Area of a Circle

A circle with radius $r$ will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is $2\pi r,$ the arc length of each semicircle is half this value, $\pi r.$

Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. As such, the area of the figure below should be the same as the circle's area.

It can be noted that if the circle is divided into more and smaller sectors, then the figure will begin to look more and more like a rectangle. Here, the shorter sides become more vertical and the longer sides become more horizontal. If the circle is divided into infinitely many sectors, the figure will become a perfect rectangle with base $\pi r$ and height $r.$ Since the area of a rectangle is the product of its $\text{height}$ and its $\text{base},$ the following formula can be derived.

It has been shown that the area of a circle is the product of $\pi$ and the square of its radius.