Find the area of a rhombus if the side is 6 cm and one of the corners is 45 °.

Let the angle A of rhombus ABCD be 45 °. Then, by the property of the rhombus, the angle C will also be equal to 45 °.

Let’s split the rhombus into two triangles: ABD and BCD. Since the triangles are equal (on two sides and the angle between them), the area of the rhombus will be equal to two areas of one of the triangles:

Srombus = 2 * Str.

The area of a triangle ABD can be found as half the product of its sides by the sine of the angle between them (angle A):

Str = 1/2 * AB * AD * sinA = 1/2 * b² * sin45 °.

Then the area of the rhombus is

Sromb = 2 * Str = 2 * 1/2 * b² * sin45 ° = b² * sin45 ° = 6² * √2 / 2 = 18√2 cm².