CTK Insights

16 Feb

Area of a Circle

Euclid did not know yet that the same constant (\pi) appears in the formula for the circumference and the area of a circle. Archimedes did, athough his method of approximating either by exausting the circle with regular polygons does not make this quite obvious.

In the 16th century, Leonardo da Vinci, and in the 17th Sato Moshun cut the circle along the diagonals of the inscribed regular polygons and rearranged the sectors to form curvilinear parallelograms equal in area to that of the circle. For any number of subdivision sectors, the total of the (curvilinear) bases is 2\pi R - the circumference of the circle of radius R. As the number of subdivisions grow, the bases flatten and become close to straight lines and the total configuration appears close to a rectangle with the height of R. So in the limit the area is found to be \displaystyle\frac{2\pi R}{2}\cdot R, giving the area of the circle as \pi R^2.

However, sometime at the end of the 11th or the beginning of the 12th century, Rabbi Abraham bar Hiyya Hanasi came up with a different method of the determination of the area of a circle. He saw the interior of a circle as a collection of smaller circles which he all cut along a radius, peeled off in sequence and flattened into straight line segments. The construction is seen to approximate a triangle of height R and base 2\pi R, again leading to the formula \pi R^2.

(The diagrams above depict different stages of Rabbi Abraham's "peeling of a circle" that were generated interactively by a Java applet.)

Rabbi Abraham's book has been written in Hebrew but then translated into Latin and is said to be very popular at the time all over Europe. It is to be regretted that nowadays it is very little known.

Both methods mentioned above make it rather clear that the same constant (\pi) appears in both relevant formulas: the length of the circumference and the area of the circle.

3 Responses to “Area of a Circle”

  1. 1
    Math Teachers at Play #35 « Let's Play Math! Says:

    [...] In a word problem Tuesday, Kitten forgot how to find the area of a circle. Time for a hands-on lesson, cutting up a paper plate — and together we’ll read Alexander Bogomolny’s post: Area of a Circle. [...]

  2. 2
    Gordy Says:

    Cool explanations to prove for area of a circle formula.

  3. 3
    fractions worksheets Says:

    Beautifully explained area of a circle formula by both the methods.

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