Question

Two rhombuses have sides with lengths of `15`. If one rhombus has a corner with an angle of `pi/12` and the other has a corner with an angle of `pi/2`, what is the difference between the areas of the rhombuses?

Answer 1

Difference between areas `=108.22`

Explanation:

Rhombus 1
d1 and d2 are the diagonals.
Half of Corner angle `=(pi/12)/2=pi/24`
`sin(pi/24)=((d1)/2)/15`
`30sin(pi/24)=d1`
`d1=3.92`
Similarly, Other corner angle `=(2pi-(pi/6))/2=11pi/12`
`sin(11pi/24)=((d2/2)/15)`
`30sin(11pi/24)=d2`
`d2=29.74`
Area of rhombus `=(d1.d2)/2=3.92*29.74=color(red)(116.78)`

Rhombus 2
Since one corner angle is `pi/2`, all the corner angles are `pi/2` each, or it is square.
Area of square `=s^2=15^2=color(red)(225)`
Difference in area between Rhombus color(red)(1 & 2) is
`=225-116.78=color(blue)(108.22)`