User:Eirik1231/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. The circle segment constitutes the part between the secant and an arc, excluding of the circle's center.

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 radians, α is the central angle in degrees, 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 $s={\frac {\alpha }{360}}\tau R={\theta }R$

• The chord length is $c=2R\sin {\frac {\theta }{2}}=R{\sqrt {2-2\cos \theta }}$

• The height is $h=R(1-\cos {\frac {\theta }{2}})=R-{\sqrt {R^{2}-{\frac {c^{2}}{4}}}}$

• 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.

$A={1 \over 2}\tau R^{2}\cdot {\frac {\theta }{\tau }}-{\frac {R^{2}\sin \theta }{2}}={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right)={\frac {R^{2}}{2}}\left({\frac {\alpha \tau }{360}}-\sin {\frac {\alpha \tau }{360}}\right)$

See also

• Circular sector
• Spherical segment
• Arc
• Conic section
• Cross section

External links

• Weisstein, Eric W. "Circular segment". MathWorld.
• Definition of a circular segment With interactive animation
• Formulae for area of a circular segment With interactive animation
• Circular segment calculator

Category:Circles