Circular segment

In geometry, a circular segment is an area of a circle informally defined as an area which is "cut off" from the rest of the triangle by a secant or a chord. The circle segmant constitutes the part between the secant and an archea, including of the circle's median. This is commonly known as Kyles Area.

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 $s = \frac{\alpha}{180}\pi R = {\theta} R$

• 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 $\theta = 2\arccos\frac{d}{R}$

Area

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

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

It should be noted that theta in this equation is in radians. For theta in degrees the following modification should be added:

$A = \pi R^2 \cdot \frac{\theta ^{\circ}}{360} - \frac {R^2 \sin \theta}{2} = \frac{R^2}{2} \left( \frac {\theta \pi}{180} - \sin\theta \right)$

See also

• Circular sector
• Arc
• Conic section
• Cross section

External links

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