A rhombus of side 20 cm has two angles of 60 degree each. Find the length of the diagonals.

Here are two properties of parallelograms with which one need be familiar:

(1) Opposite angles are of equal measure

(2) Sum of interior angles is 360º

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Visualize a rhombus (a parallelogram with all sides equal in length) with two opposite angles of 60º (these angles can be anywhere, so long as they are opposite). The lengths of the adjacent sides for the first diagonal are 20 cm and 20 cm. We can use the Law of Cosines to find the length of the first diagonal, represented by c:

c² = a² + b² – 2ab cos C
c² = (20)² + (20)² – 2(20)(20) cos 60º
c² = 400 + 400 – 800 cos 60º
c² = 800 – 800 cos 60º
c² = 800(1 – cos 60º)
c² = 800(1 – 1/2)
c² = 800(1/2)
c² = 400

Hence, c = 20 (taking principal root), so the length of the first diagonal is 20 cm. The other two angles are 30º (using Fact (2)), giving the sum of the interior angles as 360º. Letting d represent the second diagonal:

d² = e² + f² – 2ef cos D
d² = (20)² + (20)² – 2(20)(20) cos 30º
d² = 400 + 400 – 800 cos 30º
d² = 800 – 800 cos 30º
d² = 800(1 – cos 30º)
d² = 800(1 – [√(3) / 2)])
d² = 800([2 – √3] / 2)
d² = 400(2 – √3)

Hence, d = 20√(2 – √3) (taking the principal root), so the length of the second diagonal is 20√(2 – √3) cm.

d1 = 2 [ 20 sin 60 ° ] = 40 √3 / 2 = 20 √3 cm

d2 = 2 [ 20 cos 60 ° ] = 40 x 1/2 = 20 cm