What are Arc Figures?

The name says: Arc figures are figures, which are formed by (circular) arcs. 
Another name is arc shapes.

This section contains a collection of arc figures arranged by the number of corners. On this web site a corner is the point, where two arcs touch. 

Figures of two circles    top

Lense  A = [2/3*Pi - 1/2*sqrt(3)]*r² U = 4/3*Pi*r

Lense A =  [1/2*Pi - 1]*a² U = Pi*a

Mushroom A = (1/8*Pi - 1/4]*r² U = 1/2*Pi*r

Three-quarter Moon 

A = [1/2*Pi + 1]*r²
U = 2*Pi*r

Figures of three circles    top

Tricorn A = [Pi - 1/2*sqrt(3)]*r² U=2*Pi*r

Bug's Eyes A = [1/4*sqrt(3) + 3/8*Pi]*a² U = 3/2*Pi*a

Three circles A = [1/4*sqrt(3) + 5/8*Pi]*a² U = 5/2*Pi*a

Arbelos, Shoemaker's Knife (Archimedes) A = 1/4*Pi*ab = 1/4*Pi*c² U = (a+b)*Pi

Crescent Moons (Hippokrates) A = 1/2*a*b U = Pi/2*[a+b+sqrt(a²+b²)]

Ying und Yang A = 1/2*Pi*r² U = 2*Pi*r

Spinning Top  A = 2*r² U = 2*Pi*r

Golf Tee A = [1/4*sqrt(3) - 1/8*Pi]*a² U = 1/2*Pi*a

Arc Triangle  A = [1/2*Pi - 1/2*sqrt(3)]*r² U = Pi*r

Figures of four circles        top

Cruciferous Flower A = (1/2*Pi - 1]*a² U = 2*Pi*a

Orbital A = [sqrt(3) + 2/3*Pi - 3]*a² U = 4/3*Pi*a

Cross A = [1 + 1/2*Pi]*a² U = 2*Pi*a

Diamonds A = (1 - Pi/4)*a² U = Pi*a

Axe head A = 1/2*a² U = Pi*a

Arc square A = [1 + 1/3*Pi - sqrt(3)]*a² U = 2/3*Pi*a

Four-Leaf Clover A = [1 + 3/4*Pi]*a² U = 3*Pi*a

Dumb-Bell A = [1 + 1/4*Pi]*a² U = 2*Pi*a

Heart A = [1 + 1/4*Pi]*a² U = 2*Pi*a

Hook A = 3/4*Pi*r² U = 3*Pi*r

Salinon A = 1/4*PI*(a+b)² = 1/4*Pi*c² U = Pi*(2*a+b)

Worm A = 5/4*Pi*a² U = 3*Pi*a

Ring A = [1/24*Pi + 1/4*sqrt(3)]*a² U = [2/3*sqrt(3)+3/2]*Pi*a

Hour-glass A = [1 - 1/4*Pi]*a² U = Pi*a

Chicken egg A = [3*Pi - sqrt(2)*Pi - 1)]*r² U = [3-1/2*sqrt(2)]*Pi*r

Figures of five circles     top

Crescent Moons (Hippokrates) A = a² U = [sqrt(2)*+2]*a

Tulip A=Pi*r² U = (2*Pi+2)*r

Figures of six circles   top

Rosette A = [2*Pi - 3*sqrt(3)]*r² U = 4*Pi*r

Rosette A = 2*Pi *r² U = 4*Pi*r

Bear head A = [1/4*sqrt(3) + 1/16*Pi]*a² U = Pi*a

Curve with constant width A = 1/2*Pi*a² + Pi*ab + Pi*b² - 1/2*sqr(3)*a² U = Pi*(a+2b)

Humming-Top A = 2*a² U = 2*Pi*a

Figures of eight circles      top

Cross A = [1 + 1/16*Pi]*a² U = 3/2*Pi*a

Cross A = [1 + 1/16*Pi]*a² U = 5/2*Pi*a

Rings    top

Circumscribed and inscribed circle of an equilateral triangle A = 1/4*Pi*a² U = sqr(3)*Pi*a (more precise: boundary line)

Circumscribed and inscribed circle of a square A = 1/4*Pi*a² U = [1+sqrt(2)]*Pi*a (more precise: boundary line)

How to Calculate Arc Figures top
You calculate arc figures by searching basic figures, which form the figures and the areas of which you know. You must multiply, subtract, add them.

1st Example 
The only basic figure is the semi-circle, that appears four times. The area is generally 1/2*Pi*r². You replace the radius r by a/2 and/or 3a/2. 

The result is A = [1/2*Pi - 1/2*sqr(3)]*r².

The result is A = [1/2*Pi - 1]*r².

...	A collection of important basic figures

Squaring the Circle  top

In the history of mathematics Hippokrates' crescent moons were important, because you can draw a triangle (or four-sided figures in other cases) with ruler and compass having the same area. Mathematicians guessed you were able to find a square with the same area as a circle in a similar way. But since the 19th century it is known, that this is not possible, because Pi is an transcendental number (Ferdinand Lindemann 1882).

Arc Figures on the Internet   top

German

Barbara Flütsch (Mathe-Aufgaben)
Kreis und Kreisteile: Berechnungen

klassenarbeiten.net
Kreisteile - auch Segmente

S M ART
Aufgabenbereich "Kreisteile - auch Segmente"

Wikipedia
Kreis (Geometrie),  Reuleaux-Dreieck, Arbelos

Alexander Bogomolny  (cut-the-knot)
Salinon, The Shoemaker's Knife

David Eppstein    (The Geometry Junkyard)
Circles and Spheres

Eric W. Weisstein  (MathWorld)
Piecewise Circular Curve, Circle, Arc, Triangle Arcs, Semicircle, Arbelos, Lens, 
Yin-Yang, Salinon, Reuleaux Triangle

University of Cambridge (nrich mathematics)
Arclets (Shapes made from arcs)

Wikipedia
Circle,  Reuleaux triangle, Arbelos, Salinon, Semicircle, Ring (geometric)

References     top
Walter Lietzmann: Altes und Neues vom Kreis, Leipzig und Berlin 1935
Eugen Beutel: Die Quadratur des Kreises, Leipzig und Berlin 1942
Maximilian Miller: Gelöste und ungelöste mathematische Probleme, Leipzig 1973