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:

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Example

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

The area of the basis is $A_{b a s e} = 100 m^{2}$

To find the area of the laterals, first we have to find $A p$ , $A p^{2} = h^{2} + a p^{2} = 5^{2} + (\frac{10}{2})^{2} A p = 5 \sqrt{2}$

So, the area of a side face is: $A_{l a t e r a l} = \frac{10 \cdot 5 \sqrt{2}}{2} = 25 \sqrt{2} A_{t o t a l} = 4 \cdot A_{l a t e r a l} + A_{b a s i s} A_{t o t a l} = 100 \sqrt{2} + 100 = 241, 4 m ²$

Generalizing, $A_{l a t e r a l s} = \frac{p e r i m e t r e_{b a s i s} \cdot A p}{2} A_{t o t a l} = A_{l a t e r a l s} + A_{b a s i s}$

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: $V = \frac{A_{b a s i s} \cdot h e i g h t}{3}$

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