In a regular hexagonal pyramid of apothem √15 cm, and the side of the base is 2 cm. Find the volume of the pyramid.

| May 2, 2021 | education

Determine the area of ​​a regular hexagon at the base of the pyramid.

S6 = a ^ 2 * 3 * √3 / 2, where a is the length of the edge of the hexagon.

S6 = 2 ^ 2 * 3 * √3 / 2 = 6 * √3 cm2.

Let’s make an apothem of the SM pyramid. Since the lateral edges of the pyramid are isosceles triangles, the apothem SM is also the height and median of the SCD triangle.

The large diagonals of a regular hexagon divide it into six regular triangles with a side of 2 cm. Then OM in triangle OCM is the height, bisector and median of a regular triangle.

ОМ = СD * √3 / 2 = 2 * √3 / 2 = √3 cm.

From the right-angled triangle SOM, by the Pythagorean theorem, SO ^ 2 = SM ^ 2 – OM ^ 2 = 15 – 3 = 12.

SO = 2 * √3 cm.

Let’s define the volume of the pyramid. V = S6 * SO / 3 = 6 * √3 * 2 * √3 / 3 = 12 cm3.

Answer: The volume of the pyramid is 12 cm3.



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