4. Volume and Surface Area of Cylinders, Cones, and Spheres
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Chapter 10
4.
Volume and Surface Area of Cylinders, Cones, and Spheres
Calculating the volume and surface area of cylinders, cones, and spheres is crucial in various fields such as engineering, architecture, and manufacturing. These shapes frequently appear in real-world applications, and understanding their properties enables accurate measurements and efficient problem-solving. By mastering these calculations, individuals can apply them to determine material requirements, design structures, and solve practical challenges. The process involves using specific formulas for each shape, considering their unique properties, such as the radius and height for cylinders and cones, and the radius for spheres and hemispheres.
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| | 14 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Volume and Surface Area of Cylinders, Cones, and Spheres
Slide of 14
In this lesson, several popular 3D figures like cylinders, cones, and spheres will be explored. To better understand their characteristics, their volume and surface area formulas will be introduced.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Relating the Volumes of Cones and Cylinders
The year is about to end and New Year's Eve parties are right around the corner. Tiffaniqua and her grandmother decide to bake chewy chocolate chip cookies for their gathering. As a part of their baking steps, Tiffaniqua fills a cone-shaped cup with flour and pours it into a cylindrical cup.
The cylindrical cup and cone-shaped cup have the same radius of 8 centimeters. The height of the cylindrical cup is 12 centimeters. What is the height of the cone-shaped cup if it takes three of them, completely filled with flour, to fill one cylindrical cup?
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Discussion
Cylinder
The first 3D figure of this lesson to breakdown is the cylinder. Here, the method of how to calculate the volume and surface area of a cylinder will be understood.
Rule
Volume of a Cylinder
Note that the surface area of a cylinder consists of the two equal circular areas and one rectangular lateral face.
Rule
Surface Area of a Cylinder
Consider a cylinder of height h and radius r.
The surface area of this cylinder is given by the following formula.
S=2πrh+2πr2
Example
Finding the Volume and Surface Area of a Cylinder
Tiffaniqua loves fireworks. Her and her grandmother team up to buy a box full of them to celebrate the new year. 
One of the types of fireworks that came in the box Tiffaniqua bought is in the shape of a cylinder. The firework's height is 18 centimeters and its radius is 4 centimeters.
a Calculate the volume of the cylindrical-shaped firework. Round the answer to the nearest integer.
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b Calculate the surface area of the cylindrical-shaped firework. Round the answer to the nearest integer.
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Hint
a The volume of a cylinder is the product of π, the square of its radius, and its height.
b The surface area of a cylinder is the sum of the areas of the base and top circles and the lateral face.
Solution
a It is a given that the firework is in the shape of a cylinder. This means we can use the formula for the volume of a cylinder to determine its volume.
It is known that h=18 and r=4. Substitute these two values into the formula.
The volume of a firework is about 905 cubic centimeters.
b It is now time to calculate the surface area of a cylindrical firework. Notice that the surface area of a cylinder consists of the areas of the base and top circles and the lateral face.
Recall the surface area formula as the sum of these three areas.
SA=2πrh+2πr2
Once again, substitute h=18 and r=4 into the surface area formula.
The surface area of a cylindrical firework is about 553 square centimeters. Pop Quiz
Practice Finding the Surface Area and Volume of a Cylinder
Find the volume or surface area of the cylinder. The radius and height are given in centimeters.
Discussion
Cone
The second 3D figure of this lesson to understand is the cone. One by one, the formulas of the volume and surface area of a cone will be considered.
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 B is the area of the circle and the height h is measured perpendicular to the base.
V=31Bh
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 r.
V=31πr2h
Note that the surface area of a right cone consists of a circular base and a curved lateral face.
Rule
Surface Area of a Cone
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 πr2 and the lateral area is πrℓ.
SA=πr2+πrℓ
Example
Finding the Volume and Surface Area of a Cone
During her winter vacation, Tiffaniqua wants to design a paper Christmas tree that spins! The tree will be made in a conical shape. She plans to use 136-centimeter sticks to make the framework of the lateral face. In that case, the radius of the base will be 64 centimeters. 
She needs to make some calculations to finish the tree's design.
The height, radius, and slant height of the cone form a right triangle, △ABC. Use the Pythagorean Theorem to find the height of the cone, AB.
Since a negative root does not make sense for a side length, consider only the positive root. The height of the cone is 120 centimeters. Now, recall the volume formula of the cone.
The volume of the cone is about 514719 cubic centimeters.
a Help her calculate the volume of the conical Christmas tree. Round the answer to the nearest cubic centimeter.
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b Help her calculate the surface area of the conical Christmas tree. Round the answer to the nearest square centimeter.
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Hint
a The volume of a cone is one-third of the product of the area of its base and height. Use the Pythagorean Theorem to find the height of the cone.
b The surface area of a cone is the sum of the area of its base and the area of its lateral face.
Solution
a The volume of a cone is one-third of the product of the area of its base and height. The base is already known, therefore, start by calculating the height of the conical Christmas tree. Recall that the distance from the vertex of the cone to its base is the height of the cone. The slant height of the cone is 136 centimeters and a radius is 64 centimeters.
AC2=AB2+BC2
SubstituteII
AC=136, BC=64
1362=AB2+642
▼
Solve for AB
CalcPow
Calculate power
18496=AB2+4096
SubEqn
LHS−4096=RHS−4096
14400=AB2
RearrangeEqn
Rearrange equation
AB2=14400
SqrtEqn
LHS=RHS
AB2=14400
SqrtPowToNumber
a2=a
AB=120
V=31πr2h
Here, r is the radius and h is the height of the cone. Substitute r=64 and h=120 into the volume formula.
V=31πr2h
SubstituteII
r=64, h=120
V=31π(64)2(120)
▼
Evaluate right-hand side
CalcPow
Calculate power
V=31π(4096)(120)
Multiply
Multiply
V=31⋅π⋅491520
MoveRightFacToNumOne
b1⋅a=ba
V=3491520π
CalcQuot
Calculate quotient
V=163840π
UseCalc
Use a calculator
V=514718.540364…
RoundInt
Round to nearest integer
V≈514719
b This time, the surface area of the cone will be calculated. Recall that the surface area of a cone is the sum of the base area and lateral face.
SA=πr2+πrℓ
Note that the radius of the base r is 64 centimeters and the slant height of the cone ℓ is 136 centimeters. Substitute these two values into the formula.
The conical tree will be about 40212 square centimeters. Note that this means Tiffaniqua needs to have at least 40212 square centimeters green paper to design the moving Christmas tree.
Pop Quiz
Practice Finding the Volume and Surface Area of a Cone
The applet shows right cones. The dimensions of the figure are given in decimeters. Use the given information to answer the question. If necessary, round the answer to one decimal place.
Discussion
Sphere
The next 3D figure of the lesson is a perfectly rounded figure called the sphere.
Concept
Sphere
Now, how to calculate the volume and surface area of a sphere will be examined.
Rule
Volume of a Sphere
Rule
Surface Area of a Sphere
The surface area of a sphere with radius r is four times pi multiplied by the radius squared.
Example
Finding the Volume and Radius of a Sphere
It snowed a lot before winter break. Tiffaniqua's school organized an inter-class snowperson competition on the last day before the holiday. The person who makes the biggest snowperson wins. 
Tiffaniqua and her class worked together. They decided to make two huge spherical snowballs. Then, they will put them on top of each other and shape the snowmperson. They plan that the head of the snowperson will have 1 meter of radius and the body will have 32 square meters surface area.
a Calculate the volume of a spherical snowball with a radius of 1 meter. Round the answer to the nearest cubic meter.
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b Calculate the radius of a spherical snowball with a surface area of 32 square meters. Round the answer to one decimal place.
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Hint
a Recall the formula for the volume of a sphere.
b Recall the formula for the surface area of a sphere.
Solution
a Start by recalling the formula for the volume of a sphere.
V=34πr3
It is known that the radius of the spherical snowball will be 1 meter. Substitute r=1 into the formula and calculate V.
V=34πr3
Substitute
r=1
V=34π(1)3
BaseOne
1a=1
V=34π⋅1
IdPropMult
Identity Property of Multiplication
V=34π
UseCalc
Use a calculator
V=4.188790…
RoundInt
Round to nearest integer
V≈4
b In the second scenario, they plan to make a snowball that has 32 square meters surface area. Recall the formula for the surface area of a sphere.
SA=4πr2
Now, substitute SA=32 into the formula to find r.
SA=4πr2
Substitute
SA=32
32=4πr2
▼
Solve for r
DivEqn
LHS/4π=RHS/4π
4π32=r2
RearrangeEqn
Rearrange equation
r2=4π32
SqrtEqn
LHS=RHS
r2=4π32
SqrtPowToNumber
a2=a
r=4π32
UseCalc
Use a calculator
r=1.595769…
RoundDec
Round to 1 decimal place(s)
r≈1.6
Discussion
Hemisphere
The final figure is the half of a sphere which is called hemisphere.
Concept
Hemisphere
A hemisphere is a three-dimensional object formed by half of a sphere and a flat circular base. Any plane that goes through the center of a sphere divides it into two hemispheres.
VolumeV=32πr3
The surface area of a hemisphere with radius r is the circular flat area plus half the surface area of a sphere with radius r. Surface Area2πr2+πr2=3πr2
Example
Finding the Surface Area and Radius of a Hemisphere
Tiffaniqua's father bought a big snow globe as a gift for her Christmas present.
a Since this is a present for Tiffaniqua, her father wants to use beautiful wrapping paper to make it even more special. The radius of the hemispherical part is 11 centimeters. Help him to find the surface area of this present to determine the amount of the wrapping paper. Round the answer to the nearest square centimeter.
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b Her father also has a box in the shape of a hemisphere to put the gift into it. The box has the volume of 3 cubic centimeters. Does the snow globe fit this box?
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Hint
a Recall the surface area formula of a hemisphere.
b Recall the volume formula of a hemisphere.
Solution
a The snow globe has a radius of 11 centimeters. Notice that the surface area of this hemispherical globe consists of a circular base and a half of the surface area of a sphere with the same radius, 11 centimeters. Recall the sum of these formulas.
SA=2πr2+πr2=3πr2
Now, substitute r=11 centimeters into the formula and calculate it.
SA=3πr2
Substitute
r=11
SA=3π(11)2
CalcPow
Calculate power
SA=3π⋅121
Multiply
Multiply
SA=363π
UseCalc
Use a calculator
SA=1140.398133…
RoundInt
Round to nearest integer
SA≈1140
b This time, the volume of the snow globe will be calculated. Recall that the volume of a hemisphere is half of the volume of sphere with the same radius.
Notice that the box is 3 cubic decimeters. Multiply it by 100 to convert cubic decimeters to cubic centimeters.
V=32πr3
Now, substitute r=11 into the volume formula.
V=32πr3
Substitute
r=11
V=32π(11)3
CalcPow
Calculate power
V=32π(1331)
MoveRightFacToNum
ca⋅b=ca⋅b
V=32662π
UseCalc
Use a calculator
V=2787.639881…
RoundInt
Round to nearest integer
V≈2788
3 dm 3=3000 cm3
Since the volume of the box is greater than 2788 cubic centimeters, the gift fits this box. The answer is yes. Pop Quiz
Finding the Volume or Surface Area of Different Spheres and Hemispheres
In the following applet, calculate either the volume or the surface area of the given sphere or hemisphere. The radius is given in centimeters. Round the answer to the nearest integer.
Closure
Calculating the Volumes of Cones and Cylinders
The challenge presented at the beginning of this lesson said that Tiffaniqua and her grandmother filled a cone-shaped cup with flour and poured it into a cylindrical cup.
It is given that the cylindrical cup and cone-shaped cup have the same radius of 8 centimeters. The height of the cylindrical cup is 12 centimeters. What is the height of the cone-shaped cup if she fills up the the cylinder after needing to pour into it three times?
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Hint
The volume of a cylinder is equal to the product of its base area and height. The volume of a cone is equal to one third of the product of its base area and height.
Solution
It is a given that the cylindrical cup is filled up by the given cone-shaped cup after three pours. This means that the volume of the given cylinder is equal to three times the volume of the given cone.
The height of the cone-shaped cup is 12 centimeters. It can be concluded that if the volume of a cylinder is three times the volume of a cone and their radii are the same then their heights should also be the same.
Vcylinder=3×Vcone
Recall that the volume of a cylinder is equal to the product of its base area and height. The volume of a cone is equal to one-third of the product of its base area and height. Note that both have circular bases.
Vcylinder=πr12h1Vcone=31πr22h2
Notice that both circular bases have the same radius. That is, r1=r2=8 centimeters. Now, substitute the radii, h1=h, and h2=12 into the corresponding formulas. Then, multiply the volume of the cone by 3 and set it equal to the volume of the cylinder.
Vcylinder=3×Vcone⇓π(8)2(h)=3⋅31π(8)2(12)
Simplify the obtained equation and solve it for h.
π(8)2(h)=3⋅31π(8)2(12)
▼
Solve for h
MoveLeftFacToNumOne
a⋅b1=ba
π(8)2(h)=33π(8)2(12)
CalcQuot
Calculate quotient
π(8)2(h)=1π(8)2(12)
IdPropMult
Identity Property of Multiplication
π(8)2(h)=π(8)2(12)
DivEqn
LHS/(π82)=RHS/(π82)
π(8)2π(8)2(h)=π(8)2π(8)2(12)
CancelCommonFac
Cancel out common factors
π(8)2π(8)2(h)=π(8)2π(8)2(12)
SimpQuot
Simplify quotient
h=12
Volume and Surface Area of Cylinders, Cones, and Spheres
Exercises
Tiffaniqua uses a cylindrical shaped storage box with the following dimensions.
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We will find the surface area of the given cylindrical box that has a base with a radius of 18 centimeters and height of 12 centimeters. Let's examine the net of a cylinder with radius r and height h.
Notice that the rectangular lateral surface is wrapped around the circular bases. Therefore, the unwrapped lateral surface is a rectangle with length 2πr, and width h. We write its area as a product of these two values. Lateral Surface=2π r h Its total surface area also includes the top and bottom circular bases. Circular Bases=2 *(π r^2) Let's now write these expressions as a sum. Then, substitute r=18 and h= 12 into this sum to find the total surface area of the box.
The surface area of the box is about 3393 square centimeters.
This time we will calculate the volume of the cylindrical box. Recall that its volume is the product of the area of its base and its height. Volume= B h Since it has a circular base, its area is π r^2. Volume=( π r^2)h Let's substitute r= 18 and h= 12 into the formula and calculate the volume.
The volume of the box is about 12 215 cubic centimeters.
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Gnarly Gnars sells the following sized ice cream cone.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"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:0.759765625em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">About<\/span><\/span><\/span><\/span><\/span>","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\">mm<\/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":["99353"]}}
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We want to find the surface area of the ice cream cone.
We will first calculate the radius of its circular base. Since its height, the slant height of the cone and radius form a right triangle, we will use the Pythagorean Theorem.
We found the radius of the circular base. Next, we will use the formula for the surface area of a cone. Note that its surface area consists of a circular base and a curved lateral face. SA=π r^2 +π r l Now that we know the radius and slant height of the cone l, we can substitute these values into the formula.
The surface area is about 13 463 square millimeters.
This time we will calculate the volume of the given cone. Let's recall the formula for the volume of a cone. V=1/3π r^2 h Now, susbtitute r= sqrt(759) and h= 125 into the formula.
The volume of the cornet is about 99 353 cubic millimeters.
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Kenny and Janelle are kicking around the following soccer ball with a diameter of 8.2 inches.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"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:0.759765625em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">About<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.924711125em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord Roboto-Regular\">in<\/span><span class=\"mord\">.<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.924711125em;\"><span style=\"top:-3.173603125em;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":["289"]}}
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We will calculate the surface area of the given soccer ball. Since it is a sphere, we will use following formula. SA=4 π r^2 We know the diameter of the ball. We will divide it by two to find the radius of the sphere. r=8.2/2= 4.1 Now that we know the radius of the ball, we can calculate its surface area.
The surface area of the ball is about 211 square inches.
This time we will calculate the volume of the given soccer ball. Let's recall the formula for the volume of a sphere. V=4/3π r^3 We will substitute r=4.1 into this formula.
The volume of the given soccer ball is about 289 cubic inches.
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Tiffaniqua wants to buy the following decorative lamp but she is unsure if it fits in her room. She figures that she should calculate its surface area and volume.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"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:0.759765625em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">About<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8238736249999999em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">cm<\/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":["10290"]}}
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We will calculate the surface area of the given lamp. Notice that it looks like the half of a sphere which is called a hemisphere. Its surface area includes two main parts. One is the half of a sphere with the same radius and the second one is the circular area at the base. Half of a Sphere + Circular Base 2π r ^2 + π r^2 Therefore, we can add these expressions to obtain the formula for the surface area of an hemisphere. 2π r ^2 + π r^2 = 3π r^2 In order to use the above formula we need to know the radius of the hemisphere. Note that we are told that its diameter is 17 centimeters. This means that we can find the radius by finding half of this value. r=34/2= 17 Now, substitute r= 17 into the surface area formula.
The surface area of the lamp is about 2724 square centimeters.
This time we will calculate the volume of the lamp. Notice that the volume of a hemisphere is half of the sphere with the same radius. V=2/3 π r^3 Let's substitute r= 17 into the above formula.
The volume of the lamp is about 10 290 cubic centimeters.
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Volume and Surface Area of Cylinders, Cones, and Spheres
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