Mathematics Question and Solution
Let the side of the regular pentagon be 1 unit.
a² = 2-2cos108
a = 1.61803398875 units.
sin30 = b/0.5
b = 0.25 unit.
c = 2b
c = 0.5 unit.
Where c is the inscribed square side.
Area Square is;
(0.5)²
= 0.25 square units.
Inscribed circle radius d is;
d = ½(b)
d = 0.125 unit.
e = 0.5-0.125
e = 0.375 unit.
f = 2atan(0.125/0.375)
f = 36.86989764584°
tan36.86989764584 = g/0.5
g = 0.375 unit.
Shaded Area is;
0.5*0.375*0.5-π*0.125*0.125
= 0.04466261479 square units.
(a) Area Shaded ÷ Area Square to 2 decimal places is;
0.04466261479÷0.25
= 0.17865045916
≈ 0.18
Calculating Length CD.
tan18 = 0.5/h
h = 1.53884176859 units.
tan36 = i/0.25
i = 0.181635632 unit.
j = h-i-0.125
j = 1.23220613659 units.
tan30 = k/0.5
k = 0.28867513459 unit.
l = j-k
l = 0.943531002 unit.
Length CD is;
(CD)² = 0.943531002²+0.125²
CD = 0.95177505312 unit.
(b) CD ÷ AB to 2 decimal places is;
0.95177505312 ÷ 1
= 0.95177505312
≈ 0.95
Therefore;
(a) is 0.18
(b) is 0.95
