is

$SA=4\pi r^2$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt{\frac{SA}{4\pi }}$

Substitute the given surface area into this equation and solve for the radius and then double the radius to get the diameter:

$r=\sqrt{\frac{64\Pi }{4\Pi }}=\sqrt{16}=4$

$diameter=2\cdot r=2\cdot 4=8$

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

Given that the volume of a sphere is $4\pi$ , find the diameter.

Possible Answers:

$2\pi \sqrt[3]{3}$

$2\sqrt[3]{3}$

$2\sqrt[3]{4}$

$2\sqrt[3]{4\pi }$

$2\sqrt[3]{3\pi }$

Correct answer:

$2\sqrt[3]{3}$

Explanation:

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}\pi r^3$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt[3]{\frac{3V}{4\pi }}$

Substitute the given volume into this equation and solve for the radius. Double the radius to find the diameter:

$r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 4\pi }{4\pi }}=\sqrt[3]{\frac{12\pi }{4\pi }}=\sqrt[3]{3 }$

$diameter=2\cdot r=2\sqrt[3]{3}$

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

What is the diameter of a sphere with a volume of $36\pi$ ?

Possible Answers:

Correct answer:

Explanation:

The volume of a sphere is determined by the following equation:

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

$36\pi=\frac{4}{3}\pi r^3$

$(\frac{3}{4})36\pi=(\frac{3}{4})(\frac{4}{3})\pi r^3$

$\frac{27\pi}{\pi}=\frac{\pi r^3}{\pi}$

27=r^3

$\sqrt[3]{27}=\sqrt[3]{r^3}$

3=r

d=2r

d=2(3)=6

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

Find the radius of a sphere whose surface area is $SA=100cm^{2}$ .

Possible Answers:

8.22cm

Not enough information to solve

2.82cm

3.54cm

35.45cm

Correct answer:

2.82cm

Explanation:

We know that the surface area of the spere is $SA=100cm^{2}$ .

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

$100cm^{2}=4\pi r^{2}$

Rearrange and solve for .

$r^{2}=\frac{100cm^{2}}{4\pi}$

$\dpi{100} r=\sqrt{\frac{100cm^{2}}{4\pi}}$

$\dpi{100}\rightarrow 2.82cm$

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

What is the radius of a sphere that has a surface area of $16\pi$ ?

Possible Answers:

$8\pi$

$2\sqrt{\pi }$

$8\sqrt{\pi }$

$\sqrt{2\pi }$

Correct answer:

Explanation:

The standard equation to find the area of a sphere is $SA=4\pi r^2$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt{\frac{SA}{4\pi }}$

To find the answer, substitute the given surface area into this equation and solve for the radius:

$r=\sqrt{\frac{16\pi }{4\pi }}=\sqrt{4}=2$

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Example Question #3 : How To Find The Radius Of A Sphere

Given that the volume of a sphere is $12\pi$ , what is the radius?

Possible Answers:

$\sqrt{9\pi }$

$9\pi$

$\sqrt[3]{12 }$

$\sqrt[3]{9}$

Correct answer:

$\sqrt[3]{9}$

Explanation:

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}\pi r^3$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt[3]{\frac{3V}{4\pi }}$

Substitute the given volume into this equation and solve for the radius:

$r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 12\pi }{4\pi }}=\sqrt[3]{\frac{36\pi }{4\pi }}=\sqrt[3]{9 }$

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

What is the radius of a sphere with a volume of $288\pi$ ?

Possible Answers:

Correct answer:

Explanation:

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

$288\pi=\frac{4}{3}\pi r^3$

$\frac{288\pi}{\pi}=\frac{4\pi r^3}{3\pi}$

$288=\frac{4r^3}{3}$

$(288)(\frac{3}{4})=\frac{3}{4}(\frac{4r^3}{3})$

216=r^3

$\sqrt[3]{216}=\sqrt[3]{r^3}$

6=r

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High School Math : Solid Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #21 : Spheres

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

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

Example Question #3 : How To Find The Diameter Of A Sphere

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

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

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

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

Example Question #3 : How To Find The Radius Of A Sphere

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

All High School Math Resources

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