Here are some formulas:

Each parallel has a length of 2*pi*cos(theta), this is the height of the map in y.

Each meridian has a length of 2*R/cos(theta), this is the width of the map in x.

The distance of each parallel to the equator is R*sin(phi)/cos(theta).

Where R is the radius of the generating globe of the chosenarea scale, theta is the standard parallel, phi is the latitude of interest.

For a point on the globe (lat/lon), where lat is the latitude of the point and lon is the longitude of the point(in degrees) and the map is centered at latlon(0,0), west longitudes are negative values as are south latitudes,

x = 2*pi*cos(theta) * (lon + 180)/360

y = R*sin(lat)/cos(phi)

The ratio of width/height of the map is determined from,

cos(theta) = acos((ratio/pi)^0.5)

Some standard projections and their standard parallels:

Lambert - pi/1 ratio, theta=0

Behrmann - ~2.356:1 ratio, theta=30

Gall-Peters - pi/2:1 ratio, theta=45

Hopefully this would be helpful.

edit: Info from Arthur Robinson, Randall Sale, Joel Morrison, "Elements of Cartography" 4th ed., 1978, John Wiley and Sons: New York.

The wikipedia describes this well under Gall-Peters projection. Even more generally, but in less detail under Lambert cylindrical equal-area projection.