A pyramid of given volume \(V\) stands on a horizontal square base of edge \(2x\), and its vertex is vertically above the centre of the base. Show that the area \(A\) of a sloping triangular face is given by \[A^2=\frac{9V^2}{16x^2}+x^4,\]

What is the formula for the volume of a pyramid?

Sketch a square based pyramid and label its height \(h\). Can we now find the area of one of the triangular faces?

Can we create any right-angled triangles where we can use Pythagoras?

…and prove that, as \(x\) varies, \(A\) is least when the face is equilateral.

Is minimising \(A^2\) the same as minimising \(A\)? How could we go about minimising \(A^2(x)\)?

Where is the stationary point of \(A^2?\)

How do we know the stationary point we’ve found is a minimum?

Can we show that for this value of \(x\) we have equilateral faces?