Area and Perimeter

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There was once a wise man who had pleased a certain ruler with his wit and intelligence. The ruler wanted to give him a reward. "Give me some land kind Sir" said the old man. "Okay... just tell me the area and the land is yours" said the king. "No no it's not some area that I ask of you.... I will tell you the perimeter of the land and you give it to me". "That's funny! Everyone who wants to buy land uses the term 'area'. I have never heard of anyone dealing in land and using the term 'perimeter'". But the old man insisted on getting the land by using perimeter as the term for measurement. "I think you are up to something. What is it?" asked the king. The old man then asked for a chess board and a string of length 16 cm. He held the string on the board to form a square and showed the king that it covered a square 4cm x 4cm. Hence the area within the square was 16 sq. cm. Now he held the string in such a way that it formed a rectangle 5cm x 3cm. The area was 15 sq cm. Next he held the string to form a triangle and asked the king to count how many squares were in. There were roughly 12 squares in and that made the area of the triangle 12 sq cm. "so you see, O great king, the perimeter of all these figures is the same, but the area contained in them differs."

The king said "I see your point there. The perimeter may be the same but the area depends on the shape of the figure.And which of these figures do you want your land to be?" "Oh please give me a circle because that will mean maximum area." said the old man. The king was more than happy to grant his wish as he had learned something very interesting that day.

Reflection: Why did the old man ask for land in the shape of a circle?

Hint: For a string 16 cm long,

A three sided figure gave an area of approximately 15 sq cm.

A square (four sided figure) had area 16 sq cm

An octagon with eight sides each 2cm long would enclose an area of approximately 19.31 Sq cm.

Did you notice a relationship between number of sides and area of the polygon? Now answer your question for reflection