Use the slider below to divide the circle into parts (from 10 to 100 pieces):
The circle (with radius \( r \)) on the right is divided into equal sectors (cake pieces). These pieces are then “cut” and rearranged to form a shape that approximates a rectangle. The height of this rectangle is approximately the radius \( r \) of the circle, and the width is nearly equal to half of the circumference (i.e. \( \pi r \)). Thus, the area of the rectangle is \( r \times (\pi r) = \pi r^2 \), which is the area of the circle.
As the number of pieces increases, the curved edges of the sectors become nearly straight, and the approximation improves. In the limit, when the circle is divided into infinitely many pieces, the rearranged shape becomes a perfect rectangle, and the formula \( \pi r^2 \) is exactly obtained.