Knowing that sin(60) = √3 / 2 show the area of the base of the box is equal to 3√3x ^ 2/2

Question

Guiding context for the question: A hollow box of chocolates is made in the shape of a straight prism with a regular hexagonal base. The height of the prism is "h" cm, and the top and the base of the prism have sides of length "x"

Answer 1

The hexagon can be split into 6 equilateral triangles with all angles equal to 60° and all sides equal to x. If we find the area of one of these triangles, we can multiply it by 6 and we will have the area of the base of the box.

To find the area of one of these triangles we will need to find the height of the triangle. This can be found by using sin(60) = height/x = √(3)/2. The height will then be equal to √(3)x/2.

The area of a triangle is .5(base)(height), for us it is (1/2)(x)(√(3)x/2), which equals √(3)x^2/4

There are 6 triangles so 6 * √(3)x^2/4 = 3√(3)x^2/2