What is the ratio of the areas of the circle circumscribing a square and the circle which is inscribed in the square?
Let's do it using geogebra!
1)Place 2 points at (0,0) and (6,0), they will be labeled A and B.
2) Use the regular polygon tool using A and B to create a 6 x 6 square ABCD.
3) Use the midpoint tool and get the midpoints of AD, DC and CB. Those points will be labeled E,F and G.
4)Use the Circle from 3 points tool to create an inscribed circle from point E,F and G.
5) Use the Circle from 3 points tool to create a circumscribed circle from points A,D and C.
6)Use the area tool to get the area of the inscribed circle. It will be called areae in the algebra panel.
7)Use the area tool to get the area of the circumscribed circle. It will be called areaf in the algebra panel.
8)In the input bar put g = areaf / areae and press enter.
9) Now pull B around while watching g. What do you notice about the relationship of the areas of the two circles? Strong argument for 2:1 don't you think.