Mathematics Question and Solution
The side length of the congruent Hexagons is 2 unit.
Area hexagon is;
(6*2²)/(4tan30)
= 6√(3)) square units.
a² = 8-8cos30
a = √(8-4√(3)) unit.
Area ABC is;
½(√(8-4√(3)))²sin60
= (2√(3)-3) square units.
(d) Area Hexagon ÷ Area ABC exactly is;
(6√(3)) ÷ (2√(3)-3)
= 6(2+√(3))
(c) Calculating Area Green to 2 decimal places.
b = a + √(2²+2²)
b = √(8-4√(3))+2√(2)
b = (√(2)+√(6)) units.
b = 3.86370330516 units.
c² = 8-8cos120
c = 2√(3) units.
Area Green is;
2((0.5x2*2sin120) + (0.5*2√(3)*(√(2)+√(6))sin135))
= 2(√(3) + (3+√(3)))
= 2(3+2√(3)) square units.
= 12.92820323028 square units.
≈ 12.93 square units.
(b) Calculating Radius of the inscribed circle to 3 decimal places.
Let it be r.
d² = (2√(3))²+(√(2)+√(6))²-4√(3)(√(2)+√(6))cos135
d = 6.77173585283 units.
(6.77173585283/sin(135)) = (2√(3)/sine)
e = 21.206023113°
sin21.206023113 = r/(3.86370330516-√(2)r)
r = 1.39758871593-0.51155297407r
1.51155297407r = 1.39758871593
r = 0.92460452257
r ≈ 0.925 unit.
(a) Calculating Angle DEF exactly.
sin30 = f/1
f = ½ unit.
cos30 = g/1
g = ½(√(3)) unit.
h = 3√(3)-g
h = ½(5√(3)) units.
i = 6-f
i = ½(11) units.
It implies;
DEF Angle exactly is;
atan(h/i)
= atan((½(5√(3))/(½(11)))
= atan(5√(3)/11)°
Therefore the answers;
(a) is atan(5√(3)/11)°
(b) is 0.925 unit
(c) is 12.93 square units
(d) is 6(2+√(3))
