Norman Door derive formula for Area If a norman door has a perimeter of 12 feet, what would the formula be for area? A norman door is a semicircle mounted on a rectangle. The width of the rectangle is x. The formula I derived is $$A(x)= 6x-\frac {x^2}{2}+\frac {\pi x^2}8$$ I am curious to know what other people got as the formula as I have been unable to find any examples involving a perimeter of 12.

Let the other side of the rectangle be $y$, and say the circle is being built on the $x-$ side. Then $$12 = x+2y+\frac 12 \pi x\implies y = 6 - x- \frac 14\pi x$$

The area of the rectangular portion is $$xy = 6x - x^2 -\frac 14\pi x^2$$

The area of the half circle is $$\frac 12\pi \frac {x^2}4=\frac 18\pi x^2$$

Summing gives us a total area of $$xy = 6x - x^2 -\frac 18\pi x^2$$