In a regular quadrangular pyramid, the apothem forms an angle of 60 degrees with the base plane.

| July 3, 2021 | education

In a regular quadrangular pyramid, the apothem forms an angle of 60 degrees with the base plane. The height of the pyramid is 6 cm. Find the surface area of the pyramid.

Consider a right-angled triangle MOH, in which the angle MHO = 60, then the angle HOM = (90 – 60) = 30.

tgМНО = MO / ОН.

OH = MO / tg60 = 6 / √3 = 2 * √3 cm.

The OH leg lies opposite the angle 30, then MH = 2 * OH = 2 * 2 * √3 = 4 * √3 cm.

The segment OH is the middle line of the triangle ACD, then AD = 2 * OH = 2 * 2 * √3 = 4 * √3 cm.

Determine the area of the base. Sbn = АD ^ 2 = (4 * √3) ^ 2 = 48 cm2.

Determine the area of the triangle MCD. Smsd = MH * CD / 2 = 4 * √3 * 4 * √3 / 2 = 8 * √3 cm.

Then S side = Smsd * 4 = 32 * √3 cm2.

Spov = S main + S side = 48 + 32 * √3 = 16 * (3 + 2 * √3) cm2.

Answer: The area of the pyramid is 16 * (3 + 2 * √3) cm2.



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