Mathematics Question and Solution
Calculating Area Inscribed Blue Triangle (Equilateral Triangle).
Let r be the radius of the inscribed six congruent circles.
a+a+r+r = 5
2a = 5-2r
a = ½(5-2r) units.
b = ⅙*180(6-2)
b = 120°
c = ½(b)
c = 60°
Calculating r.
tan60 = r/a
√(3) = r/(½(5-2r))
(5-2r)√(3) = 2r
5√(3) = 2r+2√(3)r
5√(3) = (2+2√(3))r
r = (5√(3))/(2+2√(3))
r = ¼(5(3-√(3)) units.
r = 1.58493649054 units.
Again, r is the radius of the inscribed six congruent circles.
tan60 = d/(0.5*5)
d = 0.5*5tan60
d = 2.5√(3) units.
d = 4.33012701892 units.
d is half the height of the ascribed regular hexagon.
e = d-2r
e = 4.33012701892-2(1.58493649054)
e = 1.16025403784 units.
Notice.
The inscribed blue triangle is equilateral.
It implies;
tan(0.5*60) = e/f
tan30 = 1.16025403784/f
f = 1.16025403784√(3)
f = 2.00961894323 units.
g = 2f
g = 2*2.00961894323
g = 4.01923788646 units.
g is the side length of the inscribed blue equilateral triangle.
Therefore area blue equilateral triangle is;
½*4.01923788646*4.01923788646sin60
= 6.99500548022 square units.
