Mathematics Question and Solution
Notice;
AB = 25 units.
cos30 = 8√(3)/a
a = 16 units.
tan30 = b/(8√(3))
b = 8 units.
Let the side of the regular hexagon be c.
c = ¼(AC)
Calculating AC.
((AC)/sin46) = (25/sin74)
AC = 18.70821971213 units.
c = ¼(18.70821971213)
c = 4.67705492803 units.
Calculating r, radius of the circle.
(r/sin16) = (25/sin148)
r = 13.00374294827 units.
Calculating Shaded Area.
cos30 = d/4.67705492803
d = 4.05044838257 units.
e = 2d
e = 8.10089676514 units.
tan16 = f/(0.5*25)
f = 3.58431732199 units.
g = 8.10089676514-f
g = 4.51657944315 units.
h = ½(25)-2c
h = 3.14589014394 units.
(2*4.67705492803/sin46) = (i/sin74)
i = 12.5 units.
j = 12.5-3.14589014394-4.67705492803
j = 4.67705492803 units.
k² = 4.67705492803²+4.51657944315²
k = 6.50187147413 units.
l = atan(4.51657944315/4.67705492803)
l = 44°
cosm = 6.50187147413/13.00374294827
m = 60°
n² = 3.14589014394²+4.67705492803²-2*4.67705492803*3.14589014394cos60
n = 4.13012912592 units.
(4.13012912592/sin60) = (4.67705492803/sino)
o = 78.72716351548°
p = 90-78.72716351548
p = 11.27283648452°
Area Shaded to the nearest square units is;
Area triangle with height (2*4.67705492803) units and base 12.5sin60) + Area sector with radius 13.00374294827 units and angle 60° - Area triangle with height 13.00374294827 units and base 6.50187147413sin60 units + Area triangle with height 3.14589014394 units and base 4.67705492803sin60 units + Area triangle with height 4.13012912592 units and base 4.51657944315sin11.27283648452
= 50.63060478211 + 88.53915529297 - 36.61064601691 + 6.37113282263 + 1.82325884163
= 110.75350572243 square units.
≈ 111 square units.
