The height of a regular quadrangular pyramid is equal to two roots of three. The side faces of the pyramid

| June 16, 2021 | education

The height of a regular quadrangular pyramid is equal to two roots of three. The side faces of the pyramid are inclined to the base plane at an angle of 60 degrees. Find the length of the side edge of the pyramid.

MO is the height of the pyramid, hence the triangle MOH is rectangular.

tgMHO = OM / OH.

√3 = 2 * √3 / OH.

OH = √3 * 2 / √3 = 2 cm.

Angle ОМH = 180 – 90 – 60 = 30.

The OH leg lies opposite an angle of 30, then MH is equal to two lengths of the OH leg. MH = 2 * 2 = 4 cm.

Since there is a square at the base of the regular pyramid, AB = AC = AD = CD = 2 * OH = 2 * 2 = 4 cm.

OH is the height of the isosceles triangle COD, which is also the median, and therefore DH = CH = CD / 2 = 4/2 = 2 cm.

In a right-angled triangle МНD, the hypotenuse МD is a lateral edge of the pyramid, and by the Pythagorean theorem it is equal to: МD ^ 2 = МН ^ 2 + DH ^ 2 = 4 ^ 2 + 2 ^ 2 = 16 + 4 = 20.

МD = √20 = 2 * √5 cm.

Answer: The length of the side rib is 2 * √5 cm.



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