Mathematics Question and Solution
Notice!
The congruent triangles are equilateral.
Calculating the radius, a of the two bigger congruent circles.
tan60 = a/b
√(3) = a/b
b = (a/√(3)) units.
sin60 = 2a/c
√(3)/2 = 2a/c
√(3)c = 4a
c = (4a/√(3)) units.
It implies.
2b+c = 8
2(a/√(3))+(4a/√(3)) = 8
(2a/√(3))+(4a/√(3)) = 8
(6a/√(3)) = 8
2√(3)a = 8
a = 4/√(3)
a = ⅓(4√(3)) units.
Again, a is the radius of the two bigger congruent circles.
Calculating d, radius of the smaller circle and of the semi circle (their radii are equal).
It implies;
3d = 2a
3d = 2(⅓(4√(3)))
3d = ⅓(8√(3))
d = ⅑(8√(3)) units.
Again, d is the radius of the smaller circle and the semi circle.
Shaded Area is;
2(area circle with radius ⅓(4√(3)) units) + Area circle with radius ⅑(8√(3)) units + Area semi circle with radius ⅑(8√(3)) units.
= 2π(⅓(4√(3)))²+π(⅑(8√(3)))+½*π(⅑(8√(3)))
= 2π(⅓*16)+π(64/27)+½*π(64/27)
= ⅓(32π)+(64π/27)+(32π/27)
= (288π+64π+32π)/27
= (288π+96π)/27
= 384π/27
= 128π/9 square units.
= 44.6804288511 square units.
≈ 44.68 square units to 2 decimal places.
