The sides of the parallelogram are a and b, and the angle between them is equal to the angle

| June 3, 2021 | education

The sides of the parallelogram are a and b, and the angle between them is equal to the angle alpha. Find the area of the parallelogram.

Consider a parallelogram ABCD, where AB = CD = a, AD = BC = b and ∠BAD = α. At the request of the task, we find the area (S) of the parallelogram. As you know, the diagonal (AC or BD) of a parallelogram divides it into two equal triangles.
Consider triangle ABD and draw its height BH. We calculate the area (SΔ) of this triangle as the product of half of the base of the triangle (AD) and its height (BH), that is, SΔ = ½ * AD * BH.
Since the height HH is perpendicular to the side AD, the triangle AHB is a right-angled triangle. Let’s use the definition of sine. We have: sin∠BAH = BH / AB, whence BH = AB * sin∠BAH = a * sinα. Therefore, SΔ = ½ * b * a * sinα.
Thus, the required area of ​​the parallelogram is S = 2 * SΔ = 2 * ½ * b * a * sinα = a * b * sinα, that is, the area of ​​the parallelogram is equal to the product of its two adjacent sides by the sine of the angle between them.
Answer: The area of ​​a parallelogram is equal to the product of its two adjacent sides by the sine of the angle between them, that is, a * b * sinα.



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