Lesson Notes
The Surface Area of a Sphere
Remember that a sphere looks like this:
DIAMETER
Sometimes we will know the total surface
area of a sphere, and we will be asked to find
its radius.
Calculate the radius of the following sphere,
given the information below.
RADIUS
The formula for the total surface area of a
sphere is:
( )
A sphere = ( 4 )(π ) r 2
A sphere = 1809.56 m 2
Where “r” is the radius.
Let’s find the total surface area of a sphere.
( )
= ( 4 )(π ) ( r )
A sphere = ( 4 )(π ) r 2
Calculate the total area of the following
sphere, given the information below.
1809.56 m 2
2
1809.56 m 2
= r2
( 4 )(π )
7m
1809.56 m 2
= r2
12.57
144 m 2 = r 2
We know that the radius r = 7 m.
Use
( )
A sphere = ( 4 )(π ) r 2
= ( 4 )(π )( 7 m )
144 m 2 = r
2
r =12 m
= 615.75 m 2
Finally, we need to know how to calculate the
total surface area of a hemisphere.
615.75 m2 is the total area.
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1

Lesson Notes
Remember that a hemisphere looks like this:
Calculate the total surface area of the
following hemisphere, given the information
below.
RADIUS
5 cm
To calculate the area of the top, we will use
the equation for the area of a circle:
We know that r = 5 cm
( )
Atop = (π ) r 2
Use
= (π )( 5 cm )
To calculate the area of the bottom, we will
use the equation for the area of a sphere and
divide by 2:
A hemisphere =
( )
Atop = (π ) r 2
2
= 78.54 cm 2
( 4 )(π ) ( r 2 )
Use
2
A hemisphere =
=
( 4 )(π ) ( r 2 )
2
2
( 4 )(π )( 5 cm )
2
314.16 cm 2
=
2
= 157.08 cm 2
To calculate the total surface area, we simply
add the two above results together.
Atotal = Atop + A hemisphere
Let’s try an example of this.
Use
Atotal = Atop + A hemisphere
= 78.54 cm 2 + 157.08 cm 2
= 235.62 cm 2
235.62 cm2 is the total surface area.
We are ready to try more challenging
questions.
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2

Lesson Notes
2.
Example
1.
Calculate the total area of the following
sphere, given the information below.
Calculate the total area of the following
sphere, given the information below.
Diameter = 5 cm
Radius = 2 cm
Step 1: State the information that you
know
d = 5 cm
Step 1: State the information that you
know
Step 2: Use the total surface area
equation for spheres
r = 2 cm
( )
A sphere = ( 4 )(π ) r 2
Step 2: Use the total surface area
equation for spheres
= ( 4 )(π )( 2.5 cm )
( )
A sphere = ( 4 )(π ) r 2
= ( 4 )(π )( 2 cm )
2
= 78.54 cm 2
2
78.54 cm2 is the total surface area.
= 50.27 cm 2
50.27 cm2 is the total surface area.
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r = 2.5 cm
3

Lesson Notes
3.
4.
Calculate the total area of the following
hemisphere, given the information
below.
Radius = 3 cm
Calculate the radius of the following
hemisphere, given the information
below.
A sphere = 1520.53 cm 2
Step 1: State the information that you
know
Step 1: State the information that you
know
r = 3 cm
Step 2: Use the area of the top (circle)
A sphere = 1520.53 cm 2
( )
Atop = (π ) r 2
= (π )( 3 cm )
Step 2: Use the total surface area
equation for spheres to find the radius
2
Step 3: Calculate the area of the bottom
(hemisphere)
A hemisphere =
1520.53 cm 2
1520.53 cm 2
= r2
( 4 )(π )
( 4 )(π ) ( r 2 )
2
( 4 )(π )( 3 cm )
=
2
1520.53 cm 2
= r2
12.57
r 2 = 121 cm 2
2
113.1 cm 2
=
2
r = 121 cm 2
r = 11 cm
= 56.55 cm 2
r = 11 is the radius
Step 4: Calculate the total surface area
Atotal = Atop + A hemisphere
= 28.27 cm 2 + 56.55 cm 2
= 84.82 cm 2
84.82 cm2 is the total surface area.
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( )
= ( 4 )(π ) ( r )
A sphere = ( 4 )(π ) r 2
= 28.27 cm 2
4
2

Lesson Notes
5.
The Earth has a spherical shape with a
diameter of 12 800 Km. What is the total
surface area of the northern hemisphere
of the Earth?
Step 1: State the information that you
know
d = 12 800 Km
r = 6 400 Km
Step 2: Use the area equation for
hemispheres
A hemisphere =
( 4 )(π ) ( r 2 )
2
( 4 )(π )( 6400 Km )
=
2
2
=
514718540.4 Km 2
2
= 257 359 270.2 Km 2
257 359 270.2 Km2 is the total surface
area of the northern hemisphere.
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