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How do architects calculate the exact amount of stained glass needed for a Gothic cathedral window, or engineers determine the contact area of interlocking gears? Real-world shapes are rarely simple circles or rectangles β€” they are intricate composites of overlapping sectors, polygons, and circular segments.

Building Blocks: Arcs, Sectors, and Segments

A sector is the pie-shaped slice of a circle enclosed by two radii and an arc, while arc length is the curved distance along that outer boundary.
Because a full circle spans 360∘360^\circ with circumference 2Ο€r2\pi r and area Ο€r2\pi r^2, a sector with central angle ΞΈ\theta in degrees represents a fraction ΞΈ360∘\frac{\theta}{360^\circ} of the entire circle: s=ΞΈ360∘(2Ο€r),Asector=ΞΈ360∘(Ο€r2)s = \frac{\theta}{360^\circ}(2\pi r), \quad A_{\text{sector}} = \frac{\theta}{360^\circ}(\pi r^2)
A circular segment is the region bounded strictly between a chord and its intercepted arc. You find its area by subtracting the isosceles triangle formed by the radii from the full sector: Asegment=Asectorβˆ’12r2sin⁑θA_{\text{segment}} = A_{\text{sector}} - \frac{1}{2}r^2 \sin\theta.
In the 5th century BCE, Greek mathematician Hippocrates of Chios proved that crescent-shaped regions bounded by circular arcs (called lunes) could have exact areas equal to right triangles, launching the mathematical study of composite circular shapes.
What happens when multiple sectors overlap inside a polygon, creating overlapping regions and hollow voids?

Dissection Strategies: Addition vs. Subtraction

You can solve every composite planar figure using either additive dissection (partitioning the shape into non-overlapping standard regions and summing their areas) or subtractive dissection (subtracting empty negative spaces from an outer bounding shape).
Let's put this into practice with a classic challenge: two intersecting quarter-circles overlapping inside a square.

Worked Example: The Overlapping Lens