The base of a solid in the xy-plane is the circle x^2 + y^2 = 16. Cross sections of the solid perpendicular to the y-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

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Answer 1

The base of this solid is a circle with

radius r = 4 and diameter d = 8, then this solid is a cone.

Area = 16π

Since the cross section mentioned is equilateral triangle, then the height of the cone is also the height of the equilateral triangle with each side is 8 units.

Remember, half of an equilateral triangle is a 30-60-90 triangle. The longest leg will be the height of the pyramid. Following the rule of 30-60-90 triangle,

the height of the pyramid (h) = 4√3

Volume (V) = one third of Area of the base (B) times height of the pyramid (h)

V = (1/3) Bh

V = (1/3) 16π (4√3)

V= 64π√3 / 3 square units