Consider a right circular cone of height 4 cm and base radius measuring 3 cm. Find the volume of regular pyramid vertex coincident with the apex of the cone and whose base hexagon is inscribed on the base of the cone.
Since the problem provides the information that the base of regular pyramid, which is a regular hexagon, is inscribed in a circle whose radius measures 3cm, you may evaluate the length of side of regular hexagon using the fact that the triangle formed with the radii of circle and the side of hexagon is an equilateral triangle.
Thus, since the bottom side of equilateral triangle represents the side of regular hexagon, yields that it also measures 3cm.
You need to evaluate the volume of regular pyramid whose base is the regular hexagon, such that:
`A = (1/3)` *Area of hexagon*height
You need to evaluate the area of hexagon, hence, you need to divide the hexagon in 6 equilateral triangles, `AOB,BOC,COD,DOE,EOF,FOA` , and then you need to find the area of triangle.
`A_(AOB) = l^2sqrt3/4 => A_(AOB) = 9sqrt3/4`
You need to evaluate the area of hexagon `ABCDEF` , hence, you need to multiplicate the area of triangle `AOB` by 6, such that:
`A_(ABCDEF) = 6*(9sqrt3/4) => A_(ABCDEF) = 27sqrt3/2`
Since the problem provides the information that the vertex of pyramid coincides with the vertex of the right circular cone, yields that the height of pyramid coincides with the height of cone.
You may evaluate the volume of pyramid, such that:
`V = (1/3)*(27sqrt3/2)*4 => V = 18sqrt3 cm^3`
Hence, evaluating the volume of regular pyramid, under the given conditions, yields `V = 18sqrt3 cm^3.`