Find the volume of the regular hexagonal prism with side lengths of 4 ft, height of 3ft and apothem of approximately 3.5 ft

A (p) = 84 ft³

Step-by-step explanation:

Volume of regular prism is equal to V (p):

V (p) = Area of the face * h

we know h = 3 ft

A regular hexagonal prism has 6 equal sides forming its face.

Finding the area (A₂) of a triangle formed by center of the prism, straight lines between the center and two adjacents vertex and side between these two vertex, and then multiply that area by 6 (number of equal triangles inside hexagonal prism) we get the area of the face, but we need further consideration, the triangle described above, has doble area of (A₁), the triangle formed by apothem, half side, and straight line between center of the face and the vertex, therefore.

Area of small triangle = base * height

A₁ = (1/2) * 4 * 3,5

A₁ = 2*3,5

A₁ = 7 ft²

A₂ = 2 * 7

A₂ = 14 ft²

Finally volume of the hexagonal prism is:

V (p) = 6 * 14

A (p) = 84 ft³