Mathematics Question and
Let the square side be 2 units.
Area square is;
2²
= 4 square units.
Calculating Area A.
a² = 2²+1²-2*2*1cos45
a = √(5-2√(2)) units.
(√(5-2√(2))/sin45) = (2/sinb)
b = asin(√(2)/(√(5-2√(2))))°
c = 180-asin(√(2)/(√(5-2√(2))))
c = 106.3249499369°
Area A is;
Are sector with radius 2 units and angle 45° - Area triangle with height 2 units and base sin45° - Area sector with radius 1 unit and angle (asin(√(2)/(√(5-2√(2)))))°
= (45π*4/360) - (½*2*sin45) - (asin(√(2)/√(5-2√(2)))π/360)
= ½(π) - ½(√(2)) - (asin(√(2)/√(5-2√(2)))π/360)
= ½(π-√(2))-(asin(√(2)/√(5-2√(2)))π/360
= ((180π-180√(2)-asin(√(2)/√(5-2√(2)))π)/360) square units.
= 0.22075344552 square units.
Therefore;
Area A ÷ Area Square as a single fraction in terms of π exactly is;
((180π-180√(2)-asin(√(2)/√(5-2√(2)))π)/360) ÷ 4
= (180π-180√(2)-asin(√(2)/√(5-2√(2)))π)/1440
Exactly in decimal is;
0.22075344552 ÷ 4
= 0.05518836138
Exactly in fraction is;
= 5436903/98515391
