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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.
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 that runs perpendicularly from the vertex to the base.
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 the product of its base area and its height.
The base area is the area of the circle and the height is measured perpendicular to the base.
Since the base is a circle, its area depends on its radius. Therefore, the base area can also be expressed in terms of the radius
Proof
Informal JustificationConsider a cone and a cylinder that have the same base area and height.
A cone can be modeled as a stack of cylinders. The sum of the volumes of the small cylinders will be greater than the cone's volume. However, the higher the number of cylinders, the more the sum will approximate the volume of the cone.
Furthermore, the ratio of the sum of the volumes of each small cylinder to the volume of the big cylinder nears as the number of stacked cylinders increases.
Therefore, the volume of a cone is one third the volume of the cylinder with the same base area and height.
Since the base area is the area of a circle, that formula can be substituted for to find a more detailed formula for the volume of the cone.
| Number of Cylinders | |
|---|---|
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.
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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.
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.
If the radius of the base is centimeters, find the volume of the hat. Round the answer to the nearest cubic centimeter.
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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.
▼
Solve for
Finding the Volume of the Hat
Recall the formula for the volume of a cone.Example
Finding the Radius of a Traffic Cone
The diagram shows a traffic cone, which has a volume of cubic inches.
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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.
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.
To do so, substitute for and for into the formula and solve for
The radius of the cone is about inches.
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 ofFinding the Radius of the Cone
Finally, using the formula for the volume of a cone, the radius of the cone can be found.▼
Solve for
CommutativePropMult
Commutative Property of Multiplication
RearrangeEqn
Rearrange equation
UseCalc
Use a calculator
RoundInt
Round to nearest integer
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.
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.
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b What percent of the interior of the cylinder is not occupied by the cone? Round the answer to one decimal place.
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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.
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.
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. 
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.
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.
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.
▼
Evaluate right-hand side
CalcPow
Calculate power
Multiply
Multiply
CommutativePropMult
Commutative Property of Multiplication
CalcQuot
Calculate quotient
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.
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.
It can be concluded that the volume of the cylinder not occupied by the cone represents of the interior of the cylinder.
▼
Evaluate
▼
Convert to percent
Discussion
Surface Area of a Cone
Consider a right cone with 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 is
Proof
Informal JustificationThe surface area of a cone is made of two main components, the area of the circular base and the lateral area. 
Let be the lateral area of a cone and the area of the circular base. The surface area of a cone is made of the sum of the area of the circular base and the lateral area.
The area of the circular base is found using the formula for the area of a circle.
The lateral area can be better visualized in two dimensions. Suppose that the cone is cut and the lateral area is expanded.
It should be noted that the figure obtained is a sector of a circle of radius the slant height of the cone. Then, the lateral area can be obtained using the formula for the area of a sector of a circle of radius
But from the image it should also be noted that the arc of the sector has the same length as the circumference of the circular base of radius Using the formula for the length of an arc relative to its measure, it is possible to write an equation to equate these quantities.
Dividing both sides of the equation by this equation can be simplified.
Now it is possible to write an expression for the lateral area. First, the expression is the same as the formula for the area of a sector.
Finally, the expressions for the area of the circular base and the lateral area can be substituted to find the expression for the surface area of a cone.
Example
Cylinder With a Cone Inside
For his experiment for the science fair, Ali plans to make the figure by himself.
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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. 
In the above formula, and can be substituted for and respectively.
The lateral area of the cylinder is about square meters.
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. 
The missing value can be found by using the Pythagorean Theorem.
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.
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.▼
Evaluate right-hand side
Lateral Area of the Cone
The lateral area of a cone is the product of the radius, and the slant height.Total Cost
Finally, the sum of the lateral areas will be multiplied by the cost per square meter. To make the shape that Ali needs, he has to spend aboutExample
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.
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.
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b Find the surface area of the cone. Round the answer to the nearest square centimeter.
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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. 
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
The first equation is obtained. It is also known that the sector of circle with radius and the circle with radius form a 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.
Now, a system of equations can be written using the two values of
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.
Writing the System of Equations
Start by drawing a right triangle whose hypotenuse is centimeters.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 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.
The surface area of the cone is about square centimeters.
Pop Quiz
Practice Finding the Volume and Surface Area of a Cone
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.
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