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SAT Math : Solid Geometry

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Example Questions

Example Question #12 : Cylinders

You have tall glass with a radius of 3 inches and height of 6 inches. You have an ice cube tray that makes perfect cubic ice cubes that have 0.5 inch sides. You put three ice cubes in your glass. How much volume do you have left for soda? The conversion factor is $1\hspace{1 mm}inch^3=0.0163871\hspace{1 mm}L$ $\displaystyle 1\hspace{1 mm}inch^3=0.0163871\hspace{1 mm}L$.

Possible Answers:

$2.77\hspace{1 mm}L$ $\displaystyle 2.77\hspace{1 mm}L$

$169.27\hspace{1 mm}mL$ $\displaystyle 169.27\hspace{1 mm}mL$

$1.69\hspace{1 mm}L$ $\displaystyle 1.69\hspace{1 mm}L$

$\frac{3}{8}inches^3$ $\displaystyle \frac{3}{8}inches^3$

$54\pi\hspace{1 mm}inches^3$ $\displaystyle 54\pi\hspace{1 mm}inches^3$

Correct answer:

$2.77\hspace{1 mm}L$ $\displaystyle 2.77\hspace{1 mm}L$

Explanation:

First we will calculate the volume of the glass. The volume of a cylinder is

$V=\pi r^2 h$ $\displaystyle V=\pi r^2 h$

$V=\pi (3\hspace{1 mm}inches)^2 \times 6\hspace{1 mm}inches = 54\pi\hspace{1 mm}inches^3$ $\displaystyle V=\pi (3\hspace{1 mm}inches)^2 \times 6\hspace{1 mm}inches = 54\pi\hspace{1 mm}inches^3$

Now we will calculate the volume of one ice cube:

$V=lwh=l^3=(\frac{1}{2}\hspace{1 mm}inch)^3=\frac{1}{8}\hspace{1 mm}inches^3$ $\displaystyle V=lwh=l^3=(\frac{1}{2}\hspace{1 mm}inch)^3=\frac{1}{8}\hspace{1 mm}inches^3$

The volume of three ice cubes is $\frac{3}{8}\hspace{1 mm}inches^3$ $\displaystyle \frac{3}{8}\hspace{1 mm}inches^3$. We will then subtract the volume taken up by ice from the total volume:

$54\pi\hspace{1 mm}inches^3-\frac{1}{8}\hspace{1 mm}inches^3\approx 169.27\hspace{1 mm}inches^3$ $\displaystyle 54\pi\hspace{1 mm}inches^3-\frac{1}{8}\hspace{1 mm}inches^3\approx 169.27\hspace{1 mm}inches^3$

Now we will use our conversion factor:

$169.27\hspace{1 mm}inches^3\times \frac{0.016387\hspace{1 mm}L}{1\hspace{1 mm}inch^3}=2.77\hspace{1 mm}L$ $\displaystyle 169.27\hspace{1 mm}inches^3\times \frac{0.016387\hspace{1 mm}L}{1\hspace{1 mm}inch^3}=2.77\hspace{1 mm}L$

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Example Question #101 : Solid Geometry

A water glass has the shape of a right cylinder. The glass has an interior radius of 2 inches, and a height of 6 inches. The glass is 75% full. What is the volume of the water in the glass (in cubic inches)?

Possible Answers:

$12 \pi$ $\displaystyle 12 \pi$

$18 \pi$ $\displaystyle 18 \pi$

$24 \pi$ $\displaystyle 24 \pi$

$9 \pi$ $\displaystyle 9 \pi$

$36 \pi$ $\displaystyle 36 \pi$

Correct answer:

$18 \pi$ $\displaystyle 18 \pi$

Explanation:

The volume of a right cylinder with radius r=2 $\displaystyle r=2$ and height h=6 $\displaystyle h=6$ is:

$\pi r^2 h = \pi (2)^2 (6) = 24 \pi$ $\displaystyle \pi r^2 h = \pi (2)^2 (6) = 24 \pi$

Since the glass is only 75% full, only 75% of the interior volume of the glass is occupied by water. Therefore the volume of the water is:

$24 \pi \times ( 75 / 100 ) = 24 \pi (0.75) = 18 \pi$ $\displaystyle 24 \pi \times ( 75 / 100 ) = 24 \pi (0.75) = 18 \pi$

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

A circle has a circumference of $4\pi$ $\displaystyle 4\pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16\pi$ $\displaystyle 16\pi$. Find the volume of the cylinder.

Possible Answers:

$6\pi$ $\displaystyle 6\pi$

$2\pi$ $\displaystyle 2\pi$

$8\pi$ $\displaystyle 8\pi$

$10\pi$ $\displaystyle 10\pi$

$4\pi$ $\displaystyle 4\pi$

Correct answer:

$8\pi$ $\displaystyle 8\pi$

Explanation:

Using the circumference, we can find the radius of the circle. The equation for the circumference is $2\pi r$ $\displaystyle 2\pi r$; therefore, the radius is 2.

Now we can find the area of the circle using $\pi r^{2}$ $\displaystyle \pi r^{2}$. The area is $4\pi$ $\displaystyle 4\pi$.

Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: $2\cdot 4\pi +4\pi h=16\pi$ $\displaystyle 2\cdot 4\pi +4\pi h=16\pi$, where $h$ $\displaystyle h$ is the height of the cylinder.

Thus, $h=2$ $\displaystyle h=2$ and the volume of the cylinder is $4\pi h=4\pi \cdot 2=8\pi$ $\displaystyle 4\pi h=4\pi \cdot 2=8\pi$.

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Example Question #1372 : Concepts

A metal cylindrical brick has a height of $6\ in$ $\displaystyle 6\ in$. The area of the top is $25\pi\ in^2$ $\displaystyle 25\pi\ in^2$. A circular hole with a radius of $2\ in$ $\displaystyle 2\ in$ is centered and drilled half-way down the brick. What is the volume of the resulting shape?

Possible Answers:

$100\pi\ in^3$ $\displaystyle 100\pi\ in^3$

$125\pi\ in^3$ $\displaystyle 125\pi\ in^3$

$175\pi\ in^3$ $\displaystyle 175\pi\ in^3$

$138\pi\ in^3$ $\displaystyle 138\pi\ in^3$

$150\pi\ in^3$ $\displaystyle 150\pi\ in^3$

Correct answer:

$138\pi\ in^3$ $\displaystyle 138\pi\ in^3$

Explanation:

To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.

The volume of a cylinder is given by the product of the base area times the height: $V=A_{base}h$ $\displaystyle V=A_{base}h$.

Find the initial volume using the given base area and height.

$V_i=(25\pi)(6)=150\pi\ in^3$ $\displaystyle V_i=(25\pi)(6)=150\pi\ in^3$

Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.

$V_{hole}=(\pi r^2)h$ $\displaystyle V_{hole}=(\pi r^2)h$

$V_{hole}=(\pi(2)^2)(3)=(4\pi)(3)=12\pi\ in^3$ $\displaystyle V_{hole}=(\pi(2)^2)(3)=(4\pi)(3)=12\pi\ in^3$

Finally, subtract the volume of the hole from the total initial volume.

$150\pi-12\pi=138\pi\ in^3$ $\displaystyle 150\pi-12\pi=138\pi\ in^3$

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Example Question #21 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?

Possible Answers:

$2583.1\pi \ in^{3}$ $\displaystyle 2583.1\pi \ in^{3}$

$4647.5\ in^{3}$ $\displaystyle 4647.5\ in^{3}$

$381.44\pi \ in^{3}$ $\displaystyle 381.44\pi \ in^{3}$

$357.5\pi \ in^{3}$ $\displaystyle 357.5\pi \ in^{3}$

$1161.88\pi \ in^{3}$ $\displaystyle 1161.88\pi \ in^{3}$

Correct answer:

$1161.88\pi \ in^{3}$ $\displaystyle 1161.88\pi \ in^{3}$

Explanation:

The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.

Thus, the volume can also be expressed as V = πr²h.

The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.

Now we can easily calculate the volume:

V = 6.5²π * 27.5 = 1161.88π in³

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Example Question #141 : Solid Geometry

Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?

Possible Answers:

12"

6.25"

30"

Correct answer:

6.25"

Explanation:

Volume of cylinder = π * (base radius)² x height = π * (base diameter / 2 )² x height

Volume Cylinder 1 = π * (3 / 2 )² x 9 = π * (1.5 )² x 9 = π * 20.25

Volume Cylinder 2 = π * (4 / 2 )² x 9 = π * (2 )² x 9 = π * 36

Total Volume = π * 20.25 + π * 36

Volume of Cylinder 3 = π * (6 / 2 )² x H = π * (3 )² x H = π * 9 x H

Set Total Volume equal to the Volume of Cylinder 3 and solve for H

π * 20.25 + π * 36 = π * 9 x H

20.25 + 36 = 9 x H

H = (20.25 + 36) / 9 = 6.25”

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Example Question #841 : Geometry

Determine the volume of a cylinder if the diameter of the base is 2 and the cylinder height is 10.

Possible Answers:

$\frac{5}{2}\pi$ $\displaystyle \frac{5}{2}\pi$

$10\pi$ $\displaystyle 10\pi$

$5\pi$ $\displaystyle 5\pi$

$20\pi$ $\displaystyle 20\pi$

$40\pi$ $\displaystyle 40\pi$

Correct answer:

$10\pi$ $\displaystyle 10\pi$

Explanation:

Write the formula for the volume of the cylinder.

$V=\pi r^2 h$ $\displaystyle V=\pi r^2 h$

The base of a cylinder is a circle, and the radius is half the diameter given.

$r=\frac{d}{2}=\frac{2}{2}=1$ $\displaystyle r=\frac{d}{2}=\frac{2}{2}=1$

Substitute the radius and the given height to the volume equation.

$V=\pi (1)^2 (10)= 10\pi$ $\displaystyle V=\pi (1)^2 (10)= 10\pi$

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Example Question #842 : Geometry

The radius of the base of the cylinder is $\frac{2}{3}$ $\displaystyle \frac{2}{3}$. The height of the cylinder is $\frac{1}{5}$ $\displaystyle \frac{1}{5}$. What is the volume?

Possible Answers:

$\frac{11}{15}\pi$ $\displaystyle \frac{11}{15}\pi$

$\frac{4}{15}\pi$ $\displaystyle \frac{4}{15}\pi$

$\frac{2}{7}\pi$ $\displaystyle \frac{2}{7}\pi$

$\frac{4}{225}\pi$ $\displaystyle \frac{4}{225}\pi$

$\frac{4}{45}\pi$ $\displaystyle \frac{4}{45}\pi$

Correct answer:

$\frac{4}{45}\pi$ $\displaystyle \frac{4}{45}\pi$

Explanation:

Write the volume formula of the cylinder and substitute the values.

$V=\pi r^2 h = \pi(\frac{2}{3})^2(\frac{1}{5}) = \pi (\frac{4}{9})(\frac{1}{5})= \frac{4}{45}\pi$ $\displaystyle V=\pi r^2 h = \pi(\frac{2}{3})^2(\frac{1}{5}) = \pi (\frac{4}{9})(\frac{1}{5})= \frac{4}{45}\pi$

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Example Question #843 : Geometry

What is the volume of a cylinder with a radius of 6 and a height of 8?

Possible Answers:

$96\pi$ $\displaystyle 96\pi$

$576\pi$ $\displaystyle 576\pi$

$288 \pi$ $\displaystyle 288 \pi$

$72\pi$ $\displaystyle 72\pi$

$144\pi$ $\displaystyle 144\pi$

Correct answer:

$288 \pi$ $\displaystyle 288 \pi$

Explanation:

Write the formula for the volume of a cylinder.

$V=\pi r^2 h$ $\displaystyle V=\pi r^2 h$

Subsstitute the given radius and height into the formula.

$V=\pi (6)^2 (8) = \pi \cdot 36 \cdot 8$ $\displaystyle V=\pi (6)^2 (8) = \pi \cdot 36 \cdot 8$

$V=288 \pi$ $\displaystyle V=288 \pi$

The volume is $288 \pi$ $\displaystyle 288 \pi$.

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Example Question #841 : Geometry

Find the volume aof a cylinder whose heigh is 5 and radius is 3.

Possible Answers:

$28\pi$ $\displaystyle 28\pi$

$45\pi$ $\displaystyle 45\pi$

$14\pi$ $\displaystyle 14\pi$

$75\pi$ $\displaystyle 75\pi$

Correct answer:

$45\pi$ $\displaystyle 45\pi$

Explanation:

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

$V=\pi{r^2}h=\pi*3^2*5=45\pi$ $\displaystyle V=\pi{r^2}h=\pi*3^2*5=45\pi$

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #12 : Cylinders

Example Question #101 : Solid Geometry

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

Example Question #1372 : Concepts

Example Question #21 : How To Find The Volume Of A Cylinder

Example Question #141 : Solid Geometry

Example Question #841 : Geometry

Example Question #842 : Geometry

Example Question #843 : Geometry

Example Question #841 : Geometry

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