Question

Two rhombuses have sides with lengths of `2`. If one rhombus has a corner with an angle of `pi/12` and the other has a corner with an angle of `pi/4`, what is the difference between the areas of the rhombuses?

Answer 1

≈ 1.793 square units

Explanation:

A rhombus has 4 equal sides and is constructed from 2 congruent isosceles triangles.

The area of 1 triangle `= 1/2 a.a sintheta = 1/2 a^2sintheta`

where a is the length of side and `theta" the angle between them "`

Now the area of 2 congruent triangles ( area of rhombus ) is

area of rhombus = `2xx1/2a^2sintheta = a^2sintheta`

hence area of 1st rhombus `= 2^2sin(pi/12) ≈ 1.035`

and area of 2nd rhombus =`2^2sin(pi/4) ≈ 2.828`

Difference in area = 2.828 - 1.035 = 1.793 square units