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sin 45^{o} = 1/√2, also known as ½√2 (rationalized denominator).

To figure the area of a regular hexagon, draw one of the central angles (it measures 60^{o}) and note the triangle formed by the two radii and one of the sides of the hexagon--all 3 sides of this triangle are= r, radius of the circumscribing circle.

The equilateral triangle has area ¼r^{2}√3...(standard formula for equilateral triangle of side r). So just multiply that by six to account for the entirety of the regular hexagon.