Algebra and Geometry Facts

Here are a few helpful and neat little facts that evade most students and teachers of algebra and geometry:

a) Though everyone can factor a^2 - b^2, a^3 - b^3, a^3 + b^3 and a^4 - b^4, most folks do not know that:
a^4 + b^4 = (a^2 + ab(sqrt(2)) + b^2) (a^2 - ab(sqrt(2)) + b^2

b) An approximation exists for the factorial function (for large n) which seems hardly related but works:
Stirling's Formula ---> n! ~~ e^(-n) n^n sqrt(2 (pi) n)

c) The area of any regular polygon of n sides, each of length x, is given by:
Area = (1/4)nx^2 (cot(180°/n))

d) The radii of circumscribed (R) and inscribed (r) circles within such regular polygons are given by:
R = (x/2) csc (180°/n) and r = (x/2) cot (180°/n)

e) The radius of a circle inscribed within any triangle of sides a, b, and c with semi-perimeter s is given by:
r = (sqrt (s (s-a) (s-b) (s-c)) / s

e) The radius of a circle circumscribed about any triangle of sides a, b, and c with semi-perimeter s is given by:
R = abc / 4 (sqrt (s (s-a) (s-b) (s-c))

f) The perimeter P and area A of polygons (of n sides) inscribed in a circle of radius r is given by:
P = 2nr sin(pi/n) and A = (1/2) nr^2 sin (2pi/n)

g) The perimeter P and area A of polygons (of n sides) circumscribed about a circle of radius r is given by:
P = 2nr tan (pi/n) and A = nr^2 tan (pi/n)

h) Factoring the seemingly prime expression a^4 + 4b^4 becomes (and there are a family of these)
a^4 + 4b^4 = (a^2 + 2b^2)^2 - (2ab)^2

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