3420.73 – Crazy Baseball Diamond

Here we have a crazy baseball diamond: six bases including two home plates. Express the area of this figure in simplest radical form. There are two right angles. The remaining angles all equal.

Solution

The angles in a hexagon total $(6 - 2) \times 180 = 720$ degrees. We subtract the two 90's, divide by 4, and see that the four congruent angles B, D, E, and F are each 135°. 135 = 90 + 45, very nice.

There are doubtless many ways to slice the figure up to get its area. A simple one is to extend CD and FE to meet at G. Then we have a pentagon, like home plate.

The area, then, is I + II (that's FBCG) - III.

I's area is 1/2 (half of a 1 x 1 square).

II is a rectangle 1 x $\sqrt{2}$ , area $\sqrt{2}$ .

III is a 45-right triangle with leg, GD = $\sqrt{2}$ - 1; its area is half of a square: ( $\sqrt{2}$ - 1)( $\sqrt{2}$ - 1)/2 = (2 - 2 $\sqrt{2}$ + 1)/2 = 3/2 - $\sqrt{2}$ .

The total area is 1/2 + $\sqrt{2}$ - (3/2 - $\sqrt{2}$ ) = 1/2 + $\sqrt{2}$ - 3/2 + $\sqrt{2}$ = 2 $\sqrt{2}$ - 1.

Is there a cleverer way to go?