Question

8.[–/1 Points]DETAILSALEXGEOM7 9.2.012.MY NOTESASK YOUR TEACHERSuppose that the base of the hexagonal pyramid below has an area of 40.6 cm2 and that the altitude of the pyramid measures 3.7 cm. A hexagonal pyramid has base vertices labeled M, N, P, Q, R, and S. Vertex V is centered above the base.Find the volume (in cubic centimeters) of the hexagonal pyramid. (Round your answer to two decimal places.) cm3

Answer 1

- The base is a regular hexagon. This implies that it can be divided into equal triangles.

- These equal triangles can be depicted below:

- If each triangle subtends an angle α at the center of the hexagon, it means that we can find the value of α since all the α angles are subtended at the center of the hexagon using the sum of angles at a point which is 360 degrees.

- That is,

`\begin{gathered} α=(360)/(6) \\ \\ α=60\degree \end{gathered}`

- We also know that regular hexagon is made up of 6 equilateral triangles.

- Thus, the formula for finding the area of an equilateral triangle is:

`\begin{gathered} A=(√(3))/(4)x^2 \\ where, \\ x=\text{ the length of 1 side.} \end{gathered}`

- Thus, the area of the hexagon is:

`A=6*(√(3))/(4)x^2`

- With the above formula we can find the length of the regular hexagon as follows:

`\begin{gathered} 40.6=6*(√(3))/(4)x^2 \\ \\ \therefore x=15.626947286066 \end{gathered}`

- The formula for the volume of a hexagonal pyramid is:

`\begin{gathered} V=(√(3))/(2)b^2* h \\ where, \\ b=\text{ the base} \\ h=\text{ the height.} \end{gathered}`

- Thus, the volume of the pyramid is

`\begin{gathered} V=(√(3))/(2)*15.626947286066^2*3.7 \\ \\ V=782.49cm^3 \end{gathered}`