The apothem of the regular quadrangular pyramid KPRST is 15, the radius of the circle described

| March 15, 2021 | education

The apothem of the regular quadrangular pyramid KPRST is 15, the radius of the circle described near the base is 12. Find the cosine of the dihedral angle at the base of the pyramid.

The diameter of the circumscribed circle around the square at the base of the pyramid is the diagonal of this square, then RT = RK = 2 * R = 24 cm.

Knowing the length of the diagonal of the square, we determine its side.

2 * RK ^ 2 = RK ^ 2 = 576.

PK ^ 2 = 576/2 = 288.

RK = 12 * √2 cm.

The length of the OH segment is half the side of the square, since OH is the middle line of the RTK triangle, then OH = RK / 2 = 12 * √2 / 2 = 6 * √2 cm.

In a right-angled triangle SOH СosSHO = OH / SH = 6 * √2 / 15 = 2 * √2 / 5 = 0.4 * √2.

SHO angle = arcos (0.4 * √2).

Answer: The dihedral angle is equal to arcos (0.4 * √2).



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