# 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.

• Are you building the circle on a side of length $x$? – lulu Sep 17 '17 at 22:59
• Yes, the radius for the circle would be 1/2 x – ERIC Sep 17 '17 at 23:54
• Ok, then...I seem to get the same expression you got, only with a minus sign in front of the last term. I posted my calculation below...always possible I made an arithmetic blunder, of course. – lulu Sep 17 '17 at 23:56
• Yes you are correct it should be a minus – ERIC Sep 18 '17 at 0:04

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 $$xy = 6x - x^2 -\frac 18\pi x^2$$