How is the area of ​​the circle πr² ?

figure-(a)

Let S is a circle whose radius is r.

Hence the circumference of the circle S is 2πr.

figure-(b)

Now, divide the circumference of the circle S into two equal parts. (Like in figure-(b))

∴ Arc (ACB) = Arc (ADB) = 2πt/2 = πr

Now, divide the above circle into many small sectors, where the number of sectors in the semicircle ACB is equal to the number of sectors in the semicircle ADB. (Like in figure-(c))

        Then arrange these sectors as shown in figure (d).

figure-(c)

figure-(d)

Now, the circle is converted into a parallelogram whose length is πr and its height is r towards its base.

∴ Area of ​​the parallelogram = length × height

                                                  = πr × r

                                                  = πr²

∴ The area of the circle is πr².

(Note:- If we divide the circle into more smaller sectors, then we get a rectangle instead of a parallelogram and here too the final result is πr².)

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