Cross Products vs using geometry

Question

Say I wanted to calculate the area of a figure comprised of the coordinates (0,1,1) (1,0,1) and (0,0,0). When I used cross products I get the area as the square root of 3/2, however, when I use standard geometry I get the answer as 1/6. Can someone please point out my mistake when using cross products( I got the vectors as <0,1,1> and <-1,1,0> and I got the cross product as <-1,1,1>

Answer 1

Here is what my standard geometry tells me. You have an isosceles triangle with two sides of length $\sqrt2$ and the angle between them is $\cos^{-1}(1/2) =\pi/3$. Its area is $$\frac12(\sqrt2)^2\sin\frac\pi3=\frac{\sqrt3}2.$$

Answer 2

The sides are given by vectors $(1,0,1), (0,1,1), (1,-1,0).$ The triangle is equilateral with lengths of sides $a=\sqrt 2$ and the area $$\mathcal{A}=\frac{a}{2}\cdot\frac{a\sqrt3}{2}=\frac{\sqrt3}{2}$$