Make a shape using the 100 meter long string
Perimetras (fictitious name) was a generous king in Greece. His people liked him because they thought that he was a good protector. It helped that he did not collect very high taxes. One of his unique qualities was that Perimetras liked intellectuals, and he held a court every month. He was also very generous. One day, he called the intellectuals Acusilaus , Aeginetta, Domophon and Menaechmus.
King Perimetras: I will give each of you a 100 meter long string and any number of pegs you need. You will have to make a shape using this string and the pegs. Each one of you would have to make a different shape. The person who makes the shape with the largest area will get a prize of a house and 10 goats. Acusilaus, take the string and as many pegs as you want to make a shape.
Triangle, square and a hexagon
Acusilaus did not know of many shapes but a square. So he took 4 pegs and made a square with a parameter of 100 meters so that each side was 25 meters. The area of the square was 25 x 25 = 625 meter2.
Acusilaus: Your highness I have made a square. It is beautiful and has an area of 625 meter2.
King Perimetras: Domophon, now it is your turn. You cannot make a square because it has already been done.
Domophon made a triangle with each side being one third of the 100 meter perimeter. Thus each side was 33.3 meters.
King Perimetras: Domophon, I see that you made an equilateral triangle. What is its area?
Menaechmus: I am a mathematician and up to date on the work of Pythagoras. This triangle would have a base of 33.3 meters and a height of 28.83 meters. Therefore, area of the triangle ABC will be base AB x height CD/2 or 33.3 x 28.83/2 which equals 480.16 meter2. Your highness, I can also give you a proof of this area.
King Perimetras: So even smaller than the square. Aeginetta, now it is turn.
Aeginetta: Your highness, I need 6 pegs and I will make a hexagon – a shape with 6 equal sides. The string is 100 meters long. So each side will be 16.67 meters long.
Menaechmus: I can tell you the area of this shape. This shape is the sum of 6 equilateral triangles each with a base of 16.67 meters (100/6). The height of each triangle will be 14.42 meters. The total area of the shape will be 6 x 16.67 x 14.42 /2 meters2 which equals 720.25 meter2.
King Perimetras: This is the best shape so far. Again, I take it that you can give a proof. Menaechmus, does that mean the hexagon wins the prize?
Area to perimeter ratio increased with the number of sides
Menaechmus: Your highness as you see the 3 sided shape had the smallest area, 4 sided shape had higher area but the 6 sided shape was the highest so far. The ratio of the area to perimeter increased with the number of sides of the shape. I will make an infinite sided shape which is a circle. The circle would have a circumference of the length of the string (100 meters) which mean its radius will be 100/(2π) or 15.92 meters. The area of this circle will be π x radius2 which comes out to 795.77 meter2. That’s the largest area you can get with the 100 meter long string.
King Perimetras: You win the prize. I am impressed how you showed me that the more sides you have in a shape the larger is the ratio of the area to the perimeter.
Proofs
Use the pictures drawn in the story.
Area of the equilateral triangle: The triangle ABC would have a base of 33.3 meters. Draw a line CD which is vertical to AB. The angles ADC and BDC are both right angles, angle A and angle C are equal because it is an isosceles triangle. The side CD is common to the triangles ACD and BCD. Therefore, the two triangles are congruent. Hence AD = BD =33.3/2 =16.65 meters
ACD is a right angle triangle with a base of 16.65 meters and a hypotenuse of 33.3 meters. Therefore, the height will be 28.83 meters because according to Pythagoras hypotenuse2 = base2+ height2. Therefore, area of the triangle ABC will be base AB x height CD/2 or 33.3 x 28.83/2 qhich equals 480.16 meter2.
Area of the hexagon: Hexagon is a shape with 6 equal sides. Draw lines connecting the corners -each angle will be 360/6 or 60°. All angles of a triangle add up to 180° and hence each of the other angles of this isosceles triangle will be (180-60)/2 or 60°. All three angles being equal, it is an equilateral triangle. Therefore, this shape is the sum of 6 equilateral triangles each with a base of 16.67 meters (100/6). Again, one can prove using Pythagoras theorem that the height of each triangle will be 14.42 meters. Each of the equilateral triangle will have an area of 16.67 x 14.42 /2 meters2 which is 120.04 meter2. So the total area will be 6 x 120.04 meter2 or to be exact 720.25 meter2.
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