At the base of the pyramid lies a right-angled triangle with leg 6 and an angle of 60 degrees adjacent to it.

At the base of the pyramid lies a right-angled triangle with leg 6 and an angle of 60 degrees adjacent to it. All side edges of the pyramid are inclined to the base plane at an angle of 30 degrees. Find the volume of the pyramid.

Consider a triangle at the base. It is rectangular, one angle = 60 °, the second = 30 °. The leg opposite to the angle 30 ° = 6, which means that the hypotenuse = 12, the second leg = 6 roots of 3. The area of ​​the triangle is half the product of legs = 6/2 * 6 roots of 3 = 18 roots of 3. Since all the side edges of the pyramid form the same angle with the base, which means that they are equal and their projections are equal. Since the projections are equal, they intersect at the intersection of the medians of the triangle. Find the median drawn to the hypotenuse of the triangle from a right angle. The median is half the hypotenuse = 6. From the top of the angle to the point of intersection of the medians 2/3 of the median => the length of the projections of the lateral edges to the base = 2/3 * 6 = 4. The height is perpendicular to the base. Tangent 30 = height divided by projection => height = 4/3 roots of 3. The volume of the pyramid is equal to 1/3 of the product of the base area by the height. Volume = 1/3 * 4/3 roots from 3 * 18 roots from 3 = 24 cubic meters. units
Answer: 24 cubic meters. units