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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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Example Questions

Example Question #171 : Geometry

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 $16\pi$ , what is the height of the cylinder?

Possible Answers:

$2\sqrt{2}$

Correct answer:

Explanation:

The volume of a cylinder is:

$Volume=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder.

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

$Volume=\pi r^2h$

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

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

$16=2r^{3}$

$8=r^{3}$

r=2

So we get:

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

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Example Question #172 : Geometry

The end (base) of a cylinder has an area of $16\pi$ 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:

$8\pi$

$16\pi$

$32\pi$

Correct answer:

$32\pi$

Explanation:

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

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

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

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

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

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

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

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Example Question #173 : Geometry

We have two right cylinders. The radius of the base Cylinder 1 is $\sqrt{3}$ 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:

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

$V_{1}=V_{2}$

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

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

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

Correct answer:

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

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

where is the volume of the cylinder, is the radius of the circular end of the cylinder, and is the height of the cylinder.

So we can write:

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

and

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

Now we can summarize the given information:

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

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

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

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

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Example Question #174 : 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:

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

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

$V_{1}=V_{2}$

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

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

Correct answer:

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

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

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

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

We know that

$h_{1}=h_{2}$

and

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

So we can write:

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

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Example Question #113 : Volume Of A Three Dimensional Figure

A cylinder has a diameter of inches and a height of inches. Find the volume, in cubic inches, of this cylinder.

Possible Answers:

98.3 in^3

100.5 in^3

402.1 in^3

308.9 in^3

Correct answer:

100.5 in^3

Explanation:

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

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

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

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

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

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

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Example Question #1993 : Hspt Mathematics

Find the volume, in cubic inches, of a cylinder that has a radius of inches and a height of inches.

Possible Answers:

2695.3 in^3

737.5 in^3

4456.1 in^3

3308.1 in^3

Correct answer:

3308.1 in^3

Explanation:

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

Now, plug in the given numbers into this equation.

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

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

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Example Question #731 : Geometry

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

Possible Answers:

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

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

$\pi^3$

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

$\pi^2$

Correct answer:

$\pi^2$

Explanation:

Write the formula for volume of a cylinder.

$V=\pi r^2 h$

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

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

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

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Example Question #175 : Geometry

A car dealership wants to fill a large spherical advertising ballon with helium. It can afford to buy 1,000 cubic yards of helium to fill this balloon. What is the greatest possible diameter of that balloon (nearest tenth of a yard)?

Possible Answers:

$9.3 \;\textrm{yd}$

$7.7 \;\textrm{yd}$

$6.2 \;\textrm{yd}$

$15.4 \;\textrm{yd}$

$12.4 \;\textrm{yd}$

Correct answer:

$12.4 \;\textrm{yd}$

Explanation:

The volume of a sphere, given its radius, is

$V = \frac{4\pi r^{3}}{3}$

Set V = 1,000 , solve for , and double that to get the diameter.

$1,000= \frac{4\pi r^{3}}{3}$

$r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$

$r \approx \sqrt[3]{238.7} \approx 6.2$

The diameter is twice this, or 12.4 yards.

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Example Question #176 : Geometry

The diameter of a sphere is . Give the volume of the sphere in terms of .

Possible Answers:

$10\pi t^3$

$72\pi t^3$

$36\pi t^3$

$30\pi t^3$

$24\pi t^3$

Correct answer:

$36\pi t^3$

Explanation:

The diameter of a sphere is so the radius of the sphere would be $6t\div 2=3t$

The volume enclosed by a sphere is given by the formula:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times (3t)^3=\frac{4}{3}\pi\times 27t^3\Rightarrow Volume=36\pi t^3$

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Example Question #177 : Geometry

A spherical balloon has a diameter of 10 meters. Give the volume of the balloon.

Possible Answers:

1023.3m^3

1000m^3

500m^3

723.3m^3

523.6m^3

Correct answer:

523.6m^3

Explanation:

The volume enclosed by a sphere is given by the formula:

$Volume=\frac{4}{3}\pi r^3$

where is the radius of the sphere. The diameter of the balloon is 10 meters so the radius of the sphere would be $10\div 2=5$ meters. Now we can get:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi\times 5^3=\frac{4}{3}\times \pi \times 125\approx 523.6 m^3$

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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

All Common Core: 8th Grade Math Resources

Example Questions

Example Question #171 : Geometry

Example Question #172 : Geometry

Example Question #173 : Geometry

Example Question #174 : Geometry

Example Question #113 : Volume Of A Three Dimensional Figure

Example Question #1993 : Hspt Mathematics

Example Question #731 : Geometry

Example Question #175 : Geometry

Example Question #176 : Geometry

Example Question #177 : Geometry

All Common Core: 8th Grade Math Resources

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