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The Surface Area of a Sphere
The formula for the surface area of a sphere was discovered by Archimedes. In the
diagram below a cylinder just encloses a sphere of radius r. Archimedes was able
to determine the formula by showing that a pair of parallel planes perpendicular to
the vertical axis of the cylinder, would enclose equal areas on both shapes.
2r
Surface area
r
2r

Surface area
= 2r x 2r
2r
= 4r2
r
Archimedes did
not have
Painting
the the
surface
of a of a sophisticated
Archimedes
wasadvantage
intrigued
algebra like we
Hediscovery.
had to express relationships
sphere
uses the
same
by use
this today.
amazing
in terms of simpler
shapes.
For him the surface
amount
of paint
as
painting
Why isgeometric
the
answer
exactly
of a sphere
was
to the
area of 4 of the greatest
four
ofnot
itsequal
greatest
circles!
4 and
4.342?
2area
2
circles that it could contain.
r2
r
Surface Area
4r2
r2
Example Questions: Calculate the surface area of the spheres below. (to 1 dp)
SA = 4r2
1
2
7.3 cm
SA = 4r2
SA = 4 x  x 7.32 = 669.7cm2
12 cm
SA = 4r2
SA = 4 x  x 122 = 1809.6 cm2
Questions: Calculate the surface area of the spheres below. (to 1 dp)
SA = 4r2
1
2
2.4 m
3.2 m
SA = 4r2
SA = 4 x  x 3.22 = 128.7 m2
SA = 4r2
SA = 4 x  x 2.42 = 72.4 m2
Example Questions: Calculate the radii of the spheres shown below. (to 1 dp)
SA = 4r2
1
2
SA = 1500 cm2
4r2 = 1500
1500
r 
4
1500
r
 10.9 cm
4
2
SA = 3500 cm2
4r2 = 3500
3500
4
3500
r
 16.7 cm
4
 r2 
Questions: Calculate the radii of the spheres shown below. (to 1 dp)
SA = 4r2
2
1
SA = 8.4 m2
SA = 1200 cm2
4r2 = 1200
4r2 = 8.4
 r2 
r
8 .4
4
8.4
 0.82m
4
 r2 
r
1200
4
1200
 9.8 cm
4
Example Questions: Calculate the surface area of the spheres below. (to 1 dp)
SA = 4r2
1
2
7.3 cm
12 cm
Worksheet 1
Questions: Calculate the surface area of the spheres below. (to 1 dp)
SA = 4r2
2
1
3.2 m
Worksheet 2
2.4 m
Example Questions: Calculate the radii of the spheres shown below. (to 1 dp)
SA = 4r2
1
2
SA = 1500 cm2
SA = 3500 cm2
Worksheet 3
Questions: Calculate the radii of the spheres shown below. (to 1 dp)
SA = 4r2
2
1
SA = 8.4 m2
SA = 1200 cm2
Worksheet 4