having doubt on area of circle

Question

we know that the circumference of the circle is 2(pi)r so if we draw lines from each point to the centre then we should get the area of the circle and it should be 2*(pi)*r*r , ;) instead of (pi)*r*r , i know i am wrong but where i am wrong

please mail me the solution with this way only, i know some other way but i want to know this way also

Answer 1

You are wrong because however thin the lines are as soon as one line that is adjacent to another will start to overlap as they approach the center, so you are measuring a large part of the area multiple times.

 

For example, at one extreme consider a line consisting of a two 1" wide measuring tapes. On the circle's circumference the edges of the tape are next to each other using up 2 inches of the circumference, but as each tape has to go through the center, they overlap because you have to lay one tape over the other. At the other extreme, if the lines are infinitely thin they still overlap completely at the center, except if they have 0 thickness, in which case your equation is multiplying a lot of zeros.