Mathematics Question and Solution
Let the side length of the regular pentagon be a.
b² = a²+(0.5a)²-2*a*(0.5a)cos108
b = 1.24860602048a units.
Calculating angle AGE.
Let it be c.
(1.24860602048a/sin108) = (a/sinc)
c = 49.6138224408°
d = 180-2*49.6138224408
d = 80.7723551184°
d is the angle of the circular arc EFG with radius 5 units.
e² = 50-50cos80.7723551184
e = 6.47936163294 units.
Calculating a, side length of the regular pentagon.
e = b
6.47936163294 = 1.24860602048a
a = 6.47936163294/1.24860602048
a = 5.1892763026 units.
a is AB.
AG = ½(a)
AG = 2.5946381513 units.
Calculating Blue Area.
It is;
Area triangle with height 5.1892763026 units and base 2.5946381513sin108 units + Area sector with radius 5 units and angle 80.7723551184° - Area triangle with height 5 units and base 5sin80.7723551184 units.
= 0.5*5.1892763026*2.5946381513sin108+(80.7723551184π*5*5)/360-(0.5*5*5sin80.7723551184)
= 6.40265240252+17.62179426758-12.33823760533
= 11.68620906477 square units.
Calculating Area Green.
f = 5+2.5946381513
f = 7.5946381513 units.
cosg = 5/7.5946381513
g = 48.82514318201°
7.5946381513²-5² = h²
h = 5.71651367961 units.
Area Green is;
Area triangle with height 5.71651367961 units and base 5 units - Area sector with radius 5 units and angle 48.82514318201°
= 0.5*5*5.71651367961-(48.82514318201*π*5*5/360)
= 14.29128419902-10.65199382855
= 3.63929037047 square units.
Therefore;
Area Green ÷ Are Blue is;
3.63929037047÷11.68620906477
= 0.31141753072
≈ 0.31 to 2 decimal places.
