There's A Quick Trick!

By symmetry of the problem, we can easily deduce that the bottom part of the circle filled with red is congruent to the upper part. That means the required area is equal to the half the area of the square, $\Rightarrow A=\frac{12×12}{2}=\boxed{72}$

One must note symmetry (the line from circle's center intersects the corner of the square), from which we can infer that the two shaded sections are equal in area to the two unshaded sections. From this is follows directly that the shaded sections are half the area of a 12x12 square, or 72. FOR THOSE WHO NEED MORE CONVINCING: First examine the shaded circular sector. As its upper boundary is the diagonal of the square, the angle at the center is 45 degree; hence, this is 1/8 of a circle with radius=12, so its area is 12^2 * pi / 8 = 18 pi. As the area of the square is 144, the area outside the circle is thus 144-36 pi. By symmetry, the area of the shaded segment outside the circle is 72 - 18 pi. The sum of the two shaded areas is thus 18 pi + (72-18 pi) = 72.

This question almost tricked for a seconds until I realised the shaded area was half the square so 12*12/2=72

the shaded portion is half of the square, so the area of shaded portion is half of the area of square with 12 units side... so the area A=(12x12)/2=144/2=72

by adding shaded part we got the half of the square. area of half of square is 12*12/2=72

First find area of square that is 144 units. And then we have to imagine that the sum of area of shaded region is half of area of square as diagonal divides the cicle into 45 degrees angle sectors. so answer is 72

Move the red part included in the circle to the white part with the same shape and the red surface will be half of the square. r²/2 = 12²/2 = 72

The Shaded Region Area is half of the square area

Length of the Square=12 units

Area of Square=12*12=144 Square Units

Area of the Shaded Region=144/2=72 Square Units

1/2x12xh! +1/2x12xh!! = 1/2x12x(h! + h!!) = 1/2x12x12 = 6x12 = 72