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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Properties of Cones

Cone-shaped figures can be recognized in many places in everyday life, such as when eating ice cream. This lesson will focus on the formulas for the volume and the surface area of a cone.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Relating the Volumes of Cones and Cylinders

Heichi knows how to calculate the volume of a cylinder. He thinks that the volume of a cone can be found using cylinders. To do so, he makes a cone-shaped mold with a height and radius of r. Then, he fills it with sand and pours it into a cylinder with the same radius and height. How can the relationship between the volumes of a cone and a cylinder be expressed? What can Heichi assume at the end of this experiment?

Cones

Before proceeding to the volume of a cone, the definition of a cone and its characteristics will be examined.

A cone is a three-dimensional solid with a circular base and a point, called the vertex or apex, that is not in the same plane as the base. The altitude of a cone is the segment from the vertex perpendicular to the plane of the base. The length of the altitude is called the height of the cone. If the altitude meets the base at its center, then the cone is a right cone. In a right cone, the distance from the vertex to a point on the edge of the base is called the slant height of a cone. If a cone is not a right cone, it is called oblique. Oblique cones do not have a uniform slant height. Considering Heichi's experiment, the formula for the volume of a cone will be one third of the volume of a cylinder with the same radius and height.

Volume of a Cone

The volume of a cone is one third of the product of its base area and height. The base area B is the area of the circle, and the height h is measured perpendicular to the base.

As the base is a circle, its area is dependent on its radius. Therefore, the base area can also be expressed in terms of the radius r.

Finding the Volume of the Cathedral of Maringá

The Cathedral of Maringá, one of the tallest churches in the world, was designed in the form of a cone by José Augusto Bellucci. The cathedral reaches 114 meters in height, excluding the cross. Furthermore, its circular base has a radius of 50 meters. Calculate its volume. Round the answer to the nearest cubic meter.

Hint

The volume of a cone is one third of the product of its base area and height.

Solution

To find the volume of the cathedral, the formula for the volume of a cone will be used.
In this formula, r is the radius of the base and h is the height of the cone. The height of the conical cathedral is 114 meters and its radius is 50 meters. By substituting these values into the formula, the volume can be found.
Simplify right-hand side
The volume of the cone is approximately 298451 cubic meters.

Finding the Volume a Chinese Conical Hat

Tadeo is learning how to make a traditional Chinese conical hat. He notices that the craftsman uses 20-centimeter bamboo sticks to make the framework. If the radius of the base is 16 centimeters, find the volume of the hat. Round the answer to the nearest cubic centimeter.

Hint

Use the Pythagorean Theorem to find the height of the cone.

Solution

To find the volume of the conical hat, its height will be calculated first. Then, the formula for the volume of a cone will be used.

Finding Height of the Hat

The distance from the vertex of the cone to its base is the height of the cone. Each stick if the frame represents the slant height of the cone. Therefore, the cone has a slant height of 20 centimeters and a radius of 16 centimeters. These three segments form a right triangle, ABC. From here, the height AB can be determined by using the Pythagorean Theorem.
AC2=AB2+BC2
202=AB2+162
Solve for AB
400=AB2+256
144=AB2
AB2=144
AB=12
The principal root was taken because negative values do not make sense for these measurements.

Finding the Volume of the Hat

Recall the formula for the volume of a cone.
Here, r is the radius and h is the height of the cone. The height was found to be 12 centimeters, and the radius of the base is 16 centimeters. The volume of the hat can be found by substituting these values into the formula.
Simplify right-hand side
The volume of the cone is about 3217 cubic centimeters.

Finding the Radius of a Traffic Cone

The diagram shows a traffic cone, which has a volume of 3128 cubic inches. The height of the cone part is 28 inches. The prism below it is a square prism with side lengths of 14 inches and height of 1 inch. Find the radius of the cone. Write the answer to the nearest inch.

Hint

The volume occupied by the traffic cone is the sum of the volume of the prism base and the volume of the cone part.

Solution

The traffic cone is basically composed of two solids — a cone and a square prism. Therefore, the volume occupied by the traffic cone is the sum of the volume of the prism and the volume of the cone
First the volume of the cone will be found. Then, the volume formula of a cone will be used to calculate the radius of the base of the cone.

Finding the Volume of the Cone

The base of the prism is a square with side lengths of 14 inches, so its area is the square of 14.
Since the volume of a prism is its base area times its height, the volume of the square prism can be found as follows.
Given that the total volume of the traffic cone is 3128 cubic inches, the volume of the cone part can be found now.
Solve for

Finding the Radius of the Cone

Finally, using the formula for the volume of a cone, the radius of the cone can be found.
To do so, substitute 2932 for and 28 for h into the formula and solve for r.
Solve for r
8796=πr2(28)
8796=28πr2

Cylinder With a Cone Inside

For the Jefferson High Science Fair, Ali is thinking about a chemistry experiment in which he will need a cylinder with a radius of 20 centimeters and a height 81 centimeters, with a cone inside. The cylinder must be open on both ends, and the cone must have an open bottom. To conduct the experiment, Ali needs to answer some questions first. Help him find the answers in order to win the first prize in the fair!

a Ali will fill the cone with water and the interior portion of the cylinder not occupied by the cone with foam. How many cubic centimeters of water and foam does Ali need? Write the answers to the nearest integer.
b What percent of the interior of the cylinder is not occupied by the cone? Round the answer to one decimal place.

Hint

a The volume of a cone is one third the product of π, the square of the radius, and the height. The volume of a cylinder is the product of π, the square of the radius, and the height.
b Use the exact values for the volumes obtained in Part A.

Solution

a The volumes will be calculated one at a time.

Volume of Water

Ali will fill the cone with water. Therefore, the volume of the cone is needed. The height and radius of the cone are the same as the height and radius of the cylinder. Therefore, the height of the cone is 81 centimeters and its radius is 20 centimeters. The volume of a cone is one third the product of π, the square of the radius, and the height.
In the this formula, 81 and 20 can be substituted for h and r, respectively.
Evaluate right-hand side
Therefore, the volume of water that Ali needs is 10800π cubic centimeters. Finally, this number will be approximated to the nearest integer.
10800π
33929
Ali needs about 33929 cubic centimeters of water.

Volume of Foam

Ali will fill the part of the cylinder not occupied by the cone with foam. Therefore, the volume of this portion is needed. It is given that the height of the cylinder is 81 centimeters and that its radius is 20 centimeters. The volume of a cylinder is the product of π, the square of the radius, and the height.
One more time, 81 and 20 can be substituted for h and r, respectively.
Evaluate right-hand side
The volume of the cylinder is 32400π cubic centimeters. The volume of the space inside the cylinder that is not occupied by the cone is the difference between the volume of the cylinder and the volume of the cone, which is 10800π cubic centimeters.
The volume of the cylinder that is not occupied by the cone is 26600π cubic centimeters and is the volume of foam that Ali will need. This number will be approximated to the nearest integer.
21600π
67858
Ali needs about 67858 cubic centimeters of foam.
b In Part A it was found that the volume of the cylinder is 32400π cubic centimeters. It was also found that the volume of the portion of the cylinder not occupied by the cone is 21600π cubic centimeters. The ratio of the second value to the first value will result in the desired percentage.
Evaluate
Convert to percent
It can be concluded that the volume of the cylinder not occupied by the cone represents of the interior of the cylinder.
Now, suppose that Ali wants to make the figure by himself. How many square meters of material does he need to use? To answer that question, Ali needs to know how to calculate the area of the surfaces of a cone.

Surface Area of a Cone

Consider a right cone with radius r and slant height The surface area of a right cone is the sum of the base area and the lateral area. The area of the base is given by πr2 and the lateral area has the area

Cylinder With a Cone Inside

For his experiment for the science fair, Ali plans to make the figure by himself. Finding the Surface Area of a Cone

Tiffaniqua's teacher gives her a piece of paper on which a circle and a sector are drawn. The paper is a square of side length 10 centimeters. The teacher also gives the following set of information.

• The circle is tangent to the top and right sides of the paper as well as to the sector.
• The centers of the circle and sector are on the diagonal of the paper.
• The circle and the sector can form a cone.

Help Tiffaniqua answer the following questions.

a Find the radii of the circle and the sector. Write the answers to one decimal place.
b Find the surface area of the cone. Round the answer to the nearest square centimeter.

Hint

a The circumference of the circle is equal to the arc length of the sector.
b The surface area of a cone is the sum of the base area and the lateral area.

Solution

a Let be the radius of the sector and r be the radius of the circle. To find these values, a system of two equations should be written first. The equations can be written using the trigonometric ratios and the properties of cones. Then, the system of equations will be solved using the Substitution Method.

Writing the System of Equations

Start by drawing a right triangle whose hypotenuse is centimeters. Since the diagonal of the square bisects the angle, the right triangle is also an isosceles triangle. Using the sine ratio of an equation describing the relationship between r and can be written.
Since , the above equation can be solved for
Solve for
The first equation is obtained. It is also known that the sector of circle with radius and the circle with radius r form a cone. As shown above, will be the slant height of the cone and r will be the radius of its circular base. Recall that in a cone, the circumference of the circular base is equal to the arc length of the sector. Since the measure of the sector is its arc length can be found using the relationship between arc length and arc measure.
Because these two lengths are equal, another equation can be obtained.
Solve for
Now, a system of equations can be written using the two values of

Solving the System of Equations

The Substitution Method will be used to find the unknowns. First use the value of to find the value of r.
Solve for r
Next, the value of r can be substituted into the other equation to find the value of

The radii of the circle and the sector are about 2.2 and 8.8 centimeters, respectively.
b The surface area of a cone is the sum of the base area and the lateral area.
By substituting and r=2.2, the surface area of the cone will be determined.
S=π(2.2)2+π(2.2)(8.8)
Evaluate right-hand side
S=π(4.84)+π(2.2)(8.8)
S=π(4.84)+π(19.36)
S=π(24.2)
The surface area of the cone is about 76 square centimeters.

Practice Finding the Volume and Surface Area of a Cone

The applet shows right cones. Use the given information to answer the question. If necessary, round the answer to one decimal place. Comparing the Volumes of a Pyramid and a Cube

In this lesson, the characteristics of cones have been studied, including their relationship with cylinders. It has been shown that the volume of a cone is one third the volume of a cylinder with the same radius and perpendicular height. How can the volume of a cuboid with side lengths h be related to the volume of a pyramid with the same base and height? The answer will be found in the next lesson.