Surface Area and Volume of
Cones
Jen Kershaw
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Printed: November 17, 2013
AUTHOR
Jen Kershaw
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C ONCEPT
Concept 1. Surface Area and Volume of Cones
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Surface Area and Volume of
Cones
Here you’ll learn to find the surface area of cones using formulas.
Have you ever tried to figure out the covering of a cone? Take a look at this dilemma.
Jessica came to buy paper at the card store. She is decorating conical party hats for her party by wrapping them in
colored tissue paper. Each hat has a radius of 4 centimeters and a slant height of 8 centimeters.
If she wants to wrap 6 party hats, how much paper will she need?
This Concept will teach you how to find the surface area of cones. Then you will know how to figure out this
dilemma.
Guidance
Previously we worked on pyramids and surface area. Let’s look at cones.
Cones have different nets. Imagine you could unroll a cone.
The shaded circle is the base. Remember, cones always have circular bases. The unshaded portion of the cone
represents its side. Technically we don’t call this a face because it has a round edge.
To find the surface area of a cone, we need to calculate the area of the circular base and the side and add them
together.
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The formula for finding the area of a circle is A = πr2 , where r is the radius of the circle. We use this formula to find
the area of the circular base.
The side of the cone is actually a piece of a circle, called a sector. The size of the sector is determined by the ratio
of the cone’s radius to its slant height, or rs .
To find the area of the sector, we take the area of the portion of the circle.
A = πr2 · rs
This simplifies to πrs.
To find the area of the cone’s side, then, we multiply the radius, the slant height, and pi.
This may seem a little tricky, but as you work through a few examples you will see that this becomes easier as
you go along.
What is the surface area of the figure below?
Now that we have the measurements of the sides of the cone, let’s calculate the area of each. Remember to use
the correct area formula.
bottom face
side
A = πr2
A = πrs
2
π(5)
π(5)(11)
25π
π(55)
78.5
55π
172.7
We know the area of each side of the cone when we approximate pi as 3.14. Now we can add these together to
find the surface area of the entire cone.
bottom face
78.5
side
+
surface area
172.7 = 251.2 in.2
Now we can look at what we did to solve this problem. We used the formula A = πr2 to find the area of the
circular base. Then we found the area of the side by multiplying πrs. When we add these together, we get a surface
area of 251.2 square inches for this cone.
Cones have a different formula because they have a circular base. But the general idea is the same. The
formula is a short cut to help us combine the area of the circular base and the area of the cone’s side. Here’s
what the formula looks like.
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Concept 1. Surface Area and Volume of Cones
SA = πr2 + πrs
The first part of the formula, πr2 , is simply the area formula for circles. This represents the base area. The
second part, as we have seen, represents the area of the cone’s side. We simply put the pieces together and
solve for the area of both parts at once. Let’s try it out.
What is the surface area of the cone?
We know that the radius of this cone is 3 inches and the slant height is 9 inches. We simply put these values in
for r and s in the formula and solve for SA, surface area.
SA = πr2 + πrs
SA = π(32 ) + π(3)(9)
SA = 9π + 27π
SA = 36π
SA = 113.04 in.2
This cone has a surface area of 113.04 square inches.
A cone has a radius of 2.5 meters and a slant height of 7.5 meters. What is its surface area?
This time we have not been given a picture of the cone, so we’ll need to read the problem carefully. It tells us the
radius and the slant height of the cone, though, so we can put these numbers in for r and s and solve.
SA = πr2 + πrs
SA = π(2.52 ) + π(2.5)(7.5)
SA = 6.25π + 18.75π
SA = 25π
SA = 78.5 in.2
This cone has a surface area of 78.5 square inches.
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Take a few minutes to write this formula down in your notebooks.
Now try a few of these on your own.
Find the surface area of each cone.
Example A
Radius = 4 in, slant height = 6 in
Solution: 125.6 in2
Example B
Radius = 3.5 in, slant height = 5 in
Solution: 93.42 in2
Example C
Radius = 5 m, slant height = 7 m
Solution: 188.4 m2
You’re right! Just be sure to put the numbers in the correct place in the formula!
Now back to Jessica and the hats.
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Concept 1. Surface Area and Volume of Cones
Jessica is decorating conical party hats for her party by wrapping them in colored tissue paper. Each hat has a radius
of 4 centimeters and a slant height of 8 centimeters. If she wants to wrap 6 party hats, how much paper will she
need?
This problem involves a cone. It does not include a picture, so it may help to draw a net. In your drawing, label the
radius and the slant height of the cone. We can also use the formula. We simply put the radius and slant height in
for the appropriate variables in the formula and solve for SA.
SA = πr2 + πrs
SA = π(4)2 + π(4)(8)
SA = 16π + 32π
SA = 48π
SA = 150.72 cm2
Jessica will need 150.72 square centimeters of tissue paper to cover one hat.
But we’re not done yet! Remember, she wants to cover 6 party hats. We need to multiply the surface area of
one hat by 6 to find the total amount of paper she needs: 150.72 × 6 = 904.32.
Jessica will need 904.32 square centimeters of paper to cover all 6 hats.
Vocabulary
Cone
a three-dimensional object with a circle as a base and a side that wraps around the base connecting at one
vertex at the top.
Surface area
the measurement of the outer covering or surface of a three-dimensional figure.
Guided Practice
Here is one for you to try on your own.
Figure out the surface area of a cone with a radius of 4.5 inches and a slant height of 8 inches.
Answer
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To figure this out, we can use the formula for finding the surface area of a cone.
SA = πr2 + πrs
SA = π(4.5)2 + π(4.5)(8)
SA = 63.585 + 113.04
SA = 176.625
SA = 176.63 in2
This is our answer.
Video Review
Watch this video about the surface area of cones.
Practice
Directions: Find the surface area of each cone. Remember that sh means slant height and r means radius.
1. r = 4 in, sh = 5 in
2. r = 5 m, sh = 7 m
3. r = 3 cm, sh = 6 cm
4. r = 5 mm, sh = 8 mm
5. r = 8 in, sh = 10 in
6. r = 11 cm, sh = 14 cm
7. r = 12 in, sh = 16 in
8. r = 3.5 cm, sh = 6 cm
9. r = 4.5 mm, sh = 7 mm
10. r = 6 cm, sh = 8 cm
11. r = 7.5 cm, sh = 9 cm
12. r = 10 cm, sh = 12 cm
13. r = 16 cm, sh = 18 cm
14. r = 13 cm, sh = 20 cm
15. r = 15.5 cm, sh = 18.5 cm
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