A Pentagon, Hexagon And A Decagon Walk Into A Bar

Geometry Level 3

As shown in the image above, a pentagon, hexagon and decagon are inscribed in three congruent circles, and their endpoints are connected to form a triangle. If the radii of each of the circles is \(1\) and the area of the triangle formed by the three polygons can be written as \(\frac{\sqrt{a}-b}{c}\), where \(a\), \(b\) and \(c\) are coprime integers, what is \(a+b+c\)?

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.