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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 #54 : How To Find The Volume Of A Prism

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Find the volume of the figure.

Possible Answers:

190.24

157.70

151.24

167.23

Correct answer:

157.70

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

From the figure, you should notice that the length of the prism is also the diameter of the half cylinder. Thus, half the length of the prism will also be the radius of the half cylinder. Also, notice that the width of the prism will be the height of the cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{4}{2}=2$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (2^2)(6)}{2}=12\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(4)(6)(5)=120$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=12\pi+120=157.70$

Remember to round to places after the decimal.

Report an Error

Example Question #55 : How To Find The Volume Of A Prism

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Find the volume of the figure.

Possible Answers:

390.24

405.78

401.10

424.96

Correct answer:

401.10

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{6}{2}=3$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (3^2)(8)}{2}=36\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(6)(8)(6)=288$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=36\pi+288=401.10$

Remember to round to places after the decimal.

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Example Question #56 : How To Find The Volume Of A Prism

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Find the volume of the figure.

Possible Answers:

2221.20

2334.29

2501.20

2471.68

Correct answer:

2471.68

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{12}{2}=6$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (6^2)(14)}{2}=252\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(12)(14)(10)=1680$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=252\pi+1680=2471.68$

Remember to round to places after the decimal.

Report an Error

Example Question #81 : Prisms

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Find the volume of the figure.

Possible Answers:

867.11

897.26

901.20

822.45

Correct answer:

897.26

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{5}{2}$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (\frac{5}{2})^2(15)}{2}=\frac{375}{8}\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(5)(15)(10)=750$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=\frac{375}{8}\pi+750=897.26$

Remember to round to places after the decimal.

Report an Error

Example Question #82 : Prisms

A cylinder is cut in half and placed on top of a rectangular prism.

Find the volume of the figure.

Possible Answers:

264.08

252.21

253.01

266.78

Correct answer:

264.08

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{3}{2}$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (\frac{3}{2})^2(17)}{2}=\frac{153}{8}\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(3)(17)(4)=204$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=\frac{153}{8}\pi+204=264.08$

Remember to round to places after the decimal.

Report an Error

Example Question #83 : Prisms

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

1032.67

856.68

912.83

655.81

Correct answer:

912.83

Explanation:

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(7)(7)(13)=637$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(7^2)=\frac{49\sqrt3}{4}$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=\frac{49\sqrt3}{4}(13)=\frac{637\sqrt3}{4}$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=637+\frac{637\sqrt3}{4}=912.83$

Make sure to round to places after the decimal.

Report an Error

Example Question #83 : Prisms

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

6238.09

6363.69

6512.35

6448.56

Correct answer:

6448.56

Explanation:

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(15)(15)(20)=4500$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(15^2)=\frac{225\sqrt3}{4}$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=\frac{225\sqrt3}{4}(20)=1125\sqrt3$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=4500+1125\sqrt3=6448.56$

Make sure to round to places after the decimal.

Report an Error

Example Question #61 : How To Find The Volume Of A Prism

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

95.31

109.65

103.18

101.27

Correct answer:

103.18

Explanation:

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(3)(3)(8)=72$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(3^2)=\frac{9\sqrt3}{4}$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=\frac{9\sqrt3}{4}(8)=\frac{72\sqrt3}{4}=18\sqrt3$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=72+18\sqrt3=103.18$

Make sure to round to places after the decimal.

Report an Error

Example Question #84 : Prisms

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

183.43

187.06

191.78

192.65

Correct answer:

183.43

Explanation:

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(4)(4)(8)=128$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(4^2)=4\sqrt3$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=4\sqrt3(8)=32\sqrt3$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=128+32\sqrt3=183.43$

Make sure to round to places after the decimal.

Report an Error

Example Question #85 : Prisms

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

6.92

7.11

7.17

8.65

Correct answer:

7.17

Explanation:

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(1)(1)(5)=5$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(1^2)=\frac{\sqrt3}{4}$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=\frac{\sqrt3}{4}(5)=\frac{5\sqrt3}{4}$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=5+\frac{5\sqrt3}{4}=7.17$

Make sure to round to places after the decimal.

Report an Error

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #54 : How To Find The Volume Of A Prism

Example Question #55 : How To Find The Volume Of A Prism

Example Question #56 : How To Find The Volume Of A Prism

Example Question #81 : Prisms

Example Question #82 : Prisms

Example Question #83 : Prisms

Example Question #83 : Prisms

Example Question #61 : How To Find The Volume Of A Prism

Example Question #84 : Prisms

Example Question #85 : Prisms

All Intermediate Geometry Resources

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