A 4 m long wire is shaped into a square and into a hexagon. What is the increase in area of the figure when this done.
A square is a figure with 4 equal sides. If the length of the sides is a, the area of the square is equal to a^2. A hexagon is a figure with 6 equal sides and if the length of the sides is a the area of the hexagon is given by `(3*sqrt3*a^2)/2`
If the 4 m long wire is used to create a square, the length of each side is 4/4 = 1 m. The area of this square is 1 m^2.
When a hexagon is created with the wire the length of each side is 4/6 = 2/3 m. The area of this hexagon is `(3*sqrt3*(4/9))/2` = `3*sqrt3*2/9` = `2/sqrt 3` `~~ 1.1547` m^2.
[Notice that the area has increased. This continues as the number of sides of the figure created in increased. A circle is a figure with an infinite number of sides. If the wire is used to create a circle the area of the circle is `pi*(4/(2*pi))^2 = 4/pi ~~1.27` m^2]