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.
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A circle segment smaller than a semicircle, enclosed by an arc and a chord
A circle segment larger than a semicircle, enclosed by an arc and a chord
Essential for geometric calculations, particularly when determining the area of a segment
The area of a minor segment can be calculated using the central angle in radians with the formula A = 1/2r^2(θ - sin(θ))
The area of a major segment can be calculated by subtracting the area of the minor segment from the area of the entire circle with the formula A = πr^2 - 1/2r^2(θ - sin(θ))
When the central angle is provided in degrees, the formulas for calculating the areas of circle segments are modified to include the conversion from degrees to radians
For a circle with a radius of 9 units and a central angle of π/3 radians for the minor segment, the area is approximately 7.64 square units
For a circle with a radius of 10 cm and a central angle of 120 degrees, the minor segment's area is approximately 75.7 square units
The sum of the minor and major segment areas should be close to the total area of the circle to confirm accuracy
The measure of the distance along the arc boundary of a circle segment
The arc length is directly proportional to the central angle and can be calculated using the formula l = rθ for radians and l = rθπ/180 for degrees
A segment with a radius of 7 cm and a central angle of 20 degrees has an arc length of 7π/9 cm, and a segment with a radius of 5 cm and a central angle of 90 degrees has an arc length of approximately 7.85 cm
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Definition of a circle segment
Region within a circle bounded by an arc and a chord.
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Area calculation of circle segments
Varies by segment type; different formulas for minor and major segments.
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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
