Mathematics Question and Solution
Calculating r, radius of the inscribed circle.
Notice.
125π cm² is the total area of the five congruent green circles.
Calculating their radius, let it be x.
5*πx² = 125π
x² = 25
x = √(25)
x = 5 cm.
Again, x is the radius of each of the five congruent circles.
Notice.
The radius of the five congruent green circles will connect to form a regular pentagon with side 10 cm.
a = ⅕*180(5-2)
a = ⅕*180*3
a = 108°
a is the single interior angle of the derived regular pentagon with side 10 cm.
b = ½(a)
b = ½(108)
b = 54°
c = (5+r) cm.
Therefore r, radius of the inscribed yellow circle is;
cosb = x/c
cos54 = 5/(5+r)
5 = 5cos54+rcos54
rcos54 = 5-5cos54
r = (5-5cos54)/cos54
r = 3.50650808352 cm.
Again, r is the radius of the inscribed yellow circle.
