# Circles in Squares Patterns

Consider this pattern.

A.   Construct the pattern with straightedge and compass or with Geometers SkethPad.

B.   We want to ask questions to compare areas within the figure.   What questions might you ask?

C.  Give the a side of the outer square is 4 units, show that the total area of the yellow (unshaded) regions.   That is, prove

D.  What is the total area of the shaded region?

E.   Show that the area of the yellow regions int the four corners of the outer square is the same as the total area of the yellow regions in the inner square.

F.    What is the expression for the total yellow area if the square has side length of  s  units?

Hint:     A useful lemma.

Discussion:

There may be multiple ways to gain insight into finding an expression for the area and the lemma may not be the one some people see.

Does the act of constructing the figure help with seeing a way to determine an expression for the area?

G.    Compare the areas of the yellow regions in this figure and the one above.

Confirm or prove what you observe.

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