Lumens is a measure of light flux, so is independent of the area. It is essentially the amount of light available. So think of a bright light. It has a fixed amount of lumens, regardless of how near or far you are.

Lux is the light level at a surface. If you think of a light bulb, the lux will vary with the distance from the light source because the light spreads out.

1 lux = 1 lumen per sq meter.

Most light meters read out in lux, which is a measure of the brightness at a surface. You will measure more luxes when you are closer to the bulb, than when farther away.

Suppose you are measuring a compact flourescent lightbulb (CFL), which is roughly cylindrical in shape. You use your light meter and read 1000 foot-candles at about 1/2" from the lightbulb. How do you compute the lumens of the lightbulb?

(1) Convert the foot-candles to lux:

1000 foot-candles = 10764 lux

(2) Compute the area over which that lux value applies. This is roughly a cylinder that is 1/2 inch bigger in all directions from the bulb. So measure the bulb. We are going to convert all of our measurements into metric units. 1/2 inch = 0.0127 meters

bulb height = 9 cm or 0.09 m
bulb radius = 2 cm or 0.02 m
inflated height = h = 0.09 m + 2 * 0.0127 m = 0.1154 m
inflated radius = r = 0.02 m + 0.0127 m = 0.0327 m

area_of_cylinder = 2*pi*r*(r+h) = 2 * 3.14 * 0.0327 *(0.0327 + 0.1154) = 0.03 m^2

(3) Compute the total lumens of the bulb:

bulb_lumens = 10764 lux * 0.03 m^2 = 327.5 lumens.

Notes:
a) If your bulb is more like a sphere (like A-style Light bulbs), you should use the formula for the surface area of a sphere rather than a cylinder.
b) When measuring the lux or foot-candles of the bulb, you should measure at several locations and average the number.
c) If you measure from a large distance away from the bulb, then you can ignore the size of the bulb in your calculations.