At the base of the pyramid lies a square with a diagonal equal to 6. One of the side edges is perpendicular to the base. The larger lateral rib is inclined to the base at an angle of 45 degrees. What is the volume of the pyramid?
Let us determine the length of the side of the square by the Pythagorean theorem.
AC ^ 2 = AD ^ 2 + CD ^ 2 = 2 * AD ^ 2.
AD ^ 2 = AC ^ 2/2 = 36/2 = 18.
AD = 3 * √2 cm.
Since, by condition, one of the faces is perpendicular to the base, then the ACS triangle is rectangular, and as the larger edge is inclined to the base plane at an angle of 45, then the triangle is isosceles, AC = SC = 6 cm.
The side edge SC is the height of the pyramid, since it is perpendicular to the base, then the volume of the pyramid is:
Vpir = Sbn * h / 3 = AB * AD * SC = 3 * √2 * 3 * √2 * 6/3 = 36 cm3.
Answer: The volume of the pyramid is 36 cm3.
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