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Monday, February 11, 2013

Circle - Circumference Derivation

Category: Integral Calculus, Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

If the equation of a circle is x2 + y2 = r2, prove that the circumference of a circle is C = 2πr.

Solution:

To illustrate the problem, let's draw the graph of a circle as follows


Photo by Math Principles in Everyday Life

If the equation of a circle is



We can rewrite the above equation as a function of x as follows





Take the derivative of the above equation with respect to x as follows



To get the length of a curve or circumference of a circle, consider only a quadrant as follows


Photo by Math Principles in Everyday Life

The length of a curve is given by the formula







Substitute the value of dy/dx to the above equation, we have





















Therefore, the circumference of a circle is