In the figure below, 3 equal circles have been drawn so that each one passes through the centers of the other two. Is the area of overlap of the three triangles more or less than a quarter of the area of a circle?

Source: Julia Robinson Mathematics Festival

**SOLUTION**

**
**The figure above shows the overlap of the three triangles in close-up. Let denote the

*area*between a circle and a chord, for example between a circle and chord and let denote the

*area*of triangle . Then,

Area of overlap =

**Area of triangle **

Triangle is equilateral because its three sides are radii of congruent circles. Let be the radius. is an altitude and equals because triangle is a triangle with a hypotenuse of length .

Area of

**Calculate **

= area of sector area of

**Area of overlap**

Area of overlap =

**Compare areas**

A quarter of the area of a circle = .

Since we conclude that the area of overlap is less than a quarter of the area of a circle.

**Answer**: Area of overlap is less than a quarter of the area of a circle.