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SAT Math : Solid Geometry

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Example Question #1 : Cylinders

What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?

Possible Answers:

25812.5π in³

6453.125π in³

3831.34π in³

1032.5π in³

4813.33π in³

Correct answer:

6453.125π in³

Explanation:

The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr². Therefore, the total formula for the volume of the cylinder is: V = πr²h.

First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.

Based on this, we know that the volume of our cylinder must be: π*12.5²*41.3 = 6453.125π in³

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Example Question #7 : How To Find The Volume Of A Cylinder

An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?

Possible Answers:

354.92 in³

157.08 in³

203.34 in³

391.33 in³

462 in³

Correct answer:

354.92 in³

Explanation:

We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 2.5²* 8 = 50π in³

The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in³.

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Example Question #1 : How To Find The Volume Of A Cylinder

An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in³, what is the mass of the remaining wood after the cylinder is drilled out?

Possible Answers:

3193.33 g

4791.43 g

3813.3 g

2594.11 g

4921.4 g

Correct answer:

4791.43 g

Explanation:

We must calculate our two volumes and subtract them. Following this, we will multiply by the density.

The volume of the cube is very simple: 12 * 12 * 12, or 1728 in³.

The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in³. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.

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Example Question #1 : How To Find The Volume Of A Cylinder

A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?

Possible Answers:

187.73 in³

103.33 in³

249.73 in³

203.44 in³

237.17 in³

Correct answer:

237.17 in³

Explanation:

The general form of our problem is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 5 * 6 * 10 = 300 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 2²π = 4π. The total volume is therefore: 4π * 5 = 20π in³.

The gel volume is therefore: 300 – 20π or (approx.) 237.17 in³.

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Example Question #1 : How To Find The Volume Of A Cylinder

A hollow prism has a base 12 in x 13 in and a height of 42 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel, surrounding the can. The thickness of the can is negligible. Its diameter is 9 in and its height is one-fourth that of the prism. The can has a mass of 1.5 g per in³, and the gel has a mass of 2.2 g per in³. What is the approximate overall mass of the contents of the prism?

Possible Answers:

139.44 g

973.44 g

13.95 kg

11.48 kg

15.22 kg

Correct answer:

13.95 kg

Explanation:

We must find both the can volume and the gel volume. The formula for the gel volume is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 12 * 13 * 42 = 6552 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is one-fourth the prism height, or 42/4 = 10.5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 4.5, not 9. The base's area is: 4.5²π = 20.25π. The total volume is therefore: 20.25π * 10.5 = 212.625π in³.

The gel volume is therefore: 6552 – 212.625π or (approx.) 5884.02 in³.

The approximate volume for the can is: 667.98 in³

From this, we can calculate the approximate mass of the contents:

Gel Mass = Gel Volume * 2.2 = 5884.02 * 2.2 = 12944.844 g

Can Mass = Can Volume * 1.5 = 667.98 * 1.5 = 1001.97 g

The total mass is therefore 12944.844 + 1001.97 = 13946.814 g, or approximately 13.95 kg.

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Example Question #182 : Geometry

A vase needs to be filled with water. If the vase is a cylinder that is $\dpi{100} \small 12{}''$ tall with a $\dpi{100} \small 2{}''$ radius, how much water is needed to fill the vase?

Possible Answers:

$\dpi{100} \small 48\pi$

$\dpi{100} \small 24\pi$

$\dpi{100} \small 12\pi$

$\dpi{100} \small 64\pi$

$\dpi{100} \small 96\pi$

Correct answer:

$\dpi{100} \small 48\pi$

Explanation:

Cylinder

$\dpi{100} \small V = \pi r^{2}h$

$\dpi{100} \small V = \pi (2)^{2}\times 12$

$\dpi{100} \small V = 4\times 12\pi$

$\dpi{100} \small V = 48\pi$

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Example Question #131 : Solid Geometry

A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?

Possible Answers:

$24\pi in^{3}$

$48\pi in^{3}$

$288\pi in^{3}$

$72\pi in^{3}$

$36\pi in^{3}$

Correct answer:

$72\pi in^{3}$

Explanation:

The volume of a cylinder is $V=\pi r^{2}h$

The diameter is given, so make sure to divide it in half.

The units are inches cubed in this example

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Example Question #11 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a radius of 4 and a height of 5?

Possible Answers:

$54\pi$

$60\pi$

$40\pi$

$72\pi$

$80\pi$

Correct answer:

$80\pi$

Explanation:

$volume = \pi r^{2}h = \pi \cdot 4^{2} \cdot 5 = 80\pi$

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Example Question #184 : Geometry

Claire's cylindrical water bottle is 9 inches tall and has a diameter of 6 inches. How many cubic inches of water will her bottle hold?

Possible Answers:

$45\pi$

$54\pi$

$324\pi$

$18\pi$

$81\pi$

Correct answer:

$81\pi$

Explanation:

The volume is the area of the base times the height. The area of the base is $\pi r^{2}$ , and the radius here is 3. $\pi\cdot 3^{2}\cdot 9=81\pi$

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Example Question #185 : Geometry

What is the volume of a circular cylinder whose height is 8 cm and has a diameter of 4 cm?

Possible Answers:

$16\pi \ cm^{3}$

$32\pi \ cm^{3}$

$128\pi \ cm^{3}$

$12\pi \ cm^{3}$

$64\pi \ cm^{3}$

Correct answer:

$32\pi \ cm^{3}$

Explanation:

The volume of a circular cylinder is given by $V = \pi r^{2}h$ where is the radius and is the height. The diameter is given as 4 cm, so the radius would be 2 cm as the diameter is twice the radius.

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Cylinders

Example Question #7 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #182 : Geometry

Example Question #131 : Solid Geometry

Example Question #11 : How To Find The Volume Of A Cylinder

Example Question #184 : Geometry

Example Question #185 : Geometry

All SAT Math Resources

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