Question

Two rhombuses have sides with lengths of `5`. If one rhombus has a corner with an angle of `pi/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 the two rhombus = 4.1618

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= 5 and /_theta = pi/12`
`d_1 = 5 * sin (pi/24) = 0.6526`
`d_2 = 5 * cos (pi/24) = 4.9752`
`Area = (d_1 * d_2) / 2 = (0.6526 * 4.9752)/2 = 1.6125`

Rhombus `1 : side= 5 and /_theta = (3pi)/8`

Rhombus `1 : side= 5 and /_theta = pi/12`
`d_1 = 5 * sin ((3pi)/16) = 4.1573`
`d_2 = 5 * cos ((3pi)/16)= 2.7779`
`Area = (d_1 * d_2) / 2 = (4.1573 * 2.7779)/2 = 5.7743`

Diff. in areas between the Rhombus = 5.7743 - 1.6125 = 4.1618#