# Problems from The pyramid: Surface area and volume

The red pyramid of Dahnshur is the third biggest of Egypt. If we know that the side of the base $a$ is bigger than its height $h$, invent some possible measurements of this construction. Then, find the volume of the pyramid.

Originally, the pyramid was white, and it was covered with an ornament of red colour. What area was it necessary to cover the pyramid?

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### Development:

The measurements of the pyramid are, approximately: $a=220 \ m$, and $h=100 \ m$.

Its volume will be $$V=\dfrac{a^2\cdot h}{3}=1.613.333 \ m^3$$

To find the area of four triangles that are necessary to make the red coverture, it is necessary to find the apothem of the pyramid: $$ap^2=\Big(\dfrac{a}{2}\Big)^2+h^2$$ $$ap=\sqrt{110^2+100^2}=148,66 \ m$$ $$A_{1 \ face}=\dfrac{ap\cdot a}{2}=7433,03 \ m^2$$ $$A_{4 \ faces}=29.732 \ m^2$$

### Solution:

The measurements of the pyramid are, approximately $a=220 \ m$, and $h=100 \ m$.

The volume: $V=\dfrac{a^2\cdot h}{3}=1.613.333 \ m^3$

$A_{4 \ faces}=29.732 \ m^2$

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