# The pyramid: Surface area and volume

The pyramid is a polyhedron formed by any polygon (base) and triangular faces that coincide at the top point, called the apex.

The following figure is an example of quadrangular pyramid:

Calculate the area of a quadrangular pyramid with side of the basis $10 \ m$ and height $5 \ m$.

The area of the basis is $$A_{base} =100 \ m^2$$

To find the area of the laterals, first we have to find $Ap$, $$Ap^2=h^2+ap^2=5^2+\Big( \dfrac{10}{2}\Big)^2 \\ Ap=5\sqrt{2}$$

So, the area of a side face is: $$A_{lateral}=\dfrac{10 \cdot 5\sqrt{2}}{2}=25\sqrt{2} \\ A_{total}=4 \cdot A_{lateral}+A_{basis}\\A_{total}=100\sqrt{2}+100=241,4 \ m²$$

Generalizing,$$A_{laterals}=\dfrac{perimetre_{basis} \cdot Ap}{2} \\ A_{total}=A_{laterals}+A_{basis}$$

To find the volume of a pyramid, it is useful to remember that it is the third part of the volume of a prism of equal basis and height: $$V=\dfrac{A_{basis} \cdot height}{3}$$

In case of the previous example $100$ obtains a volume $100 \cdot \dfrac{5}{3} = 166,67$