Question

A solid oblique pyramid has an equilateral triangle as a base with an edge length of 4StartRoot 3 EndRoot cm and an area of 12StartRoot 3 EndRoot cm2.

A solid oblique pyramid has an equilateral triangle base with an edge length of 4 StartRoot 3 EndRoot centimeters and an area of 12 StartRoot 3 EndRoot centimeters squared. The triangle base has points A C D and the apex is point B. The angle of the lateral sides with the triangle base is 30 degrees.

What is the volume of the pyramid?

12StartRoot 3 EndRoot cm3
16StartRoot 3 EndRoot cm3
24StartRoot 3 EndRoot cm3
32StartRoot 3 EndRoot cm3

Answer 1

Let H be the foot of the perpendicular from the apex B to the base plane. Because each lateral edge BA, BC, BD makes the same angle 30° with the base plane, the apex is directly above the circumcenter H of the equilateral base. For each lateral edge BA we have

BA sin30° = BH = h, so BA = 2h,
and the projection AH = BA cos30° = 2h·(√3/2) = h√3.

But AH is the circumradius R of the equilateral triangle. With side a = 4√3,
R = a/√3 = (4√3)/√3 = 4, so h√3 = 4 ⇒ h = 4/√3.

The volume is V = (1/3)(base area)·h = (1/3)(12√3)·(4/√3) = 16 cm^3.

Note: 16 cm^3 is the correct volume; none of the given choices (they are all multiples of √3) equals 16.