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

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 $R\; =\; h\; +\; d\; \backslash frac\{\}\{\}$

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

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

Derivation of the area formula

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

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

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

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

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

According to trigonometry, $2\backslash sin\; x\backslash cos\; x\; =\; \backslash sin\; 2x$, therefore

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

The area is therefore:

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

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

See also

• Circular sector

External links

• MathWorld's definition of "circular segment"

Circles

Kruhová úseč | Cirkelsegment | Kreissegment | 弓形