Math CentralQuandaries & Queries


Question from Michael, a parent:

An equilateral triangle and a regular hexagon have equal perimeters.  If the area of the triangle 10, what is the area of the hexagon?

Hi Michael.

There are many ways to solve this problem, but I'm going to show you a purely algebriac way. You'll see in this reply to Dana that I derived the equation for any regular polygon's area (ARP) given its side length a (number of sides is n):

So for a regular hexagon whose perimeter P is known, n=6 and a=P/6, the area of the hexagon is:

[Note: You can see that I've substituted tan 30° with its exact value.]

But we don't yet know what P actually is, except that it is perimeter of an equalateral triangle with area 10 square units. Of course, an equilateral triangle is also a regular polygon, so the same equation applies here. Let's use this to find the value P2 which we need in the equation above.

[Again, tan 60° is replaced with its exact value.]

So when we solve this for P2, we get

Now we can substitute this value of P2 into the earlier equation to find the value AH that answers the question:

Hope this helps,
Stephen La Rocque.

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