Question

Find the volume of a pyramid with a square base, where the perimeter of the base is10.7 ft and the height of the pyramid is 9.8 ft. Round your answer to the nearesttenth of a cubic foot.

Answer 1

The volume of a pyramid is one third of the product between the base area and the height:

$V=\frac{1}{3}\cdot A_{base}\cdot h$

We already know the height of the pyramid, however we need to estimate the base area from the information given.

The base is a square. Squares have their four sides with the exat same length.

The perimeter is the sum of the length of the sides of a polygon, then, the perimeter of the square is four times its side:

$P=4\cdot L$

Where L is the length of the side. Now, knowing the perimeter we can estimate the lenght of the sides of the base, from which we can calculate the base area:

$\begin{gathered} 4\cdot L=P \\ L=\frac{P}{4} \\ L=\frac{10.7ft}{4} \\ L=2.675ft \end{gathered}$

Now we know the sides of the base have a length of 2.675 ft each. To estimate the area of the base we just need to square this lenght:

$A_{\text{base}}=L^2$

Then, to calculate the volume of the pyramid:

$\begin{gathered} V=\frac{1}{3}\cdot A_{base}\cdot h \\ \\ V=\frac{1}{3}\cdot L^2\cdot h \end{gathered}$

Let's replace values:

$\begin{gathered} V=\frac{1}{3}\cdot(2.675ft)^2\cdot9.8ft \\ V\approx23.375ft^3 \\ V\approx23.4ft^3 \end{gathered}$

The volume of the pyramid is approximately 23.4 cubic feet.