# The height of a regular triangular pyramid is 3, and the side of the base is 18. Find the apothem of this pyramid.

July 24, 2021 | education

| To solve the problem, consider the figure.

Consider an equilateral triangle ABC at the base of the pyramid. The heights of the triangle, lowered from the vertices, divide the sides into equal parts, then BE = CE = 18 / = 9 cm.

In an equilateral triangle, its height is equal to √3 / 2 of its sides, then AE = (18 * √3) / 2 = 9 * √3.

In an equilateral triangle, the heights at the point of their intersection are divided in the ratio (2/1), then FE = AE / 3 = 3 * √3.

Consider a right-angled triangle GFE, whose hypotenuse GE is the desired apothem. By the Pythagorean theorem GE ^ 2 = GF ^ 2 + FE ^ 2 = 3 ^ 2 + (3 * √3) ^ 2 = 9 + 9 * 3 = 36.

GE = √36 = 6 cm.

Answer: The apothem of the pyramid is 6 cm.

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