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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

Find the surface area of a hemisphere with a radius of $\displaystyle 8$.

Possible Answers:

623.35 $\displaystyle 623.35$

603.19 $\displaystyle 603.19$

625.88 $\displaystyle 625.88$

608.08 $\displaystyle 608.08$

Correct answer:

603.19 $\displaystyle 603.19$

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ $\displaystyle \text{Surface Area of Sphere}=4\pi r^2$, where $\displaystyle r$ is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$ $\displaystyle \text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$ $\displaystyle \text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (8)^2=603.19$ $\displaystyle \text{Surface Area of Hemisphere}=3\pi (8)^2=603.19$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #52 : Spheres

Find the surface area of a hemisphere with a radius of $\displaystyle 10$.

Possible Answers:

942.48 $\displaystyle 942.48$

958.87 $\displaystyle 958.87$

905.67 $\displaystyle 905.67$

933.62 $\displaystyle 933.62$

Correct answer:

942.48 $\displaystyle 942.48$

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ $\displaystyle \text{Surface Area of Sphere}=4\pi r^2$, where $\displaystyle r$ is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$ $\displaystyle \text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$ $\displaystyle \text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (10)^2=942.48$ $\displaystyle \text{Surface Area of Hemisphere}=3\pi (10)^2=942.48$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #55 : Spheres

Find the surface area of a hemisphere with a radius of $\displaystyle 11$.

Possible Answers:

1258.25 $\displaystyle 1258.25$

1000.27 $\displaystyle 1000.27$

1323.98 $\displaystyle 1323.98$

1140.40 $\displaystyle 1140.40$

Correct answer:

1140.40 $\displaystyle 1140.40$

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ $\displaystyle \text{Surface Area of Sphere}=4\pi r^2$, where $\displaystyle r$ is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$ $\displaystyle \text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$ $\displaystyle \text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (11)^2=1140.40$ $\displaystyle \text{Surface Area of Hemisphere}=3\pi (11)^2=1140.40$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #53 : Spheres

Find the surface area of a hemisphere that has a radius of $\frac{1}{4}$ $\displaystyle \frac{1}{4}$.

Possible Answers:

0.81 $\displaystyle 0.81$

0.59 $\displaystyle 0.59$

0.58 $\displaystyle 0.58$

0.67 $\displaystyle 0.67$

Correct answer:

0.59 $\displaystyle 0.59$

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ $\displaystyle \text{Surface Area of Sphere}=4\pi r^2$, where $\displaystyle r$ is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$ $\displaystyle \text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$ $\displaystyle \text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (\frac{1}{4})^2=0.59$ $\displaystyle \text{Surface Area of Hemisphere}=3\pi (\frac{1}{4})^2=0.59$

Make sure to round to $\displaystyle 2$ places after the decimal.

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

Find the surface area of a hemisphere that has a radius of $\frac{3}{2}$ $\displaystyle \frac{3}{2}$.

Possible Answers:

24.84 $\displaystyle 24.84$

21.21 $\displaystyle 21.21$

20.59 $\displaystyle 20.59$

23.23 $\displaystyle 23.23$

Correct answer:

21.21 $\displaystyle 21.21$

Explanation:

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ $\displaystyle \text{Surface Area of Sphere}=4\pi r^2$, where $\displaystyle r$ is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$ $\displaystyle \text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$ $\displaystyle \text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (\frac{3}{2})^2=21.21$ $\displaystyle \text{Surface Area of Hemisphere}=3\pi (\frac{3}{2})^2=21.21$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #291 : Solid Geometry

True or false: A sphere with radius 1 has surface area $3 \pi$ $\displaystyle 3 \pi$.

Possible Answers:

False

True

Correct answer:

False

Explanation:

Given radius $\displaystyle r$, the surface area of a sphere $\displaystyle A$ can be calculated according to the formula

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

Set r = 1 $\displaystyle r = 1$:

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

The statement is false.

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

The surface area of a sphere is $100\pi \; in^{2}$ $\displaystyle 100\pi \; in^{2}$. What is the diameter of the sphere?

Possible Answers:

$20\; in$ $\displaystyle 20\; in$

$15\; in$ $\displaystyle 15\; in$

$10\; in$ $\displaystyle 10\; in$

$5\; in$ $\displaystyle 5\; in$

Correct answer:

$10\; in$ $\displaystyle 10\; in$

Explanation:

The surface area of a sphere is given by $SA=4\pi r^{2}$ $\displaystyle SA=4\pi r^{2}$

So the equation to sovle becomes $4\pi r^{2}=100\pi$ $\displaystyle 4\pi r^{2}=100\pi$ or $r^{2}=25$ $\displaystyle r^{2}=25$ so $r=5\; in$ $\displaystyle r=5\; in$

To answer the question we need to find the diameter:

$d=2r=2\cdot 5=10\; in$ $\displaystyle d=2r=2\cdot 5=10\; in$

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

If the volume of a sphere is $2\:units$ $\displaystyle 2\:units$, what is the sphere's diameter?

Possible Answers:

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

$\sqrt[3]{\frac{4\pi}{3}}$ $\displaystyle \sqrt[3]{\frac{4\pi}{3}}$

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

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

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

Correct answer:

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

Explanation:

Write the formula for the volume of a sphere:

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

Plug in the volume and find the radius by solving for $\displaystyle r$:

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

Start solving for $\displaystyle r$ by multiplying both sides of the equation by $\displaystyle 3$:

$2\cdot3=\frac{4}{3}\pi r^3\cdot3$ $\displaystyle 2\cdot3=\frac{4}{3}\pi r^3\cdot3$

$6=4\pi r^3$ $\displaystyle 6=4\pi r^3$

Now, divide each side of the equation by $4\pi$ $\displaystyle 4\pi$:

$\frac{6}{4\pi}=\frac{4\pi r^3}{4\pi}$ $\displaystyle \frac{6}{4\pi}=\frac{4\pi r^3}{4\pi}$

$\frac{6}{4\pi}=r^3$ $\displaystyle \frac{6}{4\pi}=r^3$

Reduce the left side of the equation:

$\frac{3}{2\pi}=r^3$ $\displaystyle \frac{3}{2\pi}=r^3$

Finally, take the cubed root of both sides of the equation:

$\sqrt[3]{\frac{3}{2\pi}}=r$ $\displaystyle \sqrt[3]{\frac{3}{2\pi}}=r$

Keep in mind that you've solved for the radius, not the diameter. The diameter is double the radius, which is: $2\sqrt[3]{\frac{3}{2\pi}}$ $\displaystyle 2\sqrt[3]{\frac{3}{2\pi}}$.

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Example Question #292 : Solid Geometry

The circumference of a sphere is $84.823 \:in$ $\displaystyle 84.823 \:in$. Find the radius.

Possible Answers:

$27.5\:in$ $\displaystyle 27.5\:in$

$27.0\:in$ $\displaystyle 27.0\:in$

$14.0\: in$ $\displaystyle 14.0\: in$

$13.5\:in$ $\displaystyle 13.5\:in$

$13.7\:in$ $\displaystyle 13.7\:in$

Correct answer:

$13.5\:in$ $\displaystyle 13.5\:in$

Explanation:

If the circumference of a sphere is $84.823 \:in$ $\displaystyle 84.823 \:in$, that means we can easily solve for the diameter by using $C=\pi \cdot d$ $\displaystyle C=\pi \cdot d$. This is the equation to find the circumference.

By substituting in the value for circumference ($\displaystyle C$), we can solve for the missing variable $\displaystyle d$:

$84.823\:in= \pi \cdot d$ $\displaystyle 84.823\:in= \pi \cdot d$

$\frac{84.823\:in}{ \pi}= d$ $\displaystyle \frac{84.823\:in}{ \pi}= d$

$d = 27\: in$ $\displaystyle d = 27\: in$

To find the radius of the sphere, we need to divide this value by $\displaystyle 2$:

$r=\frac{d}{2}=\frac{27\:in}{2}=13.5\:in$ $\displaystyle r=\frac{d}{2}=\frac{27\:in}{2}=13.5\:in$

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Example Question #291 : Solid Geometry

Find the diameter of a sphere if the volume is $\pi$ $\displaystyle \pi$.

Possible Answers:

$\sqrt[3]{\frac{3}{4}}$ $\displaystyle \sqrt[3]{\frac{3}{4}}$

$2\sqrt[3]{\frac{3}{4}}$ $\displaystyle 2\sqrt[3]{\frac{3}{4}}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 1$

$\displaystyle 2$

Correct answer:

$2\sqrt[3]{\frac{3}{4}}$ $\displaystyle 2\sqrt[3]{\frac{3}{4}}$

Explanation:

Write the formula for the volume of a sphere.

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

Plug in the given volume and solve for the radius.

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

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

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

The diameter is double the radius.

$d=2\sqrt[3]{\frac{3}{4}}$ $\displaystyle d=2\sqrt[3]{\frac{3}{4}}$

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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 Surface Area Of A Sphere

Example Question #52 : Spheres

Example Question #55 : Spheres

Example Question #53 : Spheres

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

Example Question #291 : Solid Geometry

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

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

Example Question #292 : Solid Geometry

Example Question #291 : Solid Geometry

All Intermediate Geometry Resources

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