In a regular triangular pyramid, the side of the base is 6, the lateral ribs are inclined to the base at an angle of 45 degrees. Find the volume of the pyramid.
Since a regular triangle lies at the base of the pyramid, its area will be equal to:
Sb = AB ^ 2 * √3 / 4 = 36 * √3 / 4 = 9 * √3 cm2.
Also, the area of the triangle ABC is equal to: Sosn = СB * АН / 2, then АН = 2 * Sosn / СB = 2 * 9 * √3 / 6 = 3 * √3 cm.
The medians of the ABC triangle at point O are divided in the ratio of 2/1, then AO = AH * 2/3 = 3 * √3 * 2/3 = 2 * √3 cm.
In a right-angled triangle AOD, one of the acute angles, by condition, is equal to 45, then the legs of this triangle are equal. AO = DO = 2 * √3 cm.
Then Vpir = Sbn * DO / 3 = 9 * √3 * 2 * √3 / 3 = 18 cm3.
Answer: The volume of the pyramid is 18 cm3.
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