Each side edge of a regular triangular pyramid is equal to 3, and the planar angle

| September 27, 2021 | education

Each side edge of a regular triangular pyramid is equal to 3, and the planar angle at its apex is 90 degrees. Find the volume of the pyramid.

The side faces of the pyramid are rectangular, isosceles triangles with a side side of 3 cm. In a right-angled triangle ADC, AC ^ 2 = 2 * CD ^ 2 = 2 * 9.

AC = 3 * √2 cm.

Since the pyramid is regular, the ABC triangle is equilateral, then Sавс = AC ^ 2 * √3 / 4 = 18 * √3 / 4 = 9 * √3 / 2 cm2.

The height of an equilateral triangle is: BH = AC * √3 / 2 = 3 * √2 * √3 / 2 = 3 * √6 / 2 cm.

By the property of the median of the triangle, ОВ = 2 * ВН / = 2 * (3 * √6 / 2) / 3 = √6 cm.

In a right-angled triangle BOD, OD ^ 2 = BD ^ 2 – BO ^ 2 = 9 – 6 = 3.

OD = √3 cm.

Then V = Sn * OD / 3 = (9 * √3 / 2) * √3 / 3 = 4.5 cm3.

Answer: The volume of the pyramid is 4.5 cm3.



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