[bonanova]

Sketch four circles, two side by side and two more directly above them.

All four circles have the same diameter and touch their two nearest neighbors.

Quickly determine the area of the space between the circles.

Extra credit:

Sketch three circles, two side by side and a third resting above and between them.

All three circles have the same diameter and touch the other two.

Quickly determine the area of the space between the circles.

Enjoy, and please use spoilers.

[plainglazed]

R²(4-pi) (square whatever unit of measure)

and

r(3)^.5-1/2piR²

[bonanova]

R²(4-pi) (square whatever unit of measure)

and

r(3)^.5-1/2piR²

Yes.

There is a cute way to arrive at the answer without performing

a calculation and to phrase it without using numbers or symbols.

[plainglazed]

For the first part

the area is that of a square minus the largest possible inscribed circle

For the second part

the area of the equilateral triangle of leg twice the radius of a circle less half the area of the circle

Edit: added second part

Edited August 21, 2009 by plainglazed

[bonanova]

For the first part

the area is that of a square minus the largest possible inscribed circle

For the second part

the area of the equilateral triangle of leg twice the radius of a circle less half the area of the circle

Edit: added second part

The shapes you mention are formed by connecting the centers of the circles.

And constructing them verifies the results without calculation.

[Guest]

is it directly related to the size of the circles...e.g. its an eighth of one circle

Sorry if this is completely wrong.

[Guest]

R²(4-pi) (square whatever unit of measure)

and

r(3)^.5-1/2piR²

I believe the second part is R²(3)^.5-1/2piR², right?