#2             78. HOW MUCH CARPET?             

The grand new theatre had been built in the shape of a regular octagon.

“First they fix the seats in place, and then they decide to lay the carpet,” groaned the carpet layer. “Oh well, let’s see how much carpet is needed.”

“I have measured the size of the octagon!” volunteered his assistant brightly, waving his sketch (right).

“Well, let us both work out the area,” countered his boss. “Now an octagon of diameter a has area ... hmmm.”

“No, you did it easily this way,” said the assistant moments later, risking instant dismissal!

Youy might like to try calculating the area using the boss’s approach, but of more interest, what was the assistant’s easy way?
HINT 1

Since the measurement b is given, the assistant presumably uses this information.
HINT 2

Think jigsaws: cutting up and rearranging.
SOLUTION

The assistant thought of cutting the octagon along the dotted lines in the original diagram and rearranging them to form this green rectangle, giving area ab.

The boss would have divided the octagon into isosceles triangles with common vertex at the centre, and added their areas – a much more difficult task.
EXTENSIONS

1. Work out the area using the boss’s method, and show that you get the same result. You will need to find the length b in terms of a.

2. Although the given solution is not a tessellation, it involves similar idea. Look up ‘tessellation’, and see if you can construct a similar problem to this one, using a different shape.