# Hexagon Problem #2

AB and BC are tangents to the circle, touching it at points A and C.

If the circumference of the circle is 42πcm, find the area of the hexagon.

## 10 thoughts on “Hexagon Problem #2”

1. The diagram is geometrically impossible as drawn. If AD ⊥ AB (tangent is perpendicular to radius) and CD ⊥ CB, and ∠D is 120 (regular hexagon), then the angle indicated by the dashed lines (∠B) must be 60. Another way to describe the problem is that either a regular hexagon is circumscribed about the circle (which you don’t seem to intend), or it’s not tangent to the circle at non-adjacent vertices A and C (which the “not to scale” drawing reveals).

2. You’re absolutely right. Quite an error! In fact, therefore, the puzzle can have no angles explicitly revealed at all. I have updated it. Many thanks.

3. Not quite.
If it is a regular hexagon then the hexagon vertex opposite D “is” the center of the circle.

• In which case there is no need to tell them anything about centers of circles.

• That’s a far simpler (read: better) explanation than the blunder we came up with.

4. Howard, that comment confused me for ages! Mainly because I was staring at the (inaccurate, but now adjusted) diagram instead of thinking about the maths. Many thanks. Can I nominate you as my puzzle verifier before posting in future? My wife is giving birth soon so I fear I may have baby brain over the next few months! 🙂

5. Roughly 370cm squared?

6. Beth says:

Enjoyed doing this one – less tricky than the ones on subject days! Going to give it to my Year 12 c1 pupils tomorrow and see how they do 🙂