Problem: Find the circumference of a circle of radius R.

Solution: We position the circle with its center in the origin:

Just like when we calculated the area of a disc, we will use three approaches. Since the various formulas describing circle were already quoted there, we will be brief here.

First will use the graph of a function approach. Again, by the symmetry, the circumference of the circle is equal to the double of the length of the upper semi-circle. The appropriate function is

Thus the circumference is

If we use the parametric equations x = Rcos(t), y = Rsin(t), we can directly calculate the whole circumference by integrating for instance for t between -π and π.

Finally, in the polar coordinates, the circle is given by ϱ(φ) = R for φ for instance between 0 and 2π. The circumference is

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