We label the points on the window for convenience:

norman window

Perimeter p is given by:

p = semicircle circumference + length AB + length BC + length CD

The circumference of a semi-circle is half the circumference of a circle:

`"curve"\ AD = 1/2(2pir) = pi r`

From the question, we know

AB = BC + 30

Also, we can see from the diagram

AD = BC = 2r

So, putting it all together, we have:

p = curve AD + AB + BC + CD

p = (πr) + (2r + 30) + 2r + (2r + 30)

Simplifying, we can write our function as

p = p(r) = π r + 6r + 60

Note the radius cannot be negative, and the window would not exist if `r = 0`.

Also, when we write "p = p(r)", we mean "p, which is a function of r".