The side of the base of a regular hexagonal pyramid is 8 m, and the lateral edge forms an angle of 60

| April 30, 2021 | education

The side of the base of a regular hexagonal pyramid is 8 m, and the lateral edge forms an angle of 60 degrees with the base plane. Find a) lateral edge b) the area of the lateral surface of the pyramid

Since a regular hexagon lies at the base of the pyramid, its diagonals divide it into six equilateral triangles.

Then the triangle AOK is equilateral, AO = KO = AK = 8 m.

The AOP triangle is rectangular, in which the angle ARC = (90 – 60) = 300, then the length of the hypotenuse AP = 2 * AO = 2 * 8 = 16 m.

The AOP triangle is isosceles, AR = KR, then its height PH is also its median, then AH = KH = AK / 2 = 8/2 = 4 cm.

From the right-angled triangle APН, according to the Pythagorean theorem, PH ^ 2 = AP ^ 2 – AH ^ 2 = 256 – 16 = 240.

PH = √240 = 4 * √15 cm.

Then Sark = AK * PH / 2 = 8 * 4 * √15 / 2 = 16 * √15 cm2.

The side faces of the regular pyramid are equal, then Sside = 6 * Sark = 6 * 16 * √15 = 96 * √15 cm2.

Answer: The lateral surface area is 96 * √15 cm2.



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