Question 1210593
Reading phrase by phrase and understanding:
"The region bounded by the circle {{{x^2+y^2=a^2}}} is the base of a solid."
The base of the solid is a circle on the x-y plane, with center at the origin, radius {{{a}}} , and diameter {{{2a}}} .
So, far we know that the area of that base is {{{pi*a^2}}} .
"Cross sections taken perpendicular to the base and parallel to the y-axis are equilateral triangles."
The y-axis is the line {{{x=0}}} , and the cross sections mentioned are obtained cutting through the solid through planes {{{x=b}}} with values of {{{b}}} in (-a,a) .
Those cross sections have a base on the x-y plane, and they are equilateral triangles.
The largest such equilateral triangle has a base of length {{{2a}}} ,
extending between the points (0,-a), and (0,a), and a height of {{{(sqrt(3)/2)*(2a)=a*sqrt(3)}}} ,
with a vertex at the point with coordinates {{{x=y=0}}} and {{{z=a*sqrt(3)}}} , which is the apex of the solid.
Intuitively, we know the solid is a cone.
Hopefully we do not need to prove that through a lot of boring algebra and geometry work.
The volume of that cone with base area {{{pi*a^2}}} and height {{{a*sqrt(3)}}} is {{{(1/3)(pi*a^2)(a*sqrt(3))=highlight(a^3*pi*sqrt(3)/3)}}}