Circular segment

From Wikipedia, the free encyclopedia

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.

1 Formula
2 Derivation of the area formula
3 See also
4 External links

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 Failed to parse (Missing texvc executable; please see math/README to configure.): R = h + d \frac{}{}

The arc length is Failed to parse (Missing texvc executable; please see math/README to configure.): s = R \theta \frac{}{} , where Failed to parse (Missing texvc executable; please see math/README to configure.): \theta \frac{}{}

is in radians.

The area is Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{R^2}{2}\left(\theta-\sin\theta\right)

Derivation of the area formula

The area of the circular sector is Failed to parse (Missing texvc executable; please see math/README to configure.): \pi R^2 \cdot \frac{\theta}{2\pi} = R^2\left(\frac{\theta}{2}\right)

If we bisect angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta , and thus the triangular portion, we will get two triangles with the area Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2} R\sin \frac{\theta}{2} R\cos \frac{\theta}{2}

or

Failed to parse (Missing texvc executable; please see math/README to configure.): 2\cdot\frac{1}{2}R\sin\frac{\theta}{2} R\cos\frac{\theta}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): = 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

Failed to parse (Missing texvc executable; please see math/README to configure.): R^2\left(\frac{\theta}{2}-\sin\frac{\theta}{2}\cos\frac{\theta}{2}\right)

According to trigonometry, Failed to parse (Missing texvc executable; please see math/README to configure.): 2\sin x\cos x = \sin 2x , therefore

Failed to parse (Missing texvc executable; please see math/README to configure.): R\sin\frac{\theta}{2}R\cos\frac{\theta}{2} = \frac{R^2}{2}\sin\theta

The area is therefore:

Failed to parse (Missing texvc executable; please see math/README to configure.): R^2\left(\frac{\theta}{2}-\frac{1}{2}\sin\theta\right)

Failed to parse (Missing texvc executable; please see math/README to configure.): = \frac{R^2}{2}\left(\theta-\sin\theta\right)