Circles Collection

Concentric Circles


The length of the red segment is 2. What is the area of the blue ring?


Four Arcs and a Square








All arcs are congruent. What is the area contained by the four arcs as a fraction of the square's area?

Screenshot 2021-12-30 195023.png

Ten Circles

The orange ring is 12 pi long. What is the total length of the blue rings? Their centers lie on the orange ring's diameter.

Screenshot 2021-12-28 210313.png

Three Circles

The red point is the center of the larger circle. What is the ratio of the yellow area to the blue area?

Screenshot 2021-12-29 184129.png

Circle, Square, Circle








The blue quadrilateral is a square. What is the ratio of the two circles' areas? 

Screenshot 2022-01-02 182504.png

Circles vs. Squares







The two large circles have radii of 1. The dodecagon is made up of five squares. Which design contains the largest area: the four circles or the dodecagon?

Screenshot 2021-12-30 201400.png

3 Circles or a Square?




The brown circles are congruent. The one on the left contains three identical circles, the one on the right  inscribes a square. Which design maximizes the yellow area?

Screenshot 2021-12-30 203205.png

Two Intersecting Circle


Where should the two moving points be placed for the pink segment to be at its maximum possible length?

Area of a Quadrilateral








Determine the position of the orange chord such that the area of the blue quadrilateral is maximum.

Max Area Quad.gif