ABCDEF is a regular hexagon. The area of the BCEF figure is 40 cm ^ 2. What is the area of the hexagon ABCDEF?

Let the length of the side of the hexagon be X cm.

The inside corners of a regular hexagon are 120.

Then, in an isosceles triangle CDE, by the cosine theorem, we define the length of the segment CE.

CE ^ 2 = CD ^ 2 + DE ^ 2 – 2 * CD * DE * Cos120.

CE ^ 2 = X ^ 2 + X ^ 2 – 2 * X * X * (-1/2) = 3 * X2.

CE = X * √3 cm.

Quadrangle BCEF is a rectangle, then Svsef = BС * CE = 40 cm2.

X * X * √3 = 40.

X2 = 40 / √3 cm2.

The area of a regular hexagon is: S = 3 * √3 * X ^ 2/2 = 3 * √3 * (40 / √3) / 2 = 60 cm2.

The area of the hexagon is 60 cm2.