Circular segment

In geometry, a circular segment (also circle segment) 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. The circle segment constitutes the part between the secant and an arc, excluding the circle's center.

Formula

 

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

Let R be the radius of the circle, c the chord length, s the arc length, h the height of the segment, and d the height of the triangular portion. The area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion.

The radius is ${\displaystyle R=h+d{\frac {}{}}}$ 

The arc length is ${\displaystyle s=R\theta {\frac {}{}}}$ , where ${\displaystyle \theta {\frac {}{}}}$  is in radians.

The area is ${\displaystyle A={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right)}$ 

Derivation of the area formula

The area of the circular sector is ${\displaystyle \pi R^{2}\cdot {\frac {\theta }{2\pi }}=R^{2}\left({\frac {\theta }{2}}\right)}$ 

If we bisect angle ${\displaystyle \theta }$ , and thus the triangular portion, we will get two triangles with the area ${\displaystyle {\frac {1}{2}}R\sin {\frac {\theta }{2}}R\cos {\frac {\theta }{2}}}$  or ${\displaystyle 2\cdot {\frac {1}{2}}R\sin {\frac {\theta }{2}}R\cos {\frac {\theta }{2}}}$ 

${\displaystyle =R^{2}\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}}$ 

Since the area of the segment is the area of the sector decreased by the area of the triangular portion, we have

${\displaystyle R^{2}\left({\frac {\theta }{2}}-\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}\right)}$ 

According to trigonometry, ${\displaystyle 2\sin x\cos x=\sin 2x}$ , therefore

${\displaystyle R\sin {\frac {\theta }{2}}R\cos {\frac {\theta }{2}}={\frac {R^{2}}{2}}\sin \theta }$ 

The area is therefore:

${\displaystyle R^{2}\left({\frac {\theta }{2}}-{\frac {1}{2}}\sin \theta \right)}$ 

${\displaystyle ={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right)}$ 

See also

• Circular sector

External links

• MathWorld's definition of "circular segment"
• Definition of a circular segment With interactive animation
• Formulae for area of a circular segment With interactive animation