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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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All Common Core: 8th Grade Math Resources

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Example Questions

Example Question #3 : Volume Of A Three Dimensional Figure

The volume of a sphere is 1000 cubic inches. What is the diameter of the sphere.

Possible Answers:

$10.4\ inches$

$12.4\ inches$

$10\ inches$

$16.4\ inches$

$12\ inches$

Correct answer:

$12.4\ inches$

Explanation:

The volume of a sphere is:

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

Where is the radius of the sphere. We know the volume and can solve the formula for :

$Volume=\frac{4}{3}\pi r^3=1000\Rightarrow r^3=\frac{3\times1000 }{4\pi}\approx 238.85\Rightarrow r\approx 6.2$ inches

So we can get:

$Diameter=2r=2\times 6.2=12.4inches$

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Example Question #494 : Grade 8

A sphere has a diameter of inches. What is the volume of this sphere?

Possible Answers:

458.3 in^3

632.9 in^3

104.7 in^3

523.6 in^3

Correct answer:

523.6 in^3

Explanation:

To find the volume of a sphere, use the following formula:

$\text{Volume}=\frac{4}{3}\pi\times r^3$ , where is the radius of the sphere.

Now, because we are given the diameter of the sphere, divide that value in half to find the radius.

$r=10\div2=5$

Now, plug this value into the volume equation.

$\text{Volume}=\frac{4}{3}\pi\times 5^3$

$\text{Volume}=\frac{4}{3}\pi\times 125$

$\text{Volume}=\frac{500}{3}\pi=523.6$ in^3

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Example Question #3 : How To Find The Volume Of A Sphere

What is the volume of a sphere with a diameter of ?

Possible Answers:

$\frac{500}{3}\pi$

$100\pi$

$200\pi$

$150\pi$

$\frac{100}{3}\pi$

Correct answer:

$\frac{500}{3}\pi$

Explanation:

Write the formula for the volume of the sphere.

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

The radius is half the diameter, which is five. Substitute the value.

$V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (5)^3=\frac{4}{3}\pi (125)= \frac{500}{3}\pi$

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Example Question #181 : Geometry

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

Possible Answers:

$\frac{1}{6} \pi \textrm{ ft}^{3}$

$\frac{1}{4} \pi \textrm{ ft}^{3}$

$\frac{1}{3} \pi \textrm{ ft}^{3}$

$\frac{1}{18} \pi \textrm{ ft}^{3}$

$\frac{1}{12} \pi \textrm{ ft}^{3}$

Correct answer:

$\frac{1}{18} \pi \textrm{ ft}^{3}$

Explanation:

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = \frac{18}{12} = \frac{3}{2}$ feet

Radius: 4 inches = $4 \div 12 = \frac{4}{12} = \frac{1}{3}$

The volume of a cone is given by the formula

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

Substitute $r = \frac{1}{3}, h = \frac{3}{2}$ :

$V = \frac{1}{3}\cdot \left ( \frac{1}{3} \right ) ^{2}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{3}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{1}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{2} \cdot \pi$

$V = \frac{1}{18} \pi$

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Example Question #101 : Solid Geometry

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

Possible Answers:

$0.768 \pi \textrm{ m}^{3}$

$1.536 \pi \textrm{ m}^{3}$

$0.512 \pi \textrm{ m}^{3}$

$4.608 \pi \textrm{ m}^{3}$

Correct answer:

$0.512 \pi \textrm{ m}^{3}$

Explanation:

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute r = 0.8, h = 2.4 in the volume formula:

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

$V = \frac{1}{3} \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = \frac{1}{3}\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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Example Question #182 : Geometry

Give the volume of a cone whose height is 10 inches and whose base is a circle with circumference $6 \pi$ inches.

Possible Answers:

$45 \pi \textrm{ in}^{3}$

$360 \pi \textrm{ in}^{3}$

$30 \pi \textrm{ in}^{3}$

$90 \pi \textrm{ in}^{3}$

$120 \pi \textrm{ in}^{3}$

Correct answer:

$30 \pi \textrm{ in}^{3}$

Explanation:

A circle with circumference $6 \pi$ inches has as its radius

$r = \frac{C }{2\pi }=\frac{6\pi }{2\pi } = 3$ inches.

The area of the base is therefore

$B = \pi r^{2} = \pi \cdot 3^{2} = 9 \pi$ square inches.

To find the volume of the cone, substitute $B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

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Example Question #183 : Geometry

The height of a cone and the radius of its base are equal. The circumference of the base is $10 \pi$ inches. Give its volume.

Possible Answers:

$\frac{1,000}{3} \pi \textrm{ in}^{3}$

$\frac{125}{3} \pi \textrm{ in}^{3}$

$\frac{5,000}{3} \pi \textrm{ in}^{3}$

$1,000 \textrm{ in}^{3}$

$125 \textrm{ in}^{3}$

Correct answer:

$\frac{125}{3} \pi \textrm{ in}^{3}$

Explanation:

A circle with circumference $10 \pi$ inches has as its radius

$r = \frac{C }{2\pi }=\frac{10\pi }{2\pi } = 5$ inches.

The height is also inches, so substitute h = r = 5 in the volume formula for a cone:

$V = \frac{1}{3} \pi r ^{2}h = \frac{1}{3} \pi \cdot 5 ^{2} \cdot 5 = \frac{125}{3} \pi$ cubic inches

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Example Question #184 : Geometry

In terms of $\pi$ , give the volume, in cubic inches, of a spherical water tank with a diameter of 20 feet.

Possible Answers:

$57,600 \pi\textrm{ in}^{3}$

$230,400 \pi\textrm{ in}^{3}$

$2,304,000 \pi \textrm{ in}^{3}$

$18,432,000\pi\textrm{ in}^{3}$

$307,200\pi\textrm{ in}^{3}$

Correct answer:

$2,304,000 \pi \textrm{ in}^{3}$

Explanation:

20 feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set r = 120 , substitute in the volume formula, and solve for :

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

$V = \frac{4}{3} \pi \cdot 120^{3}$

$V = \frac{4}{3} \pi \cdot 1,728,000$

$V = 2,304,000 \pi$ cubic inches

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Example Question #185 : Geometry

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

Possible Answers:

$27,000,000\pi \textrm{ cm}^{3}$

$4,500,000\pi \textrm{ cm}^{3}$

$1,125,000\pi \textrm{ cm}^{3}$

$36,000,000\pi \textrm{ cm}^{3}$

$3,375,000\pi \textrm{ cm}^{3}$

Correct answer:

$4,500,000\pi \textrm{ cm}^{3}$

Explanation:

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute r = 150 in the volume formula:

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

$V= \frac{4}{3} \cdot \pi \cdot 150^{3}$

$V= \frac{4}{3} \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

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Example Question #11 : Volume Of A Three Dimensional Figure

A cone has a radius of inches and a height of inches. Find the volume of the cone.

Possible Answers:

$12\pi\:in^3$

$24\pi\:in^3$

$36\pi\:in^3$

$108\pi\:in^3$

Correct answer:

$36\pi\:in^3$

Explanation:

The volume of a cone is given by the formula:

$\text{Volume}=\frac{1}{3}\pi r^2h$

Now, plug in the values of the radius and height to find the volume of the given cone.

$\text{Volume}=\frac{1}{3}\pi\times3^2\times12=\frac{1}{3}\pi\times9\times12=\frac{108}{3}\pi=36\pi\:in^3$

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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 #3 : Volume Of A Three Dimensional Figure

Example Question #494 : Grade 8

Example Question #3 : How To Find The Volume Of A Sphere

Example Question #181 : Geometry

Example Question #101 : Solid Geometry

Example Question #182 : Geometry

Example Question #183 : Geometry

Example Question #184 : Geometry

Example Question #185 : Geometry

Example Question #11 : Volume Of A Three Dimensional Figure

All Common Core: 8th Grade Math Resources

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