# Find The Volume of a Square Pyramid Using Integrals

Find the formula for the volume of a square pyramid using integrals in calculus.

Problem : A pyramid is shown in the figure below. Its base is a square of side $$a$$ and is orthogonal to the y axis. The height of the pyramid is $$H$$. Use integrals and their properties to find the volume of the square pyramid in terms of $$a$$ and $$H$$.

Solution to the problem:
Let us first position the pyramid so that two opposite sides of the square base are perpendicular to the x axis and the center of its base is at the origin of the x-y system of axes. If we look at the pyramid in a direction orthogonal to the x-y plane, it will look like a two dimensional shape as shown below. AC is the slant height.

Let $$x = A'B'$$ be the length of half of the side of the square at height $$y$$. The area $$A$$ of the square at height $$y$$ is given by:
$A(x) = (2x)^2$
The volume is found by adding all the volumes $$A \, dy$$ that make the pyramid from $$y = 0$$ to $$y = H$$. Hence
Volume = $$\int_{0}^{H} A^2 \, dy$$
$= 4 \int_{0}^{H} x^2 \, dy$
We now use the fact that triangles ABC and AB'C' are similar and therefore the lengths of their corresponding sides are proportional to write:
$\frac{a/2}{x} = \frac{H}{H - y}$
We now solve the above for $$x$$ to obtain
$x = \frac{a (H - y)}{2 H}$
We now substitute $$x$$ in the integral that gives the volume to obtain
$\text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \int_{0}^{H} (H - y)^2 \, dy$
Let us define $$t$$ by
$t = H - y \quad \text{and} \quad dt = - dy$
Substitute and change the limits of integration to write the volume as follows:
$\text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \int_{H}^{0} t^2 (- dt)$
Evaluate the integral and simplify
$\text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \left[\frac{H^3}{3}\right]$
$\text{Volume} = \frac{a^2 H}{3}$
The volume of a square pyramid is given by the area of the base times the third of the height of the pyramid.

## More references and Links

integrals and their applications in calculus.
Area under a curve .
Area between two curves .
Find The Volume of a Solid of Revolution .
Volume by Cylindrical Shells Method .
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