Calculation of the Circle’s Area Explained Visually

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Have you asked why we calculate the area of a circle with radius \( R \) as \( \pi R^{2} \)? There is a good reason behind it. We can tackle this problem from an integration point of view.

So, first things first. We split our circle in \( N \in \mathbb{N} \) rings and calculate the total area as a sum of the rings' areas. For a large \( N \), these rings will start resembling like rectangles with sides \( dr \) and \( 2\pi r \). Now, we can translate the problem to integrating the function \( 2\pi rdr \) on the interval \( [0, R] \). This principle is illustrated in the animation below by using Matplotlib's Animation API:

Numerical Calculation of the circle's area
Animation: Numerically calculate the area of a circle

The source code related this visualization can be found in this Python Notebook. For more information, please follow me on Twitter.

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