The apothem of a regular triangular pyramid is 4 cm, and the radius of a circle circumscribed

| July 5, 2021 | education

The apothem of a regular triangular pyramid is 4 cm, and the radius of a circle circumscribed about the base is 3√3 cm. Find the total surface area of the pyramid.

The point of intersection of the bisectors of the triangle is the center of the circumscribed circle, and since the triangle ABC is equilateral, the center of the circle is the point of intersection of the medians and heights.

Then R = ОА = 3 * √3 cm.By the property of the medians, ОН = ОА / 2 = 3 * √3 / 2 cm, then BK = ОА + ОН = 9 * √3 / 2 cm.

The segment AH is the height of an equilateral triangle, then AH = BC * √3 / 2.

BC = 2 * BK / √3 = 2 * (9 * √3 / 2) / √3 = 9 cm.

Determine the area of ​​the side face of the pyramid.

Sдсв = ВС * ДН / 2 = 4 * 9/2 = 18 cm2.

Then S side = 3 * Sdw = 3 * 18 = 54 cm2.

Determine the area of ​​the base of the pyramid.

Sbn = ВС * АН / 2 = 9 * (9 * √3 / 2) / 2 = 81 * √3 / 2 cm2.

Spov = Sside + Sbn = 54 + 81 * √3 / 2 = 27 * (2 + 3 * √3 / 2) cm2.

Answer: The total surface area is 27 * (2 + 3 * √3 / 2) cm2. cm2.



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