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Matter of Scale

Can you prove Pythagoras' Theorem using enlargements and scale factors?

Conical Bottle

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?


Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Fit for Photocopying

Age 14 to 16
Challenge Level

You may be familiar with the standard paper size A4.

Two sheets of A4 fit together to make a sheet of A3, two sheets of A3 fit together to make a sheet of A2, and so on.

Each member of the A paper size family is an enlargement of the others - they are all similar shapes.

Can you work out the ratio of the shorter to the longer side of a piece of A paper?
If you're not sure where to start, click below for a hint.

Alison labelled the shorter side of her A4 paper $x$, and the longer side $y$:
As the rectangles are similar, she knew that $x:y = 2y:4x$.
Can you use this to work out an expression for $y$ in terms of $x$?

A sheet of A0 has an area of 1 square metre.
Can you use this fact to deduce the length and width of the different A paper sizes?

On a photocopier, approximately what percentage would you need to scale by in order to photocopy an A3 poster onto A4 paper?

Here are some challenging questions to consider:
Can you find a consistent way to define A(-1) and other negative paper sizes?
Can you find a consistent way to define A$(\frac{1}{2})$, and other fractional paper sizes?