In a regular triangular pyramid, the side face makes an angle of 60 degrees with the base plane.

| July 5, 2021 | education

In a regular triangular pyramid, the side face makes an angle of 60 degrees with the base plane. Find the total surface area of the pyramid if the apothem of the side face is 4

In a right-angled triangle DOH, the angle ODH = (90 – 60) = 30, and the leg OH located opposite it is equal to half the length of DH. OH = DH / 2 = 4/2 = 2 cm.

The medians of the equilateral triangle ABC at point O are divided in the ratio of 2/1, then AO = 2 * OH = 2 * 2 = 4 cm, then AH = 2 + 4 = 6 cm.

AH is the height and median of triangle AB, then AH = BC * √3 / 2 = 6.

BC = 2 * 6 / √3 = 4 * √3 cm.

The area of ​​the base of the prism is: Sbn = ВС ^ 2 * √3 / 4 = 48 * √3 / 4 = 12 * √3 cm2.

The area of ​​the triangle DСВ is equal to: Sдсв = ВС * DН / 2 = 4 * √3 * 4/2 = 8 * √3 cm2.

Then S side = 3 * Sdw = 24 * √3 cm2.

Spov = S main + S side = 12 * √3 + 24 * √3 = 36 * √3 cm2.

Answer: The total surface area of ​​the pyramid is 36 * √3 cm2.



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