A = (1/2) (v

_{2}- v

_{1}) ^ (v

_{3}- v

_{1})

= (1/2) (v

_{2}^v

_{3}+ v

_{3}^v

_{1}+ v

_{1}^v

_{2})

where v

_{i}are the vertices. A general n-gon can be broken into triangles sharing a vertex v

_{1}, so the total area is

A = (1/2) (v

_{1}^v

_{2}+ v

_{2}^v

_{3}+ v

_{3}^v

_{1})

+ (1/2) (v

_{1}^v

_{3}+ v

_{3}^v

_{4}+ v

_{4}^v

_{1})

+ (1/2) (v

_{1}^v

_{4}+ v

_{4}^v

_{5}+ v

_{5}^v

_{1})

+ ...

+ (1/2) (v

_{1}^v

_{n-1}+ v

_{n-1}^v

_{n}+ v

_{n}^v

_{1})

= (1/2) (v

_{1}^v

_{2}+ v

_{2}^v

_{3}+ v

_{3}^v

_{4}+ ... + v

_{n-1}^v

_{n}+ v

_{n}^v

_{1}),

a cyclic sum of wedge products of connected pairs of vertices. We could also write this as

A = (1/2) (v

_{1}^(v

_{2}- v

_{1}) + v

_{2}^(v

_{3}- v

_{2}) + ... + v

_{n}^(v

_{1}- v

_{n}))

= (1/2) (x

_{1}^Δx

_{1}+ x

_{2}^Δx

_{2}+ ... + x

_{n}^Δx

_{n})

which resembles an integral of the position vector x. Indeed, this is related to the fundamental theorem of geometric calculus. (That's int (∇F)

^{~}d

^{m}x = int F

^{~}d

^{m-1}x, where the integral on the left is over an m-dimensional region with tangent blade d

^{m}x, the integral on the right is over its boundary, and F

^{~}is the reverse of a multivector function F.) In 2 dimensions, ∇x = 2, so we can convert the integral of 1 = (1/2)∇x, over the region's area, into an integral over the boundary curve:

A = int 1 d

^{2}x = (1/2) int x dx = (1/2) int x ^ dx.

The last equality is because the integral must be a bivector (no scalar part). More generally, we can use ∇x = m to find a region's (hyper)volume:

V = int 1 d

^{m}x = (1/m) int x d

^{m-1}x = (1/m) int x ^ d

^{m-1}x.

Along one facet F of an m-polytope, the tangent blade d

^{m-1}x is constant, and x ^ d

^{m-1}x is constant because x is varying parallel to d

^{m-1}x, so we can take this to be v

_{1}^ d

^{m-1}x, with v

_{1}a vertex of F. So this part of the integral is

(1/m) int

_{F}x ^ d

^{m-1}x = (1/m) v

_{1}^ int

_{F}d

^{m-1}x

(and V is the sum of these integrals over all facets). But this term on the right is the directed volume of F, so we can calculate higher-dimensional volume recursively from lower dimensions.

Of course, if you want a scalar volume, then you need to take the magnitude of the multivector: ||V|| = sqrt{ V

^{~}V } = sqrt{ | V V | }. But we need the direction of V in order to use the recursive formula.