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Circle Segments and Their Properties

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Understanding circle segments is crucial for geometric problem-solving. This overview covers the classification of circle segments into minor and major, based on their size relative to a semicircle. It details formulas for calculating the area of these segments using central angles in both radians and degrees, and explains how to determine the arc length of a segment. Practical examples are provided to illustrate the application of these formulas in real-world scenarios.

Definitions and Classifications of Circle Segments

A circle segment is a region within a circle enclosed by an arc and a chord—the straight line joining the arc's endpoints. Circle segments are classified into two types: the minor segment, which is smaller than a semicircle, and the major segment, which is larger. This classification is essential for geometric calculations, particularly when determining the area of a segment, as the formula varies depending on the segment type.
Close-up of a wooden compass with metal hinge drawing an arc on white paper, set on a dark table with a refractive glass of water in the background.

Area Calculation of Circle Segments in Radians

The area of a circle segment can be calculated using the central angle in radians. The formula for the area of a circle, A = πr^2, where r is the radius, is the starting point for these calculations. For a minor segment with a central angle θ in radians, the area is given by A = 1/2r^2(θ - sin(θ)). To find the area of a major segment, subtract the area of the minor segment from the area of the entire circle, resulting in A = πr^2 - 1/2r^2(θ - sin(θ)).

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00

Definition of a circle segment

Region within a circle bounded by an arc and a chord.

01

Area calculation of circle segments

Varies by segment type; different formulas for minor and major segments.

02

The area of a major segment is found by subtracting the minor segment's area from the whole circle's area, leading to ______ = ______^2 - 1/2______^2(θ - sin(θ)).

A

πr

r

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