Find the height of a regular pyramid whose volume is 196 centimetres cubed and base area is 42 centimetres squared.
A regular pyramid is just a pyramid whose base is a regular polygon such as a square or a regular hexagon, for example. We have a formula for finding the volume of a pyramid. The volume is equal to one-third multiplied by the area of the base multiplied by the height of the pyramid. The height here is the perpendicular height. That’s the distance from the base of the pyramid to its point or apex. And this height will be perpendicular to the base.
In this question, we’ve been given the volume of the pyramid. It’s 196 centimetres cubed. And we’ve also been given the base area. It’s 42 centimetres squared. It’s the height that we want to work out. So we can form an equation. Substituting 196 for the volume and 42 for the base area gives 196 is equal to one-third multiplied by 42 multiplied by ℎ. And we now solve this equation to find the height of the pyramid.
First, we note that one-third multiplied by 42 or one-third of 42, which we can find by dividing 42 by three, is equal to 14. So our equation simplifies to 196 equals 14 multiplied by ℎ. We can also express this without the multiplication sign. 14 multiplied by ℎ can just be expressed as 14ℎ.
To solve this equation for ℎ, we need to divide by 14 as 14ℎ divided by 14 gives ℎ. But whatever we do to one side of the equation we must also do to the other. So on the left, we have 196 divided by 14, which is equal to 14. The units for this height will be centimetres because the volume was given in centimetres cubed and the base area was given in centimetres squared.
We’ve found that the height of this regular pyramid is equal to 14 centimetres.