A Dissection Dilemma

Here we have a bit of a puzzler...

First we take a hexagon - it's not a regular hexagon - it has a little bit of a skew to make it more interesting...

We are going to examine its area. To save work, we will split it into two equal bits....

Each half can be filled with four triangles and a rectangle...

The four triangles are (base x height):
4 x 5, 4 x 7, 4 x 7, 3 x 5
and the rectangle is:
4 x 5 = 20 square units in area.

Now, before you reach for your calculator, let's check the figures in a different way.

This time, we'll split the hexagon in half along another diagonal; again we can fill each half with four triangles and a rectangle....

Notice how we have cleverly used the same size of triangles - in fact the only thing we had to do was to turn two of the triangles over onto their backs to make them fit. And of course we've still got a rectangle in each half.

So, we can calculate the area by adding up the areas of the four triamgles and the rectangle...

As before the four triangles are (base x height):
4 x 5, 4 x 7, 4 x 7, 3 x 5 (that's just the same as before)
and this time the new rectangle is:
3 x 7 = 21 square units in area....

Hold hard there!!! That can't be right.... all we did was turn over some triangles, and somehow we've got an extra unit of area in each half of the hexagon!!!??? (that is, the hexagon's area in both examples is the area of the triangles plus the area of the rectangles - but the rectangles have different areas...)


I do hope this doesn't give you a sleepless night - think of it as a conjuring trick that was done right under your nose. You know it can't really have happened - and yet you just saw it..

Get out your pen and paper, and have a go at it for yourself. I'm not going to spill the beans here, but if you really can't sort it out, do write to me..

A good format would be:
Dear Sir,
I am humbled and brought low by my inability to solve your paradox, and beg that... blah...

Oh all right then, just a quick note will do.

Have fun...

Click to go to the Hall of Hexagons.

Comments and discussion always welcome -

David King last update 19Nov01