Final answer:
The area of the rhombus is 648√3 cm², calculated using the trigonometric properties to determine the length of the shorter diagonal and the rhombus area formula. The perimeter is found to be 144 cm by using the Pythagorean theorem to find the length of a side and multiplying by four.
Step-by-step explanation:
The question asks us to calculate the area and perimeter of a rhombus given the length of its longest diagonal of 36 cm and one of its interior angles, which measures 120°.
To find the area of the rhombus, we would need the length of the other diagonal. This can be found using the trigonometric properties of the rhombus.
Since the diagonals of a rhombus bisect each other at right angles and each diagonal bisects two opposite interior angles, we can set up a right triangle where one leg is half of the unknown diagonal, and the adjacent angle is half of 120°, which is 60°.
The other leg is half of the given diagonal, 18 cm. Using the tangent of 60° which equals √3, we can write the equation leg/opposite = √3 which leads to leg = 18√3 cm. Multiplying by 2, we find the shorter diagonal to be 36√3 cm.
To calculate the area (A) of the rhombus, the formula is A = (d1 × d2)/2, with d1 and d2 representing the lengths of the diagonals. Substituting the given values, the area is A = (36 cm × 36√3 cm)/2 = (1296√3 cm²)/2 = 648√3 cm².
To find the perimeter (P), we need to first find the length of a side (s) of the rhombus.
This can be done using the Pythagorean theorem on one of the right triangles formed by cutting the rhombus along a diagonal, with legs of 18 cm and 18√3 cm.
We get s² = (18 cm)² + (18√3 cm)², resulting in s = 36 cm.
Since all sides of a rhombus are equal, the perimeter is 4 times the length of a side, thus P = 4 × 36 cm = 144 cm.