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ACT Math : Geometry

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Example Questions

Example Question #1052 : Act Math

The surface area of a cone equals $24\pi \textup{cm}^2.$ If the radius of the cone equals $3\textup{cm}$ , what is the height of the cone?

Possible Answers:

$4\textup{cm}$

$3\textup{cm}$

$5\textup{cm}$

$8\textup{cm}$

$6\textup{cm}$

Correct answer:

$4\textup{cm}$

Explanation:

To solve this question, you need to know the surface area of a cone equation: $\textup{surface area}=\pi r(r+\sqrt{h^2 +r^2}),$ where is the radius of the cone $(3\textup{cm})$ , and is the height of the cone. We must substitute all of the values that we know, and solve for the height of the cone to get the answer.

$24\textup{cm}^2=\pi *3\textup{cm}(3\textup{cm}+\sqrt{h^2 +(3\textup{cm})^2}),$ Therefore, the height of the cone is $4\textup{cm.}$

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Example Question #3 : Cones

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Possible Answers:

$45\pi$

$90\pi$

$60\pi$

$120\pi$

$30\pi$

Correct answer:

$60\pi$

Explanation:

Cylinder Volume = $\pi*r^{2}*h$

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

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

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

$=\pi*3^{2}*10-\frac{1}{3}*\pi*3^{2}*10$

$=\pi*90-\pi*30$

$=\pi*60$

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Example Question #781 : Geometry

You have two cones, one with a diameter of and a height of and another with a diameter of and a height of . If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?

Possible Answers:

$48\pi$

$6\pi$

$24\pi$

$8\pi$

$12\pi$

Correct answer:

$24\pi$

Explanation:

1. Find the volume of each cone:

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

Cone 1:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (3^{2})(10)=30\pi$

Cone 2:

Since the diameter is , the radius is 1.5 .

$Volume=\frac{1}{3}\pi (1.5^{2})(8)=6\pi$

2. Subtract the smaller volume from the larger volume:

$30\pi-6\pi=24\pi$

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Example Question #2 : How To Find The Volume Of A Cone

You have an empty cone and a cylinder filled with water. The cone has a diameter of and a height of . The cylinder has a diameter of and a height of . If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?

Possible Answers:

$252\pi$

$300\pi$

$1176\pi$

$276\pi$

$288\pi$

Correct answer:

$252\pi$

Explanation:

1. Find the volumes of the cone and cylinder:

Cone:

Since the diameter is , the radius is .

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

$Volume=\frac{1}{3}\pi (4^{2})(9)=48\pi$

Cylinder:

Since the diameter is , the radius is .

$Volume=\pi r^{2}h$

$Volume=\pi (5^{2})(12)=300\pi$

2. Subtract the cone's volume from the cylinder's volume:

$300\pi-48\pi=252\pi$

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

What is the volume of a cone with a height of 7 cm and a radius of 4 cm? Leave your answer in terms of $\pi$ and as a fraction if need be.

Possible Answers:

$\frac{448}{3}\pi cm^3$

$\frac{56}{3} \pi cm^3$

$\frac{112}{3}\pi cm^3$

$112\pi cm^3$

$\frac{196}{3}\pi cm^3$

Correct answer:

$\frac{112}{3}\pi cm^3$

Explanation:

To find the volume of a cone plug the radius and height into the formula for the volume of a cone.

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

$\\=\frac{1}{3}\pi 4^2\cdot 7\\ \\=\frac{112}{3}\pi cm^3$

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Example Question #12 : Cones

Which of the following will quadruple the volume of a cone?

Doubling the radius
Doubling the height
Quadrupling the height

Possible Answers:

1 and 3

2 only

1, 2 and 3

3 only

1 only

Correct answer:

1 and 3

Explanation:

Bearing in mind the volume formula for a cone:

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

Because the volume varies by the square of the radius, doubling the radius will quadruple the volume (since 2^2 = 4 .) Because the volume also varies linearly by the height, quadrupling the height will quadruple the volume.

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Example Question #6 : How To Find The Volume Of A Cone

What is the volume of a cone with a radius of and a height of ? Leave your answer in terms of $\pi$ , reduce all fractions.

Possible Answers:

$8\pi$

$231.32\pi$

$24\pi$

$36\pi$

$72\pi$

Correct answer:

$24\pi$

Explanation:

To find the volume of a cone with radius , and height use the formula:

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

We plug in our given radius and height to find:

$V = \frac{1}{3}\pi * 6^2*2 = \frac{72}{3}\pi = 24\pi$

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

A conical paper cup is being used as a makeshift funnel for motor oil. If the cup is 100 millimeters deep at the center and has a radius of 70 millimeters, how many cubic millimeters of motor oil can it hold at one time? Round your final answer to the nearest integer.

Possible Answers:

$233333\pi mm^2$

$784333\pi mm^2$

$106667\pi mm^2$

$387667\pi mm^2$

$99999\pi mm^2$

Correct answer:

$233333\pi mm^2$

Explanation:

The formula for the volume of a cone is:

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

With the information we have, we can simply plug values into this equation and solve.

$V = \frac{\pi r^2h}{3} = \frac{(100^2)(70)\pi}{3} \approx 233333\pi$

So our cone can hold approximately $233333\pi$ cubic millimeters of motor oil.

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

Find the volume of a cone whose diameter is and height is .

Possible Answers:

$18\pi$

$72\pi$

$36\pi$

$6\pi$

$72\pi$

Correct answer:

$6\pi$

Explanation:

To find the volume of a cone, simply use the following formula. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}*\pi*3^2*2=\frac{1}{3}*9*2*\pi=6\pi$

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

Find the volume of a cone with radius and height of .

Possible Answers:

$36\pi$

$108\pi$

$28\pi$

$12\pi$

Correct answer:

$12\pi$

Explanation:

To solve simply use the formula for the voluma of a cone. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}*\pi*3^2*4=12\pi$

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ACT Math : Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1052 : Act Math

Example Question #3 : Cones

Example Question #781 : Geometry

Example Question #2 : How To Find The Volume Of A Cone

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

Example Question #12 : Cones

Example Question #6 : How To Find The Volume Of A Cone

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

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

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

All ACT Math Resources

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