Contents of the "Area" section:
Proof That Square Has Smallest Perimeter offers two different proofs that a non-square rectangle has a larger perimeter than a square with the same area.
Vector Area has several ways to find the area of a triangle and of a polygon using vectors, and a special measure similar to a determinant, but for a 2*n matrix.
Heron's Formula, Bretschneider's Formula, and Brahmagupta's Formula (part of the cyclic quadrilateral section) all give the areas of various polygons.
Triangle Area using Determinant
Cross Product, near the bottom of the page, explains why the magnitude of the cross product of two vectors equals twice the area of the triangle formed by the vectors.
Triangle Area using Vectors Cross Products
Heron's formula for the area of a triangle, if all you know is the lengths of its sides.
Polygon Area using Determinant
The webmaster and author of this Math Help site is Graeme McRae.