“What goes around comes around”

“What is the area of a triangle ? “. ” Half into base into height” – came out a loud monotone of all the 5th-grade students in unison. ” Area of a rectangle ?” ..”Length into breath”. ” Area of trapezium…” .. This call and answer went back and forth for another ten minutes until all the formulae were memorized by all the students except one student, who was promptly picked up by the math professor for his inattention.

The puzzled child mustered up courage somehow and questioned his professor ” Sir, I can visualize why an area of the rectangle would be LB or why an area of ……etc but I can’t understand how the area of the circle would be pir*r ? Annoyed for not contributing in the “formulae chanting” the professor punished the child, silently dismissing his original doubt and wondering himself how it was derived.

While taking her daily stroll the Vice-principal observed this child punished outside the class. Reminiscing about her rebellious childhood, she came to the child’s aid. ” Oh, so you’re having trouble understanding a formula ? Don’t worry I will explain this to you,” she said merrily.

“Consider you have a long clew ( a ball of yarn ). Now take out a thread and make a circle. The perimeter will be 2piradius.(Perimeter of a circle is proportional to its diameter and this relationship had been shown by lot of mathematicians. So not that difficult to assume this relationship )Now take out another thread slightly smaller than the previous one. Make a circle inside the bigger one completely touching it. Keep doing this till you have a complete circle. Now the area of this circle should remain same even if I rearrange the same threads in any different manner. So let me take out the outermost thread and make it straight. This will be 2piradius. Take out the next one and the next one and keep stacking them one above the other. What you will get at the end will be a right-angled triangle with base = 2pir and height = r. The area of this triangle is very easy and the formula can also be visualized which is 1/2 * base height. This, when you calculate, will be 1/2 * 2pi*r * r = PI * r * r ! The formula !”

It became all clear to him after doing this thought experiment. “Now I get it why it was pi-r-square” beamed the 5th grader.

The child was sent inside the class and the professor was called out !

– Math Lover.