Review question

# When is the area of a face of this pyramid a minimum? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6820

## Suggestion

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?