Hexagons are twice as hard as triangles
This problem came in ganesh's oral puzzles.
Find the radius of the circle inscribed in a regular hexagon of side 6 cm.
Let's say you cannot remember that there is a formula for this. Can Geogebra help? Yep!
1)Use the regular polygon tool and put A and B on (2,2) and (8,2).
2) When the input box appears enter 6 and you will get a hexagon. Adjust A and B by actually inputting the correct values if you could not place them there.
3)Get the midpoint of AB it will be called G.
4)Get the midpoint of CD it will be called H.
5)Get the midpoint of EF it will be called I.
6)Connect AD with a line segment.
7)Connect BE with a line segment.
8)Use the intersection tool to find J the intersection of AD and BE. It is also the center of the hexagon.
9)Use the circle through 3 points tool to construct circle through G,H and I.
10)Now draw line segment JH and measure it.
11) In the variable pane you will see the value of JH. Set rounding to 15 digits in options. You should get JH = 5.19615242270663. See the first picture to check your drawing.
12)Go over to the Inverse Symbolic calculator and input 5.19615242270663
You should see 3√3 cm. That is the exact answer! Another nifty example of experimental mathematics in action!
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.