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Square Quadrants


SquareQuadrants

The areas of the regions illustrated above can be found from the equations

 A+4B+4C=1
(1)
 A+3B+2C=1/4pi.
(2)

Since we want to solve for three variables, we need a third equation. This can be taken as

 A+2B+C=2E+D,
(3)

where

 D=1/4sqrt(3)
(4)
 D+E=1/6pi,
(5)

leading to

 A+2B+C=D+2E=2(D+E)-D=1/3pi-1/4sqrt(3).
(6)

Combining the equations (1), (2), and (6) gives the matrix equation

 [1 4 4; 1 3 2; 1 2 1][A; B; C]=[1; 1/4pi; 1/3pi-1/4sqrt(3)],
(7)

which can be inverted to yield

A=1-sqrt(3)+1/3pi
(8)
B=-1+1/2sqrt(3)+1/(12)pi
(9)
C=1-1/4sqrt(3)-1/6pi.
(10)

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References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 67-69, 1991.

Referenced on Wolfram|Alpha

Square Quadrants

Cite this as:

Weisstein, Eric W. "Square Quadrants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquareQuadrants.html

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