Lesson Notes
The Surface Area of a Cone
Remember that a cone looks like this:
The total surface area of a cone is calculated
by adding the lateral area to the area of the
base. The formula for the total surface area is:
Radius
Height
Height
=
Atotal L.A.cone + Abase
slant
height
slant
height
or
Diameter
( )
Atotal = (π )( r )( s ) + (π ) r 2
The base of the cone is always a disc. The
area of the base is calculated by the formula:
Let’s do some examples of finding the surface
area of a cone.
( )
Abase = (π ) r
2
Calculate the lateral area of the following
cone, given the information below.
There is a Pythagorean relationship between
the slant height (s), the radius (r), and the
height (h).
2
2
s =r + h
h
Radius = 16 cm
Slant Height = 40 cm
2
s
We know that the radius r = 16 cm, and the
slant height s = 40 cm.
r
Use
The lateral area of a cone ( L.A.cone ) is the
surface area of a cone that does not include
the area of the base. The lateral area formula
is written as:
= 2010.62 cm 2
L.A.cone = (π )( r )( s )
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L.A.cone = (π )( r )( s )
= (π )(16 cm )( 40 cm )
Let’s look at a slightly more challenging area
of a cone question.
1

Lesson Notes
Calculate the total surface area of the
following cone, given the information below.
Radius = 1.5 cm
To find the total surface area, use:
=
Atotal L.A.cone + Abase
( )
= (π )( r )( s ) + (π ) r 2
Height = 2 cm
= (π )(1.5 cm )( 2.5 cm ) + (π )(1.5 cm )
= 11.78 cm 2 + 7.07 cm 2
We know that the radius r = 1.5 cm, and the
height h = 2 cm.
= 18.85 cm 2
We need to calculate the slant height (s).
2 cm
We are ready to do some more challenging
examples
s
1.5 cm
s 2 = (1.5 cm ) + ( 2 cm )
2
2
s 2 = 2.25 cm 2 + 4 cm 2
s 2 = 6.25 cm 2
s = 6.25 cm 2
s = 2.5 cm
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2
2

Lesson Notes
2.
Example
1.
Calculate the total surface area of the
following cone, given the information
below.
Calculate the lateral area of the
following cone, given the information
below.
20 m
11 cm
9m
5 cm
Step 1: State the information that you
know
r=9m
Step 1: State the information that you
know
r = 5 cm
s = 20 m
Step 2: Calculate the lateral area
s = 11 cm
L.A.cone = (π )( r )( s )
Step 2: Use the lateral area formula for
cones
= (π )( 9 m )( 20 m )
L.A.cone = (π )( r )( s )
= 565.49 m 2
= (π )( 5 cm )(11 cm )
Step 3: Calculate the area of the base
using the area formula for circles
= 172.79 cm 2
( )
Abase = (π ) r 2
172.79 cm2 is the lateral area.
= (π )( 9 m )
2
= 254.47 m 2
Step 4: Calculate the total surface area
by adding the lateral area to the area of
the base
=
Atotal L.A.cone + Abase
= 565.49 m 2 + 254.47 m 2
= 819.96 m 2
819.96 m2 is the total surface area.
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3

Lesson Notes
3.
Calculate the total surface area of the
following cone, given the information
below.
Step 3: Calculate the total surface area
=
Atotal L.A.cone + Abase
( )
= (π )( r )( s ) + (π ) r 2
Radius = 3 cm
= (π )( 3 cm )( 5 cm ) + (π )( 3 cm )
Height = 4 cm
2
= 47.12 cm 2 + 28.27 cm 2
= 75.39 cm 2
Step 1: State the information that you
know
r = 3 cm
4.
h = 4 cm
Lateral Area = 125.66 cm²
Step 2: Use the Pythagoras’ theorem to
find the slant height
4 cm
Area of Base = 78.54 cm²
s
Step 1: State the information that you
know
L.A.cone = 125.66 cm 2
3 cm
=
s2
( 4 cm ) + ( 3 cm )
2
Abase = 78.54 cm 2
2
=
s 2 16 cm 2 + 9 cm 2
Step 2: Find the radius using the area of
the base
s 2 = 25 cm 2
s = 25 cm
Calculate the slant height of the
following cone, given the information
below.
( )
= (π ) ( r )
Abase = (π ) r 2
2
78.54 cm 2
s = 5 cm
78.54 cm 2
= r2
π
25 cm 2 = r 2
25 cm 2 = r
r = 5 cm
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4
2

Lesson Notes
Step 3: Calculate the slant height using
the lateral area
Step 2: Find the radius using the area of
the base
L.A.cone = (π )( r )( s )
125.66 cm 2 = (π )( 5 cm )( s )
50.27 cm 2
125.66 cm 2
=s
(π )( 5 cm )
50.27 cm 2
= r2
π
2
16 cm 2 = r 2
2
125.66 cm
=s
15.71
s = 8 cm
5.
( )
= (π ) ( r )
Abase = (π ) r 2
16 cm 2 = r
r = 4 cm
Calculate the total surface area of the
following cone, given the information
below.
Step 3: Calculate the total surface area
=
Atotal L.A.cone + Abase
= (π )( r )( s ) + 50.27 cm 2
Area of Base = 50.27 cm²
= (π )( 4 cm )( 9 cm ) + 50.27 cm 2
Slant Height = 9 cm
= 113.10 cm 2 + 50.27 cm 2
= 163.37 cm 2
Step 1: State the information that you
know
Abase = 50.27 cm 2
s = 9 cm
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student with an active Math Help Services account
5