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How do I find the surface area of a football?


The method depends on the tools available. With any method, you have to imagine the surface broken up into many pieces. find the area of each piece, and then add them together. Calculus gives us a mathematical operation known as integration for just this sort of thing. If you do not yet know integration through calculus, you can do it by hand.

Draw or imagine rings around the football. This breaks the football surface into many circular bands. With a string, measure the distance around each band and the width of each band. It will be most accurate if you measure at the center of the band rather than along either edge. The area of a band is (distance around)*(width). Add the bands together to find the area of the whole football. To do it like this in calculus, You need a formula for radius as a function of longitudinal position. The distance around, or circumference, is then (2*pi*radius).

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

The "shape" of a football is not mathematically simple because of its "pointy" ends, i.e. they are not rounded like an ellipse for example. To the extent that a football of length = 2a and width = 2b and is approximated by an ellipsoid of revolution the formula for the surface area is"

S = 2(pi)a^2 +[(pi)b^2/e] * ln[(1+e)/(1-e)]

where, 'e' is called the eccentricity, and: e = [ 1 - (b/a)^2]^1/2

This surface area would over-estimate the surface area a bit.

Vince Calder

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