The area of a plane in a crystal lattice can be determined using the Miller indices. Miller indices are a set of numbers used to describe the orientation of crystal planes within a crystal lattice. Each number in the Miller indices represents the intercept of the plane with the crystallographic axes.

Given that the Miller indices for the plane in question are [1 1 1] and the lattice constant is 2, we can determine the area of the plane using the following steps:

1. Determining the plane's intercepts:
- The Miller indices [1 1 1] indicate that the plane intercepts the x-axis, y-axis, and z-axis at 1 unit.
- To determine the actual intercepts, we multiply the Miller indices by the lattice constant.
- In this case, the intercepts are [2 2 2], which represent the coordinates (2, 0, 0), (0, 2, 0), and (0, 0, 2) in the crystal lattice.

2. Finding the lengths of the edges:
- The lengths of the edges of the parallelepiped formed by the intercepts can be found using the distance formula.
- The distance between two points (x1, y1, z1) and (x2, y2, z2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
- Applying this formula to the intercepts, we find that the lengths of the edges are:
a = sqrt((2 - 0)^2 + (0 - 0)^2 + (0 - 0)^2) = 2
b = sqrt((0 - 0)^2 + (2 - 0)^2 + (0 - 0)^2) = 2
c = sqrt((0 - 0)^2 + (0 - 0)^2 + (2 - 0)^2) = 2

3. Calculating the area of the plane:
- The area of the plane can be determined using the formula:
Area = (a * b * sin(theta)) / c
where theta is the angle between the two edges a and b.
- In this case, the angle between the edges a and b is 60 degrees, as the plane is a (111) plane in a cubic lattice.
- The sine of 60 degrees is sqrt(3) / 2.
- Substituting the values into the formula, we get:
Area = (2 * 2 * sqrt(3) / 2) / 2 = 2 * sqrt(3) / 2 = sqrt(3)

Therefore, the area of the plane with Miller indices [1 1 1] in a lattice with a lattice constant of 2 is equal to sqrt(3), which is approximately 1.732.