The side of the rhombus is 12 cm, one of its angles = 30 degrees, find S.

Let ABCD be given, in which the side of the rhombus is AD = 12 cm, and the angle ∠BAD = 30 °.

Let us draw the height of BH to the base of AD in a rhombus and consider the resulting right-angled triangle ΔАВН (∠ВНА = 90 °), in which the leg ВН is equal to half of the hypotenuse AB, since it lies opposite the angle ∠ВАD = 30 °; AB = AD = 12 cm – by the property of the sides of the rhombus, then BH = 12 cm: 2; BH = 6 cm.

The area of a rhombus is found as the product of the length of the base of the rhombus by the length of its height, that is, S = AD · BH or S = 12 cm · 6 cm; S = 72 cm².

Answer: the area of the rhombus is 72 cm².