AI Tutor
Imagine slicing a round pie from the center to the crust β€” that slice is a sector. But what if you make a straight cut connecting the two crust edges and only eat the outer sliver?

Slicing the Circle: Sectors

A sector is the region inside a circle enclosed by two radii and the arc between them, resembling a classic slice of pie.
Because a full circle covers 360∘360^\circ with an area of Ο€r2\pi r^2, a sector with a central angle ΞΈ\theta is simply a proportional fraction of the total circle area.
Let's test this formula: if a circle has radius r=6Β cmr = 6\text{ cm} and a central angle ΞΈ=60∘\theta = 60^\circ, its sector area is 60∘360βˆ˜Γ—Ο€(62)=16Γ—36Ο€=6Ο€β‰ˆ18.85Β cm2\frac{60^\circ}{360^\circ} \times \pi (6^2) = \frac{1}{6} \times 36\pi = 6\pi \approx 18.85\text{ cm}^2.
What happens if we draw a straight line between the two endpoints of that arc and cut off the pointy corner?

What is a Circular Segment?

A chord is a straight line segment joining two points on a circle. The region trapped between a chord and its arc is called a circular segment.
To find the area of the minor segment, you take the whole sector area and subtract the area of the inner isosceles triangle: Areasegment=Areasectorβˆ’Areatriangle\text{Area}_{\text{segment}} = \text{Area}_{\text{sector}} - \text{Area}_{\text{triangle}}.
How do we calculate the area of that inner triangle when we only know the radius rr and the angle ΞΈ\theta?

Finding the Inner Triangle Area