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Extending to Three Dimensions

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.

Explore

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 Then, he fills it with sand and pours it into a cylinder with the same radius and height.

Animation to fill up a cylinder using a cone
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?

Discussion

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.
Interactive Cone where the height and radius can be change. The Slant height is highlighted.
If a cone is not a right cone, it is called oblique. Oblique cones do not have a uniform slant height.
Interactive Cone changing from oblique to right

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.

Rule

Volume of a Cone

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

Base area of a cone and its height

The base area is the area of the circle, and the height 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

Example

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.

Cathedral of Maringá
The cathedral reaches meters in height, excluding the cross. Furthermore, its circular base has a radius of 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, is the radius of the base and is the height of the cone. The height of the conical cathedral is meters and its radius is meters. By substituting these values into the formula, the volume can be found.
Simplify right-hand side
The volume of the cone is approximately cubic meters.

Example

Finding the Volume a Chinese Conical Hat

Tadeo is learning how to make a traditional Chinese conical hat. He notices that the craftsman uses centimeter bamboo sticks to make the framework.

Framework of Conical Hat

If the radius of the base is 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 centimeters and a radius of centimeters.

Right triangle ABC formed inside conical hat
These three segments form a right triangle, From here, the height can be determined by using the Pythagorean Theorem.
Solve for
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, is the radius and is the height of the cone. The height was found to be centimeters, and the radius of the base is 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 cubic centimeters.

Example

Finding the Radius of a Traffic Cone

The diagram shows a traffic cone, which has a volume of cubic inches.

Traffic cone

The height of the cone part is inches. The prism below it is a square prism with side lengths of inches and height of 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 inches, so its area is the square of 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 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 for and for into the formula and solve for
Solve for
The radius of the cone is about inches.

Example

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 centimeters and a height centimeters, with a cone inside. The cylinder must be open on both ends, and the cone must have an open bottom.

Cylinder and a cone inside

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 the square of the radius, and the height. The volume of a cylinder is the product of 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 centimeters and its radius is centimeters.

Height and radius of the cone
The volume of a cone is one third the product of the square of the radius, and the height. In the this formula, and can be substituted for and respectively.
Evaluate right-hand side
Therefore, the volume of water that Ali needs is cubic centimeters. Finally, this number will be approximated to the nearest integer.
Ali needs about 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 centimeters and that its radius is centimeters.

Height and radius of the cylinder
The volume of a cylinder is the product of the square of the radius, and the height. One more time, and can be substituted for and respectively.
Evaluate right-hand side
The volume of the cylinder is 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 cubic centimeters. The volume of the cylinder that is not occupied by the cone is cubic centimeters and is the volume of foam that Ali will need. This number will be approximated to the nearest integer.
Ali needs about cubic centimeters of foam.
b In Part it was found that the volume of the cylinder is cubic centimeters. It was also found that the volume of the portion of the cylinder not occupied by the cone is 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.

Discussion

Surface Area of a Cone

Consider a right cone with radius and slant height

Cone with its radius 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 and the lateral area has the area

Example

Cylinder With a Cone Inside

For his experiment for the science fair, Ali plans to make the figure by himself.

Cylinder with a cone inside
The material to be used to create the figure costs per square meter. What is the cost of the material for this object? Write the answer to two decimal places.

Hint

The area of the lateral surface of a cone is the product of the radius, and the slant height. A cylinder's lateral surface area is twice the product of the radius, and the height.

Solution

Since the cost is given per square meter, all the measures will be converted from centimeters to meters. To do this, the measures need to be multiplied by a conversion factor, With this information, update the measures on the diagram.

Linear measures of cylinder and cone in centimeters

The lateral areas of the cylinder and the cone will be calculated one at a time. Then, their sum will be multiplied by the cost per square meter.

Lateral Area of the Cylinder

Recall that the cylinder is open in both ends. The lateral area of a cylinder is twice the product of the radius, and the height. In the above formula, and can be substituted for and respectively.
Evaluate right-hand side
The lateral area of the cylinder is about square meters.

Lateral Area of the Cone

The lateral area of a cone is the product of the radius, and the slant height. The slant height is the hypotenuse of the right triangle formed by the radius, the height, and the segment that connects the center of the base of the cylinder with a point on the circumference of the opposite base.

Right triangle
The missing value can be found by using the Pythagorean Theorem.
Solve for
The slant height of the cone is about meters. Now, the formula for the lateral area of a cone can be used. Substitute and for and respectively.

Total Cost

Finally, the sum of the lateral areas will be multiplied by the cost per square meter.
Evaluate
To make the shape that Ali needs, he has to spend about

Example

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 centimeters.

A square piece of paper with a circle and a sector drawn on it.

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 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.
A square paper and a right triangle with hypotenuse r+l
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 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 form a cone.
A square paper and cone
As shown above, will be the slant height of the cone and 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
Solve for
Next, the value of can be substituted into the other equation to find the value of

The radii of the circle and the sector are about and centimeters, respectively.
b The surface area of a cone is the sum of the base area and the lateral area. By substituting and the surface area of the cone will be determined.
Evaluate right-hand side
The surface area of the cone is about square centimeters.

Pop Quiz

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.

Cones

Closure

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.

Comparing the volumes of a cylinder and cone
How can the volume of a cuboid with side lengths be related to the volume of a pyramid with the same base and height? The answer will be found in the next lesson.
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