How to find the volume of a cone - ISEE Upper Level: Quantitative Reasoning

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Question

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

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Answer

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set and :

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

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = \frac{1}{3} \pi (3R)^{2}H = \frac{1}{3} \pi (9R^{2})H = 3\pi R ^{2}H$

, so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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Question

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

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Answer

Let be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $\frac{1}{2}r$ and , respectively.

(a) The volume of Cone A is $\frac{1}{3} \pi r^{2} h$ .

(b) The volume of Cone B is

$\frac{1}{3} \pi \left ( \frac{1}{2} r\right ) ^{2} \cdot 3 h = \frac{1}{3} \cdot \frac{1}{4} \cdot 3\cdot \pi r ^{2} h= \frac{1}{4} r ^{2} h$ .

Since $\frac{1}{4} r ^{2} h < \frac{1}{3} r ^{2} h$ , the cone in (a) has the greater volume.

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Question

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

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Answer

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

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

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

$r^{3}=\frac{ 1 }{ \pi}$

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi} \times 36 = \frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

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Question

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

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Answer

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height $\frac{1}{2} H$ .

In the formula for the volume of a cylinder, set and $h = \frac{1}{2} H$ :

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

$V = \pi R ^{2} \cdot \frac{1}{2}H = \frac{1}{2} \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = \frac{1}{3} \pi R^{2}H$

$\frac{1}{2} > \frac{1}{3}$ , so

$\frac{1}{2} \pi R ^{2}H > \frac{1}{3} \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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Question

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

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Answer

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 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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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.

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Answer

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 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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Question

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

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Answer

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Begin by using the formula for volume of a cone:

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

Now, we simply need to plug in our knowns.

We know the height is 30 ft.

We know that the diameter is 10ft, however, we need the radius.

Divide 10 by 2 to get a radius of 5 ft.

Now, let's go....

$V=\pi (5ft)^2\frac{30ft}{3}=\pi25ft^2*10ft=250 \pi ft^3$

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Question

Find the volume of a cone with the following measurements:

diameter = 12in

height = 6in

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Answer

To find the volume of a cone, we will use the following formula:

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

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in.

We know the height of the cone is 6in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (6\text{in})^2 \cdot \frac{6\text{in}}{3}$

$V = \pi \cdot 36\text{in}^2 \cdot 2\text{in}$

$V = \pi \cdot 72\text{in}^3$

$V = 72\pi \text{ in}^3$

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Question

Find the volume of a cone with the following measurements:

height: 12in

diameter: 6in

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Answer

To find the volume of a cone, we will use the following formula:

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

where r is the radius and h is the height of the cone.

Now, we know the height of the cone is 12in. We also know the diameter of the cone is 6in. We know the diameter is two times the radius. Therefore, the radius is 3in.

So, we get

$V = \pi \cdot (3\text{in})^2 \cdot \frac{12\text{in}}{3}$

$V = \pi \cdot 9\text{in}^2 \cdot 4\text{in}$

$V = \pi \cdot 36\text{in}^3$

$V = 36\pi \text{ in}^3$

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Question

Find the volume of a cone with the following measurements:

Diameter: 14in
Height: 9in

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Answer

To find the volume of a cone, we will use the following formula:

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

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 14in. We also know the diameter is two times the radius. Therefore, the radius is 7in.

We know the height of the cone is 9in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (7\text{in})^2 \cdot \frac{9\text{in}}{3}$

$V = \pi \cdot 49\text{in}^2 \cdot 3\text{in}$

$V = \pi \cdot 147\text{in}^3$

$V = 147\pi \text{ in}^3$

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Question

Find the volume of a cone with a base diameter of 2, and a height of 10.

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Answer

Write the formula to find the volume of a cone.

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

The diameter is 2, which means the radius is 1. Substitute the known dimensions.

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

The answer is: $\frac{10}{3}\pi$

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Question

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

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Answer

Let be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $\frac{1}{2}r$ and , respectively.

(a) The volume of Cone A is $\frac{1}{3} \pi r^{2} h$ .

(b) The volume of Cone B is

$\frac{1}{3} \pi \left ( \frac{1}{2} r\right ) ^{2} \cdot 3 h = \frac{1}{3} \cdot \frac{1}{4} \cdot 3\cdot \pi r ^{2} h= \frac{1}{4} r ^{2} h$ .

Since $\frac{1}{4} r ^{2} h < \frac{1}{3} r ^{2} h$ , the cone in (a) has the greater volume.

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Question

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

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Answer

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

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

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

$r^{3}=\frac{ 1 }{ \pi}$

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi} \times 36 = \frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

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Question

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

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Answer

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height $\frac{1}{2} H$ .

In the formula for the volume of a cylinder, set and $h = \frac{1}{2} H$ :

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

$V = \pi R ^{2} \cdot \frac{1}{2}H = \frac{1}{2} \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = \frac{1}{3} \pi R^{2}H$

$\frac{1}{2} > \frac{1}{3}$ , so

$\frac{1}{2} \pi R ^{2}H > \frac{1}{3} \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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Question

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Tap to reveal answer

Answer

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set and :

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

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = \frac{1}{3} \pi (3R)^{2}H = \frac{1}{3} \pi (9R^{2})H = 3\pi R ^{2}H$

, so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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Question

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

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Answer

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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Question

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

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Answer

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 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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Question

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

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Answer

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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