The apothem of a regular quadrangular pyramid is 6 cm, height 3√2. Find: a) the angle formed by the lateral

| July 7, 2021 | education

The apothem of a regular quadrangular pyramid is 6 cm, height 3√2. Find: a) the angle formed by the lateral edge and the plane of the base of the pyramid b) the area of the lateral surface of the pyramid

In a right-angled triangle SOH, according to the Pythagorean theorem, we determine the length of the leg OH.

OH ^ 2 = SH ^ 2 – SO ^ 2 = 36 – 18 = 18.

OH = √18 = 3 * √2 cm.

At the base of the regular pyramid lies the square ABCD, the diagonals of which are halved at the point of intersection, OA = OS = AC / 2. Apothem SH is the height and median of the isosceles triangle SCD, then DH = CH = CD / 2. Then OH is the middle line of the triangle ASD , and therefore AD = 2 * OH = 2 * 3 * √2 = 6 * √2 cm.

Determine the length of the diagonal BD at the base of the pyramid.

ВD ^ 2 = 2 * АD ^ 2 = 2 * 72 = 144.BD = 12 cm.

Then OD = BD / 2 = 12/2 = 6 cm.

In a right-angled triangle SOD, tgsto = SO / OD = 3 * √2 / 6 = √2 / 2.

Angle SDO = arctan √2 / 2.

Determine the area of ​​the triangle SСD. Sscd = СD * SH / 2 = 6 * √2 * 6/2 = 18 * √2 cm2.

Then S side = 4 * Sscd = 4 * 18 * √2 = 72 * √2 cm2.

Answer: The lateral surface area is 72 * √2 cm2, the angle is arctan √2 / 2.



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