Short answers, pulled from the story.
The standard unit of area in the International System of Units is the square metre. This unit is defined as the area of a square with sides of one metre and serves as the SI derived unit for area.
Hippocrates of Chios became the first to demonstrate that the area of a disk is proportional to the square of its diameter in the 5th century BCE. His work on the quadrature of the lune of Hippocrates laid the groundwork for future discoveries in geometry.
The formula for the surface area of a sphere is 4 pi r squared, where r is the radius of the sphere. Archimedes first obtained this formula in his work On the Sphere and Cylinder, showing that the surface area of a sphere is exactly four times the area of a flat disk of the same radius.
Brahmagupta developed a formula for the area of a cyclic quadrilateral in the 7th century CE in terms of its sides. A cyclic quadrilateral is defined as a quadrilateral inscribed in a circle.
The isoperimetric inequality states that a circle has the largest area of any closed figure with a given perimeter. For a closed curve of length L and area A, 4 pi A is less than or equal to L squared, and equality holds if and only if the curve is a circle.