Mathematics Question and Solution
Notice!
The area of the regular pentagon and the square are the same.
Let it be 4 square units.
Therefore the side length of the square a, is;
a = √(4)
a = 2 units.
Calculating b, the side length of the regular pentagon.
Notice, b = x
c = ⅕(180(5-2))
c = 108°
c is the single interior angle of the regular pentagon.
d = 108-90
d = 18°
cos18 = e/b
e = 0.9510565163b units.
sin18 = f/d
f = 0.3090169944b units.
g = 2f+b
g = 2(0.3090169944b)+b
g = 1.6180339887b units.
It implies;
½(b²)sin108+½(1.6180339887b+b)0.9510565163b = 4
0.4755282581b²+1.2449491424b² = 4
1.7204774005b² = 4
b² = 2.3249360897
b = 1.524774111 units.
b = x, the side length of the regular pentagon.
Calculating y.
h = 180-108
h = 72°
cos72 = j/1.524774111
j = 0.4711811129 units.
k = b+j
k = 1.524774111+0.4711811129
k = 1.9959552239 units.
It implies;
y = (the square side length)-k
y = 2-1.9959552239
y = 0.0040447761 units.
Therefore;
x : y is;
= x÷y
= 1.524774111÷0.0040447761
= 376.9736744982
= 377 to the nearest whole number.
