What is the surface area of a size 5 soccer ball?

Answer 1

The circumference of a Size five soccer ball is 27-28 inches according to the rules of soccer. (FIFA RULE BOOK)
Now if C= the circumference, we know that C=2Pi(r), where r is the radius and Pi is the constant Pi which we approximate with 3.14. You may use 22/7 if you prefer.
(If you want a better approximation, consider using Pi = 3.14159265, however, it does not make sense to use Pi to 4 or 5 significant digits if the measurement of the circumference is not even close to that accurate. That is why we did not do so here.)

The surface area of a sphere is 4Pir2 . A soccer ball is NOT a sphere, but this is a good approximation. (In reality it is a truncated icosahedron which is made up of regular pentagons and regular hexagons. Once you add the air to the ball, it becomes close enough to a sphere to use a spherical model as an excellent approximation. Once could easily calculate the area of 1 of the hexagons and 1 of the pentagons and find the surface area exactly. It might be fun to do so and compare to the answer using the sphere!)

So first we find that ( C2 /4Pi2 )=r2
This come from taking C=2Pi(r) which is the definition of circumference of a sphere and dividing both sides by 2xPi giving
C/2Pi =r, then square both sides and you have ( C2 /4Pi2 )=r2 . Now substitute that into the formula for surface area.

SA= 4Pir2 =4Pi( C2 /4Pi2 ).
Now note that the 4 cancel as does one of the Pi's
so we have
SA=C2 /Pi
=272 /3.14 approximately 232.2 square inches. This is about 1.6 square feet.
Now if we use C=28 we have about 249.682 square inches which is 1.7 square feet.
So the answer is between 1.6 and 1.7 square feet.

Method 2


We can also start with circumference C=27- 28 inches.
and C=2Pi(r) as mentioned above.
So r=14/Pi or about 4.456 inches. This is the radius of the soccer ball.
Now Surface area is 4XPiXr2 = 4xPix4.4562
this is 249.55 square inches.
Using 27 instead of 28 we have r=4.29718
and Surface area is 4xPix4.297182 =232.048 square inches.
When we convert 249.55 to square feet by dividing by 144 square inches per square foot, we get 1.7 square feet ( 1.73299) and for 232.048 we get 1.6 (1.61144). So the answer is the same. There are some rounding errors and the problem can be redone to any degree of accuracy needed.

The advantage of Method 1 is that you find a formula for the SA, the surface area by using the circumference. This method will allow you to plug in the circumference for a size 4 ball, for example and find its area right away. Method 2 requires you to find the radius give the circumference. This works well too, but you need to find r each time C changes.