Question

If we had a Pyramid with a hexagonal base, how many of these pyramids would it take to fill up a hexagonal Prism with a congruent base, and equal height?

Answer 1

3 of the pyramids

Explanation:

Given:

Hexagonal Prism

Hexagonal based pyramid

Required

How many of the pyramid will fill up the prism

To do this, we start by calculating the volumes of both shapes.

Let V1 be the volume of the hexagonal prism

`V_1 = (3√(3))/(2)a^2h`

Where: a = base and h = height

Let V2 be the volume of the hexagonal based pyramid

`V_2 = (√(3))/(2)a^2h`

Where: a = base and h = height

The number of the pyramid that can occupy the prism is calculated by dividing V1 by V2

`Number = (V_1)/(V_2)`

`Number = (3√(3))/(2)a^2h /(√(3))/(2)a^2h`

Convert division to multiplication

`Number = (3√(3))/(2)a^2h * (2)/(√(3)a^2h)`

`Number = (3√(3))/(2) * (2)/(√(3))`

`Number = (3√(3))/(1) * (1)/(√(3))`

`Number = (3*1)/(1) * (1)/(1)`

`Number = 3`