Mathematics Question and Solution
Let the square side, x be 1 unit.
sin67.5 = a/1
a = 0.92387953251 units.
cos67.5 = b/1
b = 0.38268343237 units.
c = 2b
c = 0.76536686473 units.
d² = 1²+1²
d = √(2) units.
sin67.5 = e/√(2)
e = 1.30656296488 units.
e is the radius of the half circle.
f = e-c
f = 1.30656296488-0.76536686473
f = 0.54119610015 units.
g = r-b
g = (r-0.38268343237) units.
r² = (r-0.38268343237)²+h²
h² = r²-r²+0.76536686474r-0.14644660941
h² = 0.76536686474r-0.14644660941
h = √(0.76536686474r-0.14644660941) units.
j = 0.92387953251-0.54119610015
j = 0.38268343236 units.
k = j+h
k = 0.38268343236+√(0.76536686474r-0.14644660941)
Calculating r, radius of the inscribed circle.
(1.30656296488-r)² = (0.38268343236+√(0.76536686474r-0.14644660941))²+r²
1.7071067812-2.61312592976r+r² = 0.1464466094+0.76536686472√(0.76536686474r-0.14644660941)+(0.76536686474r-0.14644660941)+r²
1.7071067812-0.1464466094+0.14644660941 = 0.76536686472√(0.76536686474r-0.14644660941)+0.76536686474r+2.61312592976r
1.70710678121-3.3784927945r = 0.76536686472√(0.76536686474r-0.14644660941)
(2.23044249745-4.41421356245r)² = 0.76536686474r-0.14644660941
4.97487373443-19.69129904502r+19.48528137492r² = 0.76536686474r-0.14644660941
19.48528137492r²-20.45666590976r+5.12132034384 = 0
Therefore, resolving the quadratic equation.
r ≠ 0.637697 units.
r = 0.412155 units.
Therefore;
x÷r is;
1÷0.412155
= 2.42627166964
