If the height of the triangular roof is half its base, then the height of the roof is:
h = (1/2) * 32 = 16 feet
The area of a triangle is given by the formula:
A = (1/2) * base * height
Plugging in the values we have:
A = (1/2) * 32 * 16
A = 256 square feet
Therefore, the area of the triangular roof is 256 square feet.