Mathematics Question and Solution
Calculating red area, area regular hexagon.
Let x be the side length of the red regular hexagon.
a² = 2x²-2x²cos120
a² = 3x²
a = √(3)x units.
b²+3² = x²
b = √(x²-9) units.
sinc = √(x²-9)/x --- (1).
sinc = 5/√(3)x --- (2).
Calculating x.
Equating (1) and (2).
√(x²-9)/x = 5/√(3)x
Cross Multiply.
5x = x√(3x²-27x)
5 = √(3x²-27)
25+27 = 3x²
52 = 3x²
x² = 52/3
x = √(52/3)
x = 2√(13)/√(3)
x = ⅓(2√(39)) units.
x = 4.16333199893 units.
Again, x is the side length of the red regular hexagon.
Recall.
a = √(3)x
And x = ⅓(2√(39)) units.
a = √(3)*⅓(2√(39))
a = 2√(13) units.
Therefore;
Area red is;
(⅓(2√(39))*2√(13))+2(½*(⅓(2√(39)))²sin120)
= ⅓(52√(3))+⅓(26√(3))
= ⅓(78√(3)
= 26√(3) square units.
