The base of the straight prism ABCDA1B1C1D1 is a parallelogram ABCD with sides of 6 cm and 6 √3

| June 1, 2021 | education

The base of the straight prism ABCDA1B1C1D1 is a parallelogram ABCD with sides of 6 cm and 6 √3 and an angle of 150 degrees. The diagonal B1D of the prism forms an angle of 60 degrees with the base plane. Find the total surface area of the prism.

In a rhombus, the sum of adjacent angles is 180, then the angle BAD = (180 – 150) = 30.

In triangle ABD, by the cosine theorem, we define the length of the side BD.

BD ^ 2 = AD ^ 2 + AB ^ 2 – 2 * AD * AB * Cos30 = 108 + 36 – 2 * 6 * √3 * 6 * (√3 / 2) = 144 – 108 = 36

ВD = 6 cm.

In a right-angled triangle BB1D tg60 = BB1 / BD.

BB1 = BD * tg60 = 6 * √3 cm.

Determine the area of ​​the base of the rhombus.

Sbn = AB * AD * Sin30 = 6 * 6 * √3 * (1/2) = 18 * √3 cm2.

Determine the total surface area of ​​the prism.

Spov = 2 * Sb + S side = 2 * Sb + Ravsd * BB1 = 2 * 18 * √3 + (12 + 12 * √3) * 6 * √3 = 36 * √3 + 72 * √3 + 216 = 108 * √3 + 216 = 108 * (√3 + 2) cm2.

Answer: The total surface area is 108 * (√3 + 2) cm2.



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