In a regular quadrangular pyramid, the side edge is 5 cm, and the height

| September 11, 2021 | education

In a regular quadrangular pyramid, the side edge is 5 cm, and the height of the pyramid is 3 cm. Find: 1) the side of the base, 2) the apothem, 3) the total surface area.

Let us construct the diagonals AC and BD of the base of the pyramid.

The SOC triangle is rectangular, then, by the Pythagorean theorem, OC ^ 2 = SC ^ 2 – SO ^ 2 = 25 – 9 = 16.

OS = 4 cm.

The OS segment is half of the AC diagonal, then AC = 2 * OC = 2 * 4 = 8 cm.

Determine the area of ​​the base of the pyramid.

Since ABCD is a square, then Sbn = AC ^ 2/2 = 64/2 = 32 cm2.

Then the side of the square will be equal to: ВС = √32 = 4 * √2 cm.

The SBC triangle is isosceles, then its apothem SH is the median of the triangle, then CH = BC / 2 = 4 * √2 / 2 = 2 * √2 cm.

Then SH ^ 2 = SC ^ 2 – CH ^ 2 = 25 – 8 = 17.

SH = √17 cm.

Determine the area of ​​the side face of the pyramid. Ssвс = ВС * SH / 2 = 4 * √2 * √17 / 2 = 2 * √34 cm2.

Then Sпов = Sсн + 4 * Ssвс = 32 + 4 * 2 * √34 = 8 * (4 + √32) cm2.

Answer: The side of the base is 4 * √2 cm, the apothem is √17 cm, the surface area is 8 * (4 + √32) cm2.



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