Area of Circle Visualized
In this applet, we consider a circle, and divide the circle into slices of sectors. Then, we visualize the circle's area by rearranging the slices to form a rectangle.
We know that
1. the formula of Area(Rectangle) = Length x Breath
2. the formula of Circumference (Circle)=2⋅π⋅r.
3. In this rearrangement,
(i) area of circle =area of rectangle
(ii) the approximate length of the rectangle is half of the circumference.
Now,
Combining these three facts/formula,
Area of circle = area of rectangle
= ( π⋅r)⋅(r)
= π⋅r^2.
Conclusion, although we use approximations for the length and breath, the resulting formula for the circle's area is exact, because we can continue the approximation with large number of slices.