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Intermediate Geometry : Solid Geometry

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Example Questions

Example Question #11 : How To Find The Diameter Of A Sphere

Find the diameter of a sphere if it has a volume of $650\pi$ .

Possible Answers:

16.19

12.28

12.11

15.74

Correct answer:

15.74

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(650\pi)}{\pi}}=15.74$

Make sure to round to places after the decimal.

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Example Question #15 : How To Find The Diameter Of A Sphere

Find the diameter of a sphere if it has a volume of $887\pi$ .

Possible Answers:

17.46

18.11

17.55

16.28

Correct answer:

17.46

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(887\pi)}{\pi}}=17.46$

Make sure to round to places after the decimal.

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Example Question #12 : How To Find The Diameter Of A Sphere

Find the diameter of a sphere if its volume is .

Possible Answers:

4.86

4.95

4.20

4.44

Correct answer:

4.86

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(60)}{\pi}}=4.86$

Make sure to round to places after the decimal.

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Example Question #13 : How To Find The Diameter Of A Sphere

Find the diameter of a sphere if it has a volume of 908 .

Possible Answers:

6.58

12.62

6.448.75

12.01

Correct answer:

12.01

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(908)}{\pi}}=12.01$

Make sure to round to places after the decimal.

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Example Question #14 : How To Find The Diameter Of A Sphere

A company wants to construct an advertising balloon spherical in shape. It can afford to buy 28,000 square meters of material to make the balloon. What is the largest possible diameter of this balloon (nearest whole meter)?

Possible Answers:

$68\;m$

$94\;m$

$76\;m$

$47\;m$

$38\;m$

Correct answer:

$94\;m$

Explanation:

This is equivalent to asking the diameter of a balloon with surface area 28,000 square meters.

The relationship between the surface area and the radius is:

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

To find the radius, substitute for the surface area, then solve:

$28,000=4\pi r^{2}$

$r^{2}=\frac{28,000}{4\pi }$

$r=\sqrt{\frac{28,000}{4\pi }}\approx47$

To find the diameter , double the radius—this is 94.

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Example Question #71 : Spheres

If the volume of a sphere is $120\ ft^{3}$ , what is the approximate length of its diameter?

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

Possible Answers:

$4.48\ ft$

$6.12\ ft$

$8.96\ ft$

$3.06\ ft$

$32.25\ ft$

Correct answer:

$6.12\ ft$

Explanation:

The correct answer is 6.12 ft.

Plug the value of V=120 into the equation so that

$120=\frac{4}{3}\pi r^{3}$

Multiply both sides by 3 to get

$360=4\pi r^{3}$

Then divide both sides by $4\pi$ to get

$28.65=r^{3}$

Then take the 3^rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

D= 6.12 ft

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

The volume of a sphere is $57\:ft^3$ . What is its radius?

Possible Answers:

$2.2\:ft$

$2.4\:ft$

$4.8\:ft$

$2.3\:ft$

$2.1\:ft$

Correct answer:

$2.4\:ft$

Explanation:

The formula for the volume of a sphere is: $\frac{4}{3} \pi r^3 = V$

The only given information in the problem is the sphere's final volume. If the volume is $57\:ft^3$ , the formula for volume can be used to calculate the sphere's radius.

In this case, , the radius, is the only unknown variable that needs to be solved for.

$\frac{4}{3} \pi r^3 = 57$

$\frac{3}{4}\cdot \frac{1}{\pi}\cdot\frac{4}{3} \pi r^3 = 57 \cdot\frac{3}{4}\cdot \frac{1}{\pi}$

$r^3 = \frac{171}{4\cdot \pi}$

$\sqrt[3]{r^3}=\sqrt[3]{\frac{171}{4\cdot \pi}} = \sqrt[3]{13.608}$

$r=\sqrt[3]{13.608}$

r={2.387 ft}

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

The area of a sphere is $87\:in^2$ . What is its radius?

Possible Answers:

$2.6\:in$

$2.64\:in$

$2.7\:in$

$2.3\:in$

$2.5\:in$

Correct answer:

$2.6\:in$

Explanation:

The only information given is the area of $87\:in^2$ .

This problem may be approached "backwards," where the area formula for a sphere can be used to solve for the radius. This is possible because the formula for area is $Area = 4 \cdot \pi \cdot r^2$ , where (the radius) is what we're looking for. After is substituted in for the area, the goal is to solve for by getting it by itself on one side of the equals sign.

$87=4 \cdot \pi \cdot r^2$

$\frac{87}{4\cdot\pi}=\frac{4 \cdot \pi \cdot r^2}{4\cdot\pi}$

$\sqrt{\frac{87}{4 \cdot \pi}} = \sqrt{r^2} = r$

$r = 2.63\: in$

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

If the volume of a sphere is $1\:unit$ , what is the sphere's exact radius?

Possible Answers:

$\sqrt{\frac{1}{\pi}}\:units$

$\sqrt{\frac{2}{3\pi}}\:units$

$\sqrt[3]{\frac{4\pi}{3}}\:units$

$\sqrt{\pi}\:units$

$\sqrt[3]{\frac{3}{4\pi}}\:units$

Correct answer:

$\sqrt[3]{\frac{3}{4\pi}}\:units$

Explanation:

Write the formula for the volume of a sphere:

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

Plug in the given volume and solve for the radius, .

$1=\frac{4}{3}\pi r^3$

Start by multiplying each side of the equation by $\frac{3}{4}$ :

$\frac{3}{4}\cdot1=\frac{4}{3}\pi r^3\cdot\frac{3}{4}$

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

Now, divide each side of the equation by $\pi$ :

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

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

Finally, take the cubed root of each side of the equation:

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

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

Given the volume of a sphere is $\frac{4}{3}\pi\:cm^3$ , what is the radius?

Possible Answers:

$\frac{\sqrt2}{3}\:cm$

$\frac{4}{3}\:cm$

$1\:cm$

$\frac{1}{2}\:cm$

$\frac{1}{4}\:cm$

Correct answer:

$1\:cm$

Explanation:

The equation for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$ , where is the length of the sphere's radius.

Plug in the given volume and solve for to calculate the sphere's radius:

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

1=r^3

r=1

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #11 : How To Find The Diameter Of A Sphere

Example Question #15 : How To Find The Diameter Of A Sphere

Example Question #12 : How To Find The Diameter Of A Sphere

Example Question #13 : How To Find The Diameter Of A Sphere

Example Question #14 : How To Find The Diameter Of A Sphere

Example Question #71 : Spheres

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

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

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

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

All Intermediate Geometry Resources

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