Mathematics Question and Solution
Let the length and width of the ascribed rectangle be 3 units and 2 units respectively.
Calculating red area.
It is;
Area quarter circle with radius 2 units - Area sector with radius 1.5 units and angle 2atan(2/1.5)° + Area triangle with height 1.5 units and base 1.5sin(2atan(2/1.5)) units - Area sector with radius 2 units and angle 2atan(1.5/2)° + Area triangle with height 2 units and base 2sin(2atan(1.5/2)) units + Area semi with radius 1 unit - Area sector with radius 1 unit and angle 2atan(1.5) + Area triangle with height 1 unit is base sin(2atan(1.5)) units - Area sector with radius 1.5 unit and angle 2atan(2/3) + Area triangle with height 1.5 unit is base 1.5sin(2atan(2/3)) units - Area sector with radius 1 unit and angle 2atan(0.5) + Area triangle with height 1 unit is base sin(2atan(0.5)) units - Area sector with radius 0.5 unit and angle 2atan(2) + Area triangle with height 0.5 unit is base 0.5sin(2atan(2)) units
(¼(2²π))-(2atan(2/1.5)*1.5²π/360)+(0.5*1.5²sin(2atan(2/1.5)))-(2atan(1.5/2)*2²π/360)+(0.5*2²sin(2atan(1.5/2)))+(½(1²π))-(2atan(1.5)*1²π/360)+(0.5*1²sin(2atan(1.5)))-(2atan(2/3)*1.5²π/360)+(0.5*1.5²sin(2atan(2/3)))-(2atan(0.5)*1²π/360)+(0.5*1²sin(2atan(0.5)))-(2atan(2)*0.5²π/360)+(0.5*0.5²sin(2atan(2)))
= 3.1415926536-2.0864142405+1.08-2.5740044352+1.92+1.5707963268-0.9827937232+0.4615384615-1.323005858+1.0384615385-0.463647609+0.4-0.2767871794-0.1
= 1.8057359351 square units.
Calculating blue area.
It is;
Area semi circle with radius 1.5 units - Area sector with radius 1.5 units and angle 2atan(2/1.5)° + Area triangle with height 1.5 units and base 1.5sin(2atan(2/1.5)) units - Area sector with radius 2 units and angle 2atan(1.5/2)° + Area triangle with height 2 units and base 2sin(2atan(1.5/2)) units-- Area sector with radius 1 unit and angle 2atan(1.5) + Area triangle with height 1 unit is base sin(2atan(1.5)) units - Area sector with radius 1.5 unit and angle 2atan(2/3) + Area triangle with height 1.5 unit is base 1.5sin(2atan(2/3)) units + Area semi with radius 0.5 unit-Area sector with radius 1 unit and angle 2atan(0.5) + Area triangle with height 1 unit is base sin(2atan(0.5)) units - Area sector with radius 0.5 unit and angle 2atan(2) + Area triangle with height 0.5 unit is base 0.5sin(2atan(2)) units.
= (0.5π*1.5²)-(2atan(2/1.5)*1.5²π/360)+(0.5*1.5²sin(2atan(2/1.5)))-(2atan(1.5/2)*2²π/360)+(0.5*2²sin(2atan(1.5/2)))-(2atan(1.5)*1²π/360)+(0.5*1²sin(2atan(1.5)))-(2atan(2/3)*1.5²π/360)+(0.5*1.5²sin(2atan(2/3)))+(0.5*0.5²π)-(2atan(0.5)*1²π/360)+(0.5*1²sin(2atan(0.5)))-(2atan(2)*0.5²π/360)+(0.5*0.5²sin(2atan(2)))
= 3.5342917353-2.0864142405+1.08-2.5740044352+1.92-0.9827937232+0.4615384615-1.323005858+1.0384615385+0.3926990817-0.463647609+0.4-0.2767871794-0.1
= 1.0203377717 square units.
Notice!
Area Red = 1.8057359351 square units.
Area Blue = 1.0203377717 square units.
Therefore, Area Red is Bigger.
