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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 .

Possible Answers:

625.88

608.08

603.19

623.35

Correct answer:

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:

$\text{Surface Area of Sphere}=4\pi r^2$ , where 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:

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

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.

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

Plug in the given radius to find the surface area.

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

Make sure to round to places after the decimal.

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Example Question #51 : Spheres

Find the surface area of a hemisphere with a radius of .

Possible Answers:

905.67

942.48

958.87

933.62

Correct answer:

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:

$\text{Surface Area of Sphere}=4\pi r^2$ , where 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:

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

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.

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

Plug in the given radius to find the surface area.

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

Make sure to round to places after the decimal.

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Example Question #55 : Spheres

Find the surface area of a hemisphere with a radius of .

Possible Answers:

1258.25

1000.27

1323.98

1140.40

Correct answer:

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:

$\text{Surface Area of Sphere}=4\pi r^2$ , where 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:

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

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.

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

Plug in the given radius to find the surface area.

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

Make sure to round to places after the decimal.

Report an Error

Example Question #56 : Spheres

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

Possible Answers:

0.59

0.58

0.67

0.81

Correct answer:

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:

$\text{Surface Area of Sphere}=4\pi r^2$ , where 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:

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

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.

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

Plug in the given radius to find the surface area.

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

Make sure to round to places after the decimal.

Report an Error

Example Question #291 : Solid Geometry

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

Possible Answers:

20.59

21.21

23.23

24.84

Correct answer:

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:

$\text{Surface Area of Sphere}=4\pi r^2$ , where 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:

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

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.

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

Plug in the given radius to find the surface area.

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

Make sure to round to places after the decimal.

Report an Error

Example Question #52 : Spheres

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

Possible Answers:

True

False

Correct answer:

False

Explanation:

Given radius , the surface area of a sphere can be calculated according to the formula

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

Set r = 1 :

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

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 $100\pi \; in^{2}$ . What is the diameter of the sphere?

Possible Answers:

$5\; in$

$20\; in$

$15\; in$

$10\; in$

Correct answer:

$10\; in$

Explanation:

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

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

To answer the question we need to find the diameter:

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

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Write the formula for the volume of a sphere:

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

Plug in the volume and find the radius by solving for :

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

Start solving for by multiplying both sides of the equation by :

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

$6=4\pi r^3$

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

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

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

Reduce the left side of the equation:

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

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

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

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

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Example Question #61 : Spheres

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

Possible Answers:

$27.0\:in$

$27.5\:in$

$13.7\:in$

$13.5\:in$

$14.0\: in$

Correct answer:

$13.5\:in$

Explanation:

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

By substituting in the value for circumference (), we can solve for the missing variable :

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

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

$d = 27\: in$

To find the radius of the sphere, we need to divide this value by :

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

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

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

Possible Answers:

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

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

$\frac{1}{2}$

Correct answer:

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

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.

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

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

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

The diameter is double the radius.

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

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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 #51 : Spheres

Example Question #55 : Spheres

Example Question #56 : Spheres

Example Question #291 : Solid Geometry

Example Question #52 : 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 #61 : Spheres

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

All Intermediate Geometry Resources

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