Area of a Circle
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.
Now, it is teacher's turn to let student to explore the applet by changing the number of sectors using the slider "Make more slices". Then student will notice that as the number of sectors increases.
Next, student will unfold the sector components using the slider "Unfold the circle".
Then, students will resemble both semi circle into a rectangle closely using the slider "Move up/Down". This helps to fit the both parts into a single rectangle.