Circular segment

In geometry, a circle segment (also circular 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

 

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. From this, we know that the radius is

${\displaystyle R=s+d}$ ,

the arc length is

${\displaystyle s=R\theta }$ 

The area of the circular sector is

${\displaystyle \pi R^{2}\cdot {\frac {\theta }{2\pi }}}$ 

or

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

If we bisect angle ${\displaystyle \theta }$ , and thus the triangular portion, we will get

 

2 triangles with the area

${\displaystyle {\frac {1}{2}}\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}}$ 

or

${\displaystyle 2\cdot {\frac {1}{2}}\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}}$ 

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

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

${\displaystyle \sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}={\frac {1}{2}}\sin \theta }$ 

The area is therefore:

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

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

Related topics

• Circular sector

External links

• MathWorld's definition of "circular segment"