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If you slice a pizza straight from the center, you get a wedge, but slice straight across between the crust tips, and you cut off a curved crust cap. How do we find the exact area of both shapes using pure radians?
Let's start with the full wedge first.

Area of a Circular Sector

A circular sector is a portion of a circle enclosed by two radii and an arc. Because the total area of a circle with radius rr is Ο€r2\pi r^2 for a complete 2Ο€2\pi radian turn, a sector with central angle ΞΈ\theta represents the exact fraction ΞΈ2Ο€\frac{\theta}{2\pi} of the whole area.
Multiplying this fraction by the full circle's area eliminates Ο€\pi entirely: Areasector=(ΞΈ2Ο€)(Ο€r2)=12r2ΞΈ\text{Area}_{\text{sector}} = \left(\frac{\theta}{2\pi}\right)(\pi r^2) = \frac{1}{2}r^2\theta
Let's put this into practice on a concrete problem.
Example: Find the exact area of a circular sector with radius r=6Β cmr = 6\text{ cm} and central angle ΞΈ=3Ο€4Β rad\theta = \frac{3\pi}{4}\text{ rad}.
Substitute the values directly into the formula: A=12(62)(3Ο€4)=12(36)(3Ο€4)=18(3Ο€4)=27Ο€2Β cm2A = \frac{1}{2}(6^2)\left(\frac{3\pi}{4}\right) = \frac{1}{2}(36)\left(\frac{3\pi}{4}\right) = 18\left(\frac{3\pi}{4}\right) = \frac{27\pi}{2}\text{ cm}^2
What happens if we draw a straight line between the outer tips of the sector?

Area of a Circular Segment

A chord is a straight line segment joining any two points on a circle's circumference. The region bounded between a chord and its subtended arc is called a circular segment.
To find the segment's area, subtract the central triangle formed by the two radii from the total sector area. The area of a triangle with two sides rr and included angle θ\theta is 12r2sin⁑θ\frac{1}{2}r^2\sin\theta.