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SSAT Upper Level Math : Volume of a Three-Dimensional Figure

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Example Questions

Example Question #3 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

The height of a cylinder is 3 inches and the radius of the circular end of the cylinder is 3 inches. Give the volume and surface area of the cylinder.

Possible Answers:

$\displaystyle V=74.78 in^3, A=113.10in^2$

$\displaystyle V=84.82 in^3, A=113.10in^2$

$\displaystyle V=84.78 in^3, A=123.04in^2$

$\displaystyle V=74.78 in^3, A=103.04in^2$

$\displaystyle V=84.78 in^3, A=103.04in^2$

Correct answer:

$\displaystyle V=84.82 in^3, A=113.10in^2$

Explanation:

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height or:

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

where $\displaystyle r$ is the radius of the circular end of the cylinder and $\displaystyle h$ is the height of the cylinder. So we can write:

$Volume=\pi r^2h=\pi \times 3^2\times 3=84.82in^3$ $\displaystyle Volume=\pi r^2h=\pi \times 3^2\times 3=84.82in^3$

The surface area of the cylinder is given by:

$A=2\pi r^2+2\pi rh$ $\displaystyle A=2\pi r^2+2\pi rh$

where $\displaystyle A$ is the surface area of the cylinder, $\displaystyle r$ is the radius of the cylinder and $\displaystyle h$ is the height of the cylinder. So we can write:

$A=2\pi r^2+2\pi rh$ $\displaystyle A=2\pi r^2+2\pi rh$

$A=2\pi (3)^2+2\pi \times 3\times 3$ $\displaystyle A=2\pi (3)^2+2\pi \times 3\times 3$

$A=18\pi+18\pi$ $\displaystyle A=18\pi+18\pi$

$A=36\pi$ $\displaystyle A=36\pi$

$\displaystyle A=113.10$

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Example Question #111 : Volume Of A Three Dimensional Figure

The height of a cylinder is two times the length of the radius of the circular end of a cylinder. If the volume of the cylinder is $16\pi$ $\displaystyle 16\pi$, what is the height of the cylinder?

Possible Answers:

$\displaystyle 2$

$\displaystyle 1$

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$

$\displaystyle 3$

$\displaystyle 4$

Correct answer:

$\displaystyle 4$

Explanation:

The volume of a cylinder is:

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

where $\displaystyle r$ is the radius of the circular end of the cylinder and $\displaystyle h$ is the height of the cylinder.

Since h=2r $\displaystyle h=2r$, we can substitute that into the volume formula. So we can write:

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

$16\pi=\pi r^2\cdot (2r)$ $\displaystyle 16\pi=\pi r^2\cdot (2r)$

$16\pi =2\pi r^{3}$ $\displaystyle 16\pi =2\pi r^{3}$

$16=2r^{3}$ $\displaystyle 16=2r^{3}$

$8=r^{3}$ $\displaystyle 8=r^{3}$

r=2 $\displaystyle r=2$

So we get:

$h=2r=2\times 2=4$ $\displaystyle h=2r=2\times 2=4$

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Example Question #1 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

The end (base) of a cylinder has an area of $16\pi$ $\displaystyle 16\pi$ square inches. If the height of the cylinder is half of the radius of the base of the cylinder, give the volume of the cylinder.

Possible Answers:

$16\pi$ $\displaystyle 16\pi$

$\displaystyle 32$

$32\pi$ $\displaystyle 32\pi$

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

$\displaystyle 16$

Correct answer:

$32\pi$ $\displaystyle 32\pi$

Explanation:

The area of the end (base) of a cylinder is $\pi r^2$ $\displaystyle \pi r^2$, so we can write:

$\pi r^2=16\pi\Rightarrow r^2=16\Rightarrow r=4\ inches$ $\displaystyle \pi r^2=16\pi\Rightarrow r^2=16\Rightarrow r=4\ inches$

The height of the cylinder is half of the radius of the base of the cylinder, that means:

$h=\frac{r}{2}=\frac{4}{2}=2\ inches$ $\displaystyle h=\frac{r}{2}=\frac{4}{2}=2\ inches$

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height:

$Volume=Area\times h=16\pi\times 2=32\pi$ $\displaystyle Volume=Area\times h=16\pi\times 2=32\pi$

$Volume=\pi r^2h=\pi\times 4^2\times 2=32\pi$ $\displaystyle Volume=\pi r^2h=\pi\times 4^2\times 2=32\pi$

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Example Question #112 : Volume Of A Three Dimensional Figure

We have two right cylinders. The radius of the base Cylinder 1 is $\sqrt{3}$ $\displaystyle \sqrt{3}$ times more than that of Cylinder 2, and the height of Cylinder 2 is 4 times more than the height of Cylinder 1. The volume of Cylinder 1 is what fraction of the volume of Cylinder 2?

Possible Answers:

$V_{1}=\frac{3}{4}V_{2}$ $\displaystyle V_{1}=\frac{3}{4}V_{2}$

$V_{1}=V_{2}$ $\displaystyle V_{1}=V_{2}$

$V_{1}=\frac{3}{2}V_{2}$ $\displaystyle V_{1}=\frac{3}{2}V_{2}$

$V_{1}=\frac{2}{3}V_{2}$ $\displaystyle V_{1}=\frac{2}{3}V_{2}$

$V_{1}=\frac{4}{3}V_{2}$ $\displaystyle V_{1}=\frac{4}{3}V_{2}$

Correct answer:

$V_{1}=\frac{3}{4}V_{2}$ $\displaystyle V_{1}=\frac{3}{4}V_{2}$

Explanation:

The volume of a cylinder is:

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

where $\displaystyle V$ is the volume of the cylinder, $\displaystyle r$ is the radius of the circular end of the cylinder, and $\displaystyle h$ is the height of the cylinder.

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$ $\displaystyle V_{1}=\pi (r_{1})^2h_{1}$

and

$V_{2}=\pi (r_{2})^2h_{2}$ $\displaystyle V_{2}=\pi (r_{2})^2h_{2}$

Now we can summarize the given information:

$r_{1}=\sqrt{3}\cdot r_{2}$ $\displaystyle r_{1}=\sqrt{3}\cdot r_{2}$

$h_{2}=4\cdot h_{1}\Rightarrow h_{1}=\frac{h_{2}}{4}$ $\displaystyle h_{2}=4\cdot h_{1}\Rightarrow h_{1}=\frac{h_{2}}{4}$

Now substitute them in the $V_{1}$ $\displaystyle V_{1}$ formula:

$V_{1}=\pi (\sqrt{3}r_{2})^2\times \frac{h_{2}}{4}\Rightarrow V_{1}=\frac{3}{4}\pi (r_{2})^2\times h_{2}\Rightarrow V_{1}=\frac{3}{4}V_{2}$ $\displaystyle V_{1}=\pi (\sqrt{3}r_{2})^2\times \frac{h_{2}}{4}\Rightarrow V_{1}=\frac{3}{4}\pi (r_{2})^2\times h_{2}\Rightarrow V_{1}=\frac{3}{4}V_{2}$

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Example Question #731 : Geometry

Two right cylinders have the same height. The radius of the base of the first cylinder is two times more than that of the second cylinder. Compare the volume of the two cylinders.

Possible Answers:

$V_{1}=V_{2}$ $\displaystyle V_{1}=V_{2}$

$V_{1}=\sqrt{2}\cdot V_{2}$ $\displaystyle V_{1}=\sqrt{2}\cdot V_{2}$

$V_{1}=3\cdot V_{2}$ $\displaystyle V_{1}=3\cdot V_{2}$

$V_{1}=4\cdot V_{2}$ $\displaystyle V_{1}=4\cdot V_{2}$

$V_{1}=2\cdot V_{2}$ $\displaystyle V_{1}=2\cdot V_{2}$

Correct answer:

$V_{1}=4\cdot V_{2}$ $\displaystyle V_{1}=4\cdot V_{2}$

Explanation:

The volume of a cylinder is:

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

where $\displaystyle r$ is the radius of the circular end of the cylinder and $\displaystyle h$ is the height of the cylinder. So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$ $\displaystyle V_{1}=\pi (r_{1})^2h_{1}$

$V_{2}=\pi (r_{2})^2h_{2}$ $\displaystyle V_{2}=\pi (r_{2})^2h_{2}$

We know that

$h_{1}=h_{2}$ $\displaystyle h_{1}=h_{2}$

and

$r_{1}=2r_{2}$ $\displaystyle r_{1}=2r_{2}$.

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}=\pi (2r_{2})^2\cdot h_{2}=4\pi (r_{2})^2h_{2}\Rightarrow V_{1}=4V_{2}$ $\displaystyle V_{1}=\pi (r_{1})^2h_{1}=\pi (2r_{2})^2\cdot h_{2}=4\pi (r_{2})^2h_{2}\Rightarrow V_{1}=4V_{2}$

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Example Question #113 : Volume Of A Three Dimensional Figure

A cylinder has a diameter of $\displaystyle 8$ inches and a height of $\displaystyle 2$ inches. Find the volume, in cubic inches, of this cylinder.

Possible Answers:

308.9 $\displaystyle 308.9$ in^3 $\displaystyle in^3$

98.3 $\displaystyle 98.3$ in^3 $\displaystyle in^3$

100.5 $\displaystyle 100.5$ in^3 $\displaystyle in^3$

402.1 $\displaystyle 402.1$ in^3 $\displaystyle in^3$

Correct answer:

100.5 $\displaystyle 100.5$ in^3 $\displaystyle in^3$

Explanation:

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

Since we are given the diameter, divide that value in half to find the radius.

$\text{radius}=8\div2=4$ $\displaystyle \text{radius}=8\div2=4$ $\displaystyle in$

Now plug this value into the equation for the volume of a cylinder.

$\text{Volume}=\pi \times 4^4 \times 2$ $\displaystyle \text{Volume}=\pi \times 4^4 \times 2$

$\text{Volume}=\pi \times 32=100.5$ $\displaystyle \text{Volume}=\pi \times 32=100.5$ in^3 $\displaystyle in^3$

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Example Question #212 : Geometry

Find the volume, in cubic inches, of a cylinder that has a radius of $\displaystyle 9$ inches and a height of $\displaystyle 13$ inches.

Possible Answers:

4456.1 $\displaystyle 4456.1$ in^3 $\displaystyle in^3$

3308.1 $\displaystyle 3308.1$ in^3 $\displaystyle in^3$

737.5 $\displaystyle 737.5$ in^3 $\displaystyle in^3$

2695.3 $\displaystyle 2695.3$ in^3 $\displaystyle in^3$

Correct answer:

3308.1 $\displaystyle 3308.1$ in^3 $\displaystyle in^3$

Explanation:

The formula to find the volume of a cylinder is $\pi r^2\times height$ $\displaystyle \pi r^2\times height$.

Now, plug in the given numbers into this equation.

$\text{Volume}=\pi \times 9^2 \times 13$ $\displaystyle \text{Volume}=\pi \times 9^2 \times 13$

$\text{Volume}=\pi \times 1053=3308.1$ $\displaystyle \text{Volume}=\pi \times 1053=3308.1$ in^3 $\displaystyle in^3$

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Example Question #114 : Volume Of A Three Dimensional Figure

Find the volume of the cylinder if the circular base has an area of $\pi$ $\displaystyle \pi$, and the height of the cylinder is also $\pi$ $\displaystyle \pi$.

Possible Answers:

$\pi^3$ $\displaystyle \pi^3$

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

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

$\frac{\pi^3}{2}$ $\displaystyle \frac{\pi^3}{2}$

$\frac{\pi^3}{4}$ $\displaystyle \frac{\pi^3}{4}$

Correct answer:

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

Explanation:

Write the formula for volume of a cylinder.

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

Remember that area of a circle is $A=\pi r^2$ $\displaystyle A=\pi r^2$

Since the area of the circle is known, substitute the area into the formula.

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

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SSAT Upper Level Math : Volume of a Three-Dimensional Figure

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #3 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

Example Question #111 : Volume Of A Three Dimensional Figure

Example Question #1 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

Example Question #112 : Volume Of A Three Dimensional Figure

Example Question #731 : Geometry

Example Question #113 : Volume Of A Three Dimensional Figure

Example Question #212 : Geometry

Example Question #114 : Volume Of A Three Dimensional Figure

All SSAT Upper Level Math Resources

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