Express the area of this figure in simplest radical form. There are two right angles, and four other angles that are marked as being congruent.
The angles in a hexagon total (6 – 2) x 180 = 720°. We subtract the two 90’s (A and C), divide by 4, and see that the four congruent angles B, D, E, and F are each 135°. And 135 = 90 + 45, very nice.
There are doubtlessly many ways to slice and dice the figure 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(FBCG) – III.
I’s area is ½ (half a 1 x 1 square).
II is a rectangle 1 x √2, whose area is √2.
III is a 45-right triangle with leg √2 – 1; its area is half a square: ½(√2 – 1)2 =
½(2 – 2√2 + 1) = 1 – √2 + ½ = 1½ – √2.
The total area is ½ + √2 – (1½ – √2) =
½ + √2 – 1½ + √2 = 2√2 – 1.