Mathematics Question and Solution
Let the side length of the ascribed regular pentagon be 1 unit.
Let a be the side length of the inscribed regular hexagon.
b = 180-108-(108-120)
b = 72-60
b = 12°
Where;
108° is the single interior angle of the regular ascribed pentagon.
120° is the single interior angle of the inscribed regular hexagon.
180° is the sum of the interior angles of a triangle.
(a/sin108) = (c/sin12)
c = 0.2186112889a units.
d = a+c
d = a+0.2186112889a
d = 1.2186112889a units.
e² = 2a²-2a²cos120
e = √(3)a unit.
Calculating a.
1² = e²+d²
1 = (√(3)a)²+(1.2186112889a)
1 = 4.4850134734a²
a² = 1/4.4850134734
a = 0.4721914553 units.
Again, a is the side length of the inscribed regular hexagon.
e = √(3)a
And a = 0.4721914553 units.
e = √(3)*0.4721914553
e = 0.8178595916 units.
c = 0.2186112889a
And a = 0.4721914553 units.
c = 0.2186112889*0.4721914553
c = 0.1032263827 units.
Shaded Area is;
Area sector with radius 1 unit and angle 108° - Area regular hexagon with side length 0.4721914553 units - Area triangle with height 0.4721914553 units and base 0.1032263827sin60 units.
= (108π*1²/360)-((0.4721914553*0.8178595916)+2(0.5*0.4721914553²sin120))-(0.2*0.4721914553*0.1032263827sin60)
= 0.9424777961-(0.3861863108+0.1930931554)-0.0211061718
= 0.3420921581 square units.
Calculating the area of the inscribed circle.
Let it's radius be f.
sin54 = f/g
g = 1.2360679775f units.
tan72 = h/0.5
h = 1.5388417686 units.
j = h-g
j = (1.5388417686-1.2360679775f) units.
k = (1+f) units.
Observing Pythagoras Rule to get f, radius of the inscribed circle.
(1+f)² = 0.5²+(1.5388417686-1.2360679775f)²
1+2f+f² = 0.25+2.3680339888-3.8042260652f+1.527864045f²
0.527864045f²-5.8042260652f+1.6180339888 = 0
Resolving the above quadratic equation to get f.
f = 0.286219 units.
Again, f is the radius of the inscribed circle.
Area circle is;
πr²
= π(0.286219)²
= 0.2573634044 square units.
It implies;
Area Circle ÷ Area Shaded to 2 decimal places is;
0.2573634044÷0.3420921581
= 0.7523218475
≈ 0.75
