Help with the pythagorean theorem?

Question

  How long is one blade of this windmill (x), if they are all the same length and the distance between the tips of any two adjacent blades is 6 yards?

 

(reduce the radical to simplest form)

Answer 1

There could be 3,4,5,6 ...  n blades. To generalize: the number of blades is the same as the number of sides in the regular polygon formed by joining the tips of adjacent blades. All of these polygons have side length 6 yards. Three blades, triangle. Four blades, square. Five blades, pentagon. Six blades, hexagon and so on. Let the number of blades (Number of sides in the polygon) be n. The central angle formed by the center of the polygon and two adjacent blades is  360°/n. All of these polygons have an apothem (a_n). Right triangles are formed with center as vertex, three  yards as one side, apothem as one side, and blade as the third side. Let x be the length of the blade. All these right triangles will have an angle equal to 360°/2n. The length of the blade is the hypotenuse of this angle.

 

sin (180°/n) = 3/x     or      x =3 /sin (180°/n)

Using this equation and knowing the number of blades (n), you can find the length of a blade, no matter how many blades there are.