Question

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?

Answer 1

Calculate the height of an equilateral triangle with sides all `4cm`, then an isosceles triangle with two sides of that length and a `4cm` base.

Explanation:

A triangular face of the pyramid is equilateral, so divides into two right angled triangles with sides `4cm`, `2cm` and `2sqrt(3)cm`.

Then looked at from one side, the pyramid looks like a isosceles triangle with base `4cm` and sides `2sqrt(3)cm`. This divides into two right angled triangles with sides `2sqrt(3)cm`, `2cm` and `2sqrt(2)cm`, 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 `4cm`, i.e. `4sqrt(2)cm`. Then the height of the pyramid is half this, i.e. `2sqrt(2)cm`.