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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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.**

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?

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.

If a cone is not a right cone, it is called

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.

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.

$V=31 Bh$

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 $r.$

$V=31 πr_{2}h$

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord Roboto-Regular\">m<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["298451"]}}

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

To find the volume of the cathedral, the formula for the volume of a cone will be used.
$V=31 πr_{2}h $
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.
The volume of the cone is approximately $298451$ cubic meters.

$V=31 πr_{2}h$

SubstituteII

$r=50$, $h=114$

$V=31 π(50)_{2}(114)$

Simplify right-hand side

CalcPow

Calculate power

$V=31 π(2500)(114)$

Multiply

Multiply

$V=31 285000π$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$V=3285000π $

UseCalc

Use a calculator

$V=298451.302091…$

RoundInt

Round to nearest integer

$V≈298451$

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.

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

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.

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.$AC_{2}=AB_{2}+BC_{2}$

SubstituteII

$AC=20$, $BC=16$

$20_{2}=AB_{2}+16_{2}$

Solve for $AB$

CalcPow

Calculate power

$400=AB_{2}+256$

SubEqn

$LHS−256=RHS−256$

$144=AB_{2}$

RearrangeEqn

Rearrange equation

$AB_{2}=144$

SqrtEqn

$LHS =RHS $

$AB_{2} =144 $

SqrtPowToNumber

$a_{2} =a$

$AB=12$

$V=31 πr_{2}h$

SubstituteII

$r=16$, $h=12$

$V=31 π(16_{2})(12)$

Simplify right-hand side

CalcPow

Calculate power

$V=31 π(256)(12)$

Multiply

Multiply

$V=31 3072π$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$V=33072π $

UseCalc

Use a calculator

$V=3216.990877…$

RoundInt

Round to nearest integer

$V≈3217$

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.

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

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 $V_{p}$ and the volume of the cone $V_{c}.$ $V=V_{p}+V_{c} $ 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.

$V_{c}=31 πr_{2}h$

SubstituteII

$V_{c}=2932$, $h=28$

$2932=31 πr_{2}(28)$

Solve for $r$

MultEqn

$LHS⋅3=RHS⋅3$

$8796=πr_{2}(28)$

CommutativePropMult

Commutative Property of Multiplication

$8796=28πr_{2}$

DivEqn

$LHS/28π=RHS/28π$

$28π8796 =r_{2}$

RearrangeEqn

Rearrange equation

$r_{2}=28π8796 $

SqrtEqn

$LHS =RHS $

$r_{2} =28π8796 $

SqrtPowToNumber

$a_{2} =a$

$r=28π8796 $

UseCalc

Use a calculator

$r=9.999738…$

RoundInt

Round to nearest integer

$r≈10$

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.

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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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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.

a The volumes will be calculated one at a time.
### Volume of Water

Therefore, the volume of water that Ali needs is $10800π$ cubic centimeters. Finally, this number will be approximated to the nearest integer.
Ali needs about $33929$ cubic centimeters of water. ### Volume of Foam

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. $V_{cone}=31 πr_{2}h $ In the this formula, $81$ and $20$ can be substituted for $h$ and $r,$ respectively.$V_{cone}=31 πr_{2}h$

SubstituteII

$h=81$, $r=20$

$V_{cone}=31 π(20_{2})(81)$

Evaluate right-hand side

CalcPow

Calculate power

$V_{cone}=31 π(400)(81)$

Multiply

Multiply

$V_{cone}=31 π(32400)$

CommutativePropMult

Commutative Property of Multiplication

$V_{cone}=31 (32400)π$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$V_{cone}=332000 π$

CalcQuot

Calculate quotient

$V_{cone}=10800π$

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. $V_{cylinder}=πr_{2}h $ One more time, $81$ and $20$ can be substituted for $h$ and $r,$ respectively. 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. $32400π−10800π=21600π $ 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. 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.
It can be concluded that the volume of the cylinder not occupied by the cone represents $66.7%$ of the interior of the cylinder.

$32400π21600π $

Evaluate

CancelCommonFac

Cancel out common factors

$32400π 21600π $

SimpQuot

Simplify quotient

$3240021600 $

ReduceFrac

$ba =b/10800a/10800 $

$32 $

Convert to percent

FracToDiv

$ba =a÷b$

$0.666666…$

WritePercent

Convert to percent

$66.666666…%$

RoundDec

Round to ${\textstyle 1 \, \ifnumequal{1}{1}{\text{decimal}}{\text{decimals}}}$

$66.7%$

Consider a right cone with radius $r$ and slant height $ℓ.$

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 $πr_{2}$ and the lateral area has the area $πrℓ.$

$S=πr_{2}+πrℓ$

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

The material to be used to create the figure costs $$125$ per{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["192.76"]}}

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

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, $100cm1m .$ $Height:81cm ⋅100cm 1m Radius:20cm ⋅100cm 1m =0.81m=0.2m $ With this information, update the measures on the diagram.

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.

$LA_{cylinder}=2πrh$

SubstituteII

$h=0.81$, $r=0.2$

$LA_{cylinder}=2π(0.2)(0.81)$

Evaluate right-hand side

CommutativePropMult

Commutative Property of Multiplication

$LA_{cylinder}=2(0.2)(0.81)π$

Multiply

Multiply

$LA_{cylinder}=0.324π$

The lateral area of a cone is the product of $π,$ the radius, and the slant height. $LA_{cone}=πrℓ $ The slant height $ℓ$ 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.$a_{2}+b_{2}=c_{2}$

SubstituteValues

Substitute values

$0.81_{2}+0.2_{2}=ℓ_{2}$

Solve for $ℓ$

CalcPow

Calculate power

$0.6561+0.04=ℓ_{2}$

AddTerms

Add terms

$0.6961=ℓ_{2}$

SqrtEqn

$LHS =RHS $

$0.6961 =ℓ$

RearrangeEqn

Rearrange equation

$ℓ=0.6961 $

$LA_{cone}=πrℓ$

SubstituteII

$ℓ=0.6961 $, $r=0.2$

$LA_{cone}=π(0.2)(0.6961 )$

CommutativePropMult

Commutative Property of Multiplication

$LA_{cone}=0.20.6961 π$

$(0.324π+0.20.6961 π)125$

$192.76$

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.

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b Find the surface area of the cone. Round the answer to the nearest square centimeter.

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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.

a Let $ℓ$ 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 $ℓ+r$ centimeters.
Now, a system of equations can be written using the two values of $ℓ.$ ${ℓ=102 −r2 −rℓ=4r (I)(II) $ ### Solving the System of Equations

The Substitution Method will be used to find the unknowns. First use the value of $ℓ$ to find the value of $r.$
Next, the value of $r$ can be substituted into the other equation to find the value of $ℓ.$
The radii of the circle and the sector are about $2.2$ and $8.8$ centimeters, respectively.

Since the diagonal of the square bisects the angle, the right triangle is also an isosceles triangle. Using the sine ratio of $45_{∘},$ an equation describing the relationship between $r$ and $ℓ$ can be written. $sin45_{∘}=ℓ+r10−r $
Since $sin45_{∘}=2 1 $, the above equation can be solved for $ℓ.$
The first equation is obtained. It is also known that the sector of circle with radius $ℓ$ and the circle with radius $r$ form a cone.

As shown above, $ℓ$ 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 $90_{∘},$ its arc length can be found using the relationship between arc length and arc measure.
$Arc Length:Circumference of Base: 36090 ⋅2πℓ2πr $
Because these two lengths are equal, another equation can be obtained.

$36090 ⋅2πℓ=2πr$

Solve for $ℓ$

$ℓ=4r$

${ℓ=102 −r2 −rℓ=4r (I)(II) $

Substitute

$(I):$ $ℓ=4r$

${4r=102 −r2 −rℓ=4r $

Solve for $r$

AddEqn

$(I):$ $LHS+r2 +r=RHS+r2 +r$

${5r+r2 =102 ℓ=4r $

FactorOut

$(I):$ Factor out $r$

${r(5+2 )=102 ℓ=4r $

DivEqn

$(I):$ $LHS/(5+2 )=RHS/(5+2 )$

$⎩⎪⎨⎪⎧ r=5+2 102 ℓ=4r $

UseCalc

$(I):$ Use a calculator

${r=2.204812…ℓ=4r $

RoundDec

$(I):$ Round to ${\textstyle 1 \, \ifnumequal{1}{1}{\text{decimal}}{\text{decimals}}}$

${r≈2.2ℓ=4r $

b The surface area of a cone is the sum of the base area and the lateral area.
$S=πr_{2}+πrℓ $
By substituting $ℓ=8.8$ and $r=2.2,$ the surface area of the cone will be determined.
The surface area of the cone is about $76$ square centimeters.

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.

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