# Circular segment

In geometry, a circular segment (symbol: ) is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord.

## Formulas

A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).

Let R be the radius of the circle, θ is the central angle in degrees, α is the central angle in radians, c the chord length, s the arc length, h the height of the segment, and d the height of the triangular portion.

• The radius is $R = h + d = h/2+c^2/8h \frac{}{}$
• The arc length is $s = \frac{\theta}{180}\pi R = {\alpha} R$
• The chord length is $c = 2R\sin\frac{\theta}{2} = R\sqrt{2-2\cos\theta}$
• The height is $h = R\left(1-\cos\frac{\theta}{2}\right) = R - \sqrt{R^2 - \frac{c^2}{4}}$
• The angle is $\theta = 2\arccos\frac{d}{R} = 2\arcsin\frac{c}{2R}$

### Area

The area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion.

Area of segment is $A = \frac{R^2}{2} ({\alpha}-\sin \alpha) = \frac{R^2}{2} \left(\frac{\theta\pi}{180} - \sin \theta \right)$