Heron's formula
The following question is prompted by Hero of Alexandria, Brahmagupta, Al Cuoco and Paul Goldenberg
A rectangle with length a and width b has a semi-perimeter of S = (a +b).
The area ab of the rectangle can be written as (S - a)(S -b).
The square root of this number can be thought of as the side length
of a square that has the same area as the rectangle.
Brahmagupta, a 7th century Indian mathematician, generalized this result to cyclic
quadrilaterals - given a semi-perimeter of S = (a + b + c + d)/2 the area of the
quadrilateral can be written as the square root of (S - a)(S - b)(S - c)(S - d)
The square root of this number can be thought of as the side length
of a square that has the same area as the quadrilateral.
Now Imagine a four dimensional rectangular parallelapiped.
The four mutually perpendicular side lengths are
S, S-a, S-b and S-c units long.
Can you think of the area of the triangle with sides a, b and c as the length
of the side of a 4D hypercube?
What questions could/would you put to your students based on this applet?