Calculate the area of a circular segment
Step 1: Determine the Radius
Measure the distance from the center of the circle to any point on the circumference.
Why: Radius scales all area calculations quadratically; accuracy here is critical.
Step 2: Measure the Central Angle
Find the angle subtended at the center by the chord bounding the segment.
Why: The angle determines the size of the sector that includes the segment.
Step 3: Convert to Radians if Needed
If angle is in degrees, convert using θ(rad) = θ(°) × π / 180.
Why: The segment formula requires radians; degrees would produce incorrect results.
Step 4: Calculate Using the Segment Formula
Apply A = ½ × r² × (θ - sin(θ)) where θ is in radians.
Why: This formula directly subtracts the triangular part from the sector, isolating the curved segment.
Step 5: Verify Reasonableness
Confirm the segment area is less than the corresponding sector area.
Why: A segment (curved region) must be smaller than its containing sector.
A circular segment is the region bounded by a chord and the arc it intercepts. Its area is calculated as:
A = ½ × r² × (θ - sin(θ))Where θ is the central angle in radians.
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