Question

Find the volume of a right prism with height h=10 cm, The base is equilateral triangle with side 4 cm, base altitude of ≈ 3.5 cm.

Answer 1

`\boxed{V=40√(3)cm^3}`

Explanation:

A prism is a solid object having two identical bases, hence the same cross section along the length. Prism are called after the name of their base. In a right prism every edge that connects the base and the opposite face makes right angles with both faces. Moreover, a triangular prism is a solid whose base is a triangle. In general, a prism's volume is always equal to the area of its base, which has the same area of the top face, times its height. In mathematical language:

`V=A_(b)* H \\ \\ Where: \\ \\ A_(b):Base \ Area \\ \\ H:Height`

Since the base is an equilateral triangle with side 4 cm, then we'll use Heron's formula to find the area. This formula uses triangle's side lengths and the semiperimeter. A polygon's semiperimeter s is half its perimeter. So the area of a triangle can be found by
`A=√(s(s-a)(s-b)(s-c))` being
`a,\:b\:and\:c` the corresponding sides of the triangle. So:

`s=(4+4+4)/(2)=6cm \\ \\ a=b=c=4cm \\ \\ A_(b)=√(6(6-4)^3)=4√(3)cm^2`

Finally:

`V=4√(3)* 10 \\ \\ \boxed{V=40√(3)cm^3}`