Question

Two rhombuses have sides with lengths of `4`. If one rhombus has a corner with an angle of `(7pi)/12` and the other has a corner with an angle of `(3pi)/8`, what is the difference between the areas of the rhombuses?

Answer 1

Difference in areas between two rhombus = 4.9947

Explanation:

Area of a rhombus = `(d_1 * d_2) / 2`
Where `d_1 , d_2` are diagonals.
`side = a`

Diagonals bisect each other at right angles.

`:. d_1/2 = (a/2 )* sin (theta /2)`
`d_2 /2 = (a/2) * cos (theta/2)`

Rhombus `1 : side= 4 and /_theta = (7pi)/12`
`d_1 = 4 * sin ((7pi)/24) = 3.1734`
`d_2 = 4 * cos ((7pi)/24) = 3.9658`
`Area = (d_1 * d_2) / 2 = (3.1734 * 3.9658)/2 = 6.2925`

Rhombus `2 : side= 4 and /_theta = (3pi)/8`
`d_1 = 4* sin ((3pi)/16) = 0.7804`
`d_2 = 4 * cos ((3pi)/16)= 3.3259`
`Area = (d_1 * d_2) / 2 = (0.7804 * 3.3259)/2 = 1.2978`

Diff. in areas between the Rhombus = 6.2925 - 1.2978 = 4.9947#