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Imagine slicing the rounded top off a loaf of artisan round bread with a single straight knife cut. That crusty piece you cut off is what mathematicians call a segment.
Unlike a sector (which looks like a wedge of pizza leading to the center), a segment is the region bounded strictly between a straight line called a chord and the curved arc connecting its endpoints.
How do we find the exact area of this isolated cap without complicated calculus?

The Area of a Segment

Because the segment is simply the wedge of the circle minus the central triangle, its area is always the sector area minus the triangle area.
For a sector with radius rr and central angle heta heta in radians, the sector area is 12r2θ\frac{1}{2}r^2\theta. The triangle formed by two radii of length rr with included angle heta heta has an area of 12absin⁑(C)=12r2sin⁑θ\frac{1}{2}ab\sin(C) = \frac{1}{2}r^2\sin\theta.
Factoring out 12r2\frac{1}{2}r^2 gives the standard segment area formula: A=12r2(ΞΈβˆ’sin⁑θ)A = \frac{1}{2}r^2(\theta - \sin\theta)
Area is only half the picture β€” what if we need to build a fence around this shape and calculate its perimeter?

The Perimeter of a Segment

The perimeter of a segment consists of two distinct boundaries: the curved arc length (ss) and the straight chord length (cc).
We know the arc length is s=rΞΈs = r\theta. To find the chord length cc, we can apply the Law of Cosines to the isosceles triangle OABOAB: c2=r2+r2βˆ’2(r)(r)cos⁑θ=2r2(1βˆ’cos⁑θ)c^2 = r^2 + r^2 - 2(r)(r)\cos\theta = 2r^2(1 - \cos\theta)