Contest problem, interestingly perplexing... ?

Question

A coin with a diameter 1 cm rolls around the outside of a regular hexagon with a side length of 1 cm, as shown. What is the length (in cm) of the path traced out by the centre of the coin after one complete rotation around the hexagon?

ok so my answer was 6+pi but the contest answer said 6+2pi

how did they get that? or is it wrong?

thanks in advance

Answer 1

If you treated it like a circle I'm guessing by the answer (though I think the path would be a hexagon) then you know the following

Hexagon side = 1 cm

That means the hexagon total width is 2.732 cm

So its radius is 1.366 cm

The diameter of the coin is 1cm

So its radius is 0.5 cm

Total of the two radii is 1.866 cm

Circumference of a circle 2Pir

So the total path is 2 * Pi * 1.866

This equals 3.73* Pi, which is neither answer

Looking at your answers I'm not sure what you are going for and neither seem right

If you look at the experiment as 6 flat sides of the hexagon and the coin rolls along the edges then it does not matter how big the coin is, it is just an offset of the surface.  Take the 6 edges and lay them end to end and you get a 6 cm long line, roll the coin and you get a 6 cm path.  

I think there is a mistake in the answer and for some reason the circular nature of the coin is being used, this must be where 2pi comes from. Circumference of a circle is 2 Pi r, and since the coin has a DIAMETER of 1 its r is 0.5, so the circumference of the coin is Pi

Maybe you mean.....the total distance the coin rolls along its edge, this would still be 6 cm

I'm clueless as to how you got either answer

Answer 2

well i must be reading the question wrong cuz it seems like you're asking what the length of the path around the hexagon is. To find that you want to know the circumference of a circle if a hexagon is inscribed in it. For any even sided shape that means the diameter will be the distance from one vertex to the opposite vertex. A hexagon is made up of 6 equilateral triangles so the length from the centre to any given vertex is equal to the length of the side (1cm). therefore the diameter of the circle is 2cm. ANd thus the circumference is piD or 2pi. ..

Now I don't knopw exactly what it is that you're finding so I have no clue about the "+6" but hopefully you now know why it's 2pi and not just pi

Answer 3

The contest's answer is wrong, 6 + pi is correct.  (The center covers a length equal to the sum of all the sides of the hexagon (6) as well as one circumference of a circle of diameter 1 (pi).)  Your description of the path (a hexagon with rounded circular corners) is correct and your answer of 6 + pi is also correct... don't be confused by the other responses.