The Mathematics of Arches

Arches are part of the design feature set of our house — so is a mansard roof, but I am not as keen to replicate it — and as I add or replace various features on the house, I would like to add the same kind of flattened arch that features on facade of the house and in some of the cabinetry. For that, I need math. In particular, given the width of a given opening and how high I would like the arch to be, I need to be able to calculate the length of material of the resulting arch.

For those who missed this particular part of geometry, here are the parts involved:

Arch

For the math, we need the following:

(x - x[0])^2 + (y - y[0])^2 = r^2

Or:

x[0] = c/2

y[0] = (s - x[0]^2/s) / 2

r^2 = x[0]^2 + y[0]^2

Y = y[0] + sqrt(r^2 - (x - x[0])^2)