Mathematics Question and Solution
Calculating blue area.
Notice.
36π cm² is the inscribed circle's area.
πr² = 36π
r² = 36
r = 6 cm.
r is the radius of the inscribed red circle.
a = ⅙*180(6-2)
a = ⅙*180*4
a = 120°
a is the single interior angle of the regular ascribed hexagon.
Let x be the side length of the ascribed regular hexagon.
tan60 = b/(½(x))
√(3) = 2b/x
b = ½√(3)x cm.
b is the height of the inscribed equilateral triangle.
c = 2r+b
c = 2(6)+½√(3)x
c = ½(24+√(3)x) cm.
c is the height of the ascribed regular hexagon.
d = ½(c)
d = ½*½(24+√(3)x)
d = ¼(24+√(3)x) cm.
Calculating x.
sin60 = d/x
½√(3) = ¼(24+√(3)x)/x
½√(3) = (24+√(3)x)/(4x)
√(3) = (24+√(3)x)/(2x)
2√(3)x = 24+√(3)x
√(3)x = 24
x = 24/√(3)
x = ⅓(24√(3))
x = 8√(3) cm.
Again, x is the side length of the ascribed regular hexagon.
Recall.
c = ½(24+√(3)x) cm.
And x = 8√(3) cm.
c = ½(24+√(3)(8√(3)))
c = ½(24+24)
c = 24 cm.
Again, c is the height of the ascribed regular hexagon.
It implies the blue area is;
Area regular hexagon with side length 8√(3) cm - Area inscribed circle - Area inscribed equilateral triangle.
((2*½*(8√(3))²sin120)+(8√(3)*24))-(36π)-(½*(8√(3))²sin60)
= (96√(3)+192√(3))-36π-(48√(3))
= 240√(3)-36π
= 12(20√(3)-3π) cm²
= 302.594858287 cm²
