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SAT Math : Solid Geometry

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Example Questions

Example Question #846 : Geometry

Find the volume of a cylinder with radius 4 and height 5.

Possible Answers:

$9\pi$

$80\pi$

$20\pi$

$100\pi$

Correct answer:

$80\pi$

Explanation:

To solve, simply use the formula for the volume of a cylinder.

$V=\pi{r^2}h=\pi*{4^2}*5=\pi*16*5=80\pi$

If you don't have the formula memorized, remember that a cylinder is made from a circular base, just with a third dimension of height. So, simply just take the formula for the area of a circle and multiply is by the height.

$A=\pi{r^2}$

h=height

$V=A*h=\pi{r^2}*h=\pi{r^2}h$

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Example Question #22 : Cylinders

Find the volume of a cone with radius 2 and height 4.

Possible Answers:

$\frac{32\pi}{3}$

$\frac{16\pi}{3}$

$60\pi$

$30\pi$

Correct answer:

$\frac{16\pi}{3}$

Explanation:

To solve, simply use the formula for the volume of a cone. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi*{2^2}*4=\frac{1}{3}\pi*4*4=\frac{16\pi}{3}$

To remember the formula for volume of a cone, it helps to break it up into it's base and height. The base is a circle and the height is just h. Now, just multiplying those two together would give you the formula of a cylinder (see problem 3 in this set). So, our formula is going to have to be just a portion of that. Similarly to volume of a pyramid, that fraction is one third.

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Example Question #22 : Cylinders

The volume of a right cylinder is $16\pi.$ It's radius is one-half of its height. What is the height of the cylinder?

Possible Answers:

$6\sqrt\pi$

$4\sqrt\pi$

Correct answer:

Explanation:

If the radius is one-half of the height of the cylinder, we can say that

$r=\frac{1}{2}h$

Then, we can substitute this expression into our equation for the volume of a right cylinder.

$V=\pi r^2h$

$16\pi=\pi (\frac{1}{2}h)^2h$

Now, we can simplify and solve for h.

$16\pi=\pi (\frac{1}{4}h^2)h$

$16\pi=\pi \cdot \frac{1}{4}h^3$

$16=\frac{1}{4}h^3$

64=h^3

$h=\sqrt[3]{64}=4$

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Example Question #23 : Cylinders

Which of the following will quadruple a cylinder's volume?

Doubling the radius
Quadripling the radius
Quadrupling the height

Possible Answers:

2 only

1 only

1 and 2

1 and 3

3 only

Correct answer:

1 and 3

Explanation:

Remember our volume equation: $V_{cyl} = \pi r^2 h$

This means that the volume of a cylinder varies directly by the square of its radius and directly by its height. That means you need to quadruple the height to quadruple the volume but also that you only have to double the radius to quadruple the volume because

$\pi (2r)^2 h = \pi(4r^2)h = 4\pi r^2 h = 4V_{cyl}$

where $V_{cyl}$ is the original cylinder volume.

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Example Question #24 : Cylinders

What is the volume of a cylinder with height of 10cm and diameter of 4cm?

Possible Answers:

$20\pi$

$40\pi cm^3$

$4\pi cm^3$

$80\pi cm^3$

$10\pi cm^3$

Correct answer:

$40\pi cm^3$

Explanation:

$Volume=\pi r^2h$

The height is given: 10cm

The diameter is 4cm so the radius is half of that: 2cm

$Volume=\pi 2^2*10$

$Volume=40\pi cm^3$

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Example Question #151 : Solid Geometry

ABC Pipeworks manufactures a 12 foot copper alloy pipe that has an internal diameter of 0.76 inches and an external diameter of 0.88 inches. If two grams of copper are in every 3 cubic inches of pipe material. How many grams of copper are in this pipe?

Possible Answers:

$\textup{15.36 grams}$

$\textup{12.32 grams}$

$\textup{4.34 grams}$

$\textup{17.12 grams}$

$\textup{6.51 grams}$

Correct answer:

$\textup{15.36 grams}$

Explanation:

First convert all units to inches.

internal diameter: 0.76 inches

external diameter: 0.88 inches

length: 144 inches

You are going to need to subtract the interior volume of the empty space inside the pipe from the external volume of the of the pipe.

Copper pipe

$[(\pi \ast .44^2)-(\pi \ast.38^2)] (144)=$

(.61-.45)(144)=

$(.16)(144)=23.04\, in^3$

Now, we need to determine how many grams of copper are in the pipe.

$(23.04\, in^3)\left ( \frac{2\, grams}{3\, in^3} \right )=$

$15.36\, grams\, of\, copper$

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Example Question #851 : Sat Mathematics

Find the volume of the figure.

Possible Answers:

2036.33

2073.45

2158.25

2200.94

Correct answer:

2073.45

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 8^2 \times 12=768\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 3^2 \times 12=108\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=768\pi-108\pi=660\pi=2073.45$

Make sure to round to places after the decimal.

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Example Question #113 : Solid Geometry

Find the volume of the figure.

Possible Answers:

1006.37

1225.22

1109.87

1206.37

Correct answer:

1206.37

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 6^2 \times 12=432\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 2^2 \times 12=48\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=432\pi-48\pi=384\pi=1206.37$

Make sure to round to places after the decimal.

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Example Question #29 : How To Find The Volume Of A Cylinder

Find the volume of the figure.

Possible Answers:

1099.56

1124.52

1025.86

1233.45

Correct answer:

1099.56

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 6^2 \times 10=360\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 1^2 \times 10=10\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=360\pi-10\pi=350\pi=1099.56$

Make sure to round to places after the decimal.

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Example Question #42 : Cylinders

Find the volume of the figure.

Possible Answers:

100658.51

116992.91

112551.20

122445.73

Correct answer:

116992.91

Explanation:

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 34^2 \times 40=46240\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 15^2 \times 40=9000\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=46240\pi-9000\pi=37240\pi=116992.91$

Make sure to round to places after the decimal.

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #846 : Geometry

Example Question #22 : Cylinders

Example Question #22 : Cylinders

Example Question #23 : Cylinders

Example Question #24 : Cylinders

Example Question #151 : Solid Geometry

Example Question #851 : Sat Mathematics

Example Question #113 : Solid Geometry

Example Question #29 : How To Find The Volume Of A Cylinder

Example Question #42 : Cylinders

All SAT Math Resources

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