The cross-section of a regular triangular prism passing through the side of the base and the opposite

| September 3, 2021 | education

The cross-section of a regular triangular prism passing through the side of the base and the opposite apex of the other base forms an angle of 30 degrees with the base plane. The height is 3 cm, find the volume of the prism.

Since the prism is regular, its lateral faces are equal rectangles, then the section BCA1 is an isosceles triangle.

Let us construct the height A1H of the isosceles triangle BCA1, which is its median, then AH is the height and median of the equilateral triangle ABC, and the angle AHA1 is the angle of 300 between the planes

In a right-angled triangle АА1Н, tgAHA1 = АА1 / АН.

AH = AA1 / tg30 = 3 / (1 / √3) = 3 * √3 cm.

The height of an equilateral triangle is: AH = BC * √3 / 2.

BC = 2 * AН / √3 = 2 * 3 * √3 / √3 = 6 cm.

Then Sbn = ВС * АН / 2 = 6 * 3 * √3 / 2 = 9 * √3 cm2.

V = Sbase * АА1 = 9 * √3 * 3 = 27 * √3 cm3.

Answer: The volume of the prism is 27 * √3 cm3.



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