In a regular quadrangular pyramid, the side edge is 5√5. The dihedral angle made

| August 1, 2021 | education

In a regular quadrangular pyramid, the side edge is 5√5. The dihedral angle made by the side face with the base plane is 60 °. The total surface of the pyramid is?

Since the angle between the base and the side face is 60, the HMS triangle is equilateral.

Let the side of the base be equal to X cm, then НМ = SH = SM = X cm.

Then the height of an equilateral triangle will be: SO = MH * √3 / 2 = X * √3 / 2 cm.

Diagonal AC ^ 2 = 2 * X ^ 2.

AC = X * √2 cm.

Then OC = AC / 2 = X * √2 / 2 cm.

In a right-angled triangle SOC SC ^ 2 = OC ^ 2 + SO ^ 2.

125 = X ^ 2/2 + 3 * X ^ 2/4 = 5 * X ^ 2/4.

X ^ 2 = 500/5 = 100.

X = AB = BC = CD = AD = 10 cm.

Sbn = AB ^ 2 = 10 ^ 2 = 100 cm2.

SO = 10 * √3 / 2 = 5 * √3 cm2.

Sside = 4 * Ssds = 4 * SH * CD / 2 = 4 * 10 * 10/2 = 200 cm2.

Sпов = Sсн + S side = 100 + 200 = 300 cm2.

Answer: The surface area of ​​the pyramid is 300 cm2.



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