A cone is inscribed in a regular quadrangular pyramid. Find the lateral surface of the cone if the bases

| August 6, 2021 | education

A cone is inscribed in a regular quadrangular pyramid. Find the lateral surface of the cone if the bases of the pyramid are equal to a and the height is h.

The diameter of the circle at the base of the cone is equal to the side of the square at the base of the pyramid.

KM = D = AB = a cm.

Then R = OH = a / 2 cm.

The apothem of the side face of the pyramid is equal to the generatrix of the cone inscribed in it.

Then in the right-angled triangle MOH, by the Pythagorean theorem, MH ^ 2 = MO ^ 2 + OH ^ 2 = h ^ 2 + (a / 2) ^ 2.

MH = √ (h ^ 2 + a ^ 2/4) cm.

Let us determine the area of the lateral surface of the cone.

S side = n * OH * MH = n * (a / 2) * √ (h ^ 2 + a ^ 2/4) = n * (a / 2) * √ (4 * h ^ 2 + a ^ 2) / 4 = n * a * √ (4 * h ^ 2 + a ^ 2).

Answer: The area of the lateral surface of the cone is = n * a * √ (4 * h ^ 2 + a ^ 2) cm2.



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