A Christmas decoration is in the shape of a square based pyramid. All edges in the pyramid have a measurement of 4cm how do you find the height of the pyramid?

Oct 24, 2015

Calculate the height of an equilateral triangle with sides all $4 c m$, then an isosceles triangle with two sides of that length and a $4 c m$ base.
A triangular face of the pyramid is equilateral, so divides into two right angled triangles with sides $4 c m$, $2 c m$ and $2 \sqrt{3} c m$.
Then looked at from one side, the pyramid looks like a isosceles triangle with base $4 c m$ and sides $2 \sqrt{3} c m$. This divides into two right angled triangles with sides $2 \sqrt{3} c m$, $2 c m$ and $2 \sqrt{2} c m$, this last length being the height of the pyramid.
Alternatively, note that stacking two of these pyramids back to back will result in a regular octohedron, who diagonal measurement is the diagonal of a square with sides of length $4 c m$, i.e. $4 \sqrt{2} c m$. Then the height of the pyramid is half this, i.e. $2 \sqrt{2} c m$.