Question

a regular hexagon is composed of 12 congruent 30-60-90 degree triangles. if the length of the hypotenuse of one of those triangles is 18sqrt3, find perimeter of hexagon.

Answer 1

Each side of the hexagon must be split into 2 exactly in the middle to make 12 congruent 30-60-90 triangles. That point (the midpoint) will form 2 line segments with the other endpoints of one of the vertexes of the hexagon and the center of the hexagon. Those two segments are the 2 legs of the triangle, and the hypotenuse is the segment connecting the center of the hexagon to the vertex. This segment splits the vertex in half perfectly, and because a regular hexagon has angle measures of 120°, the measure of the new angle created must be 60°. The length of the leg on top of the side of the hexagon must be across from the 30° angle because it is adjacent to the 60° angle. Using the proportions explained in 1, if the hypotenuse is 18√(3), than the side length across from the 30° angle is half the length of the hypotenuse, or 9√(3). Because this is only half the length of that side of the hexagon, multiply that by 2 to get that the length of one side of the hexagon is 18√(3). You want the perimeter of the hexagon, so multiply that by 6 (for each side) to get that the perimeter of the hexagon is 108√(3) (about 187.061) units.