King Perimetras

fig.g10.2

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.

Fig,g10.1            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.