Segment Area Calculator

A circular segment is the region bounded by a chord and the arc connecting its endpoints. To calculate its area, our tool applies foundational trigonometric and geometric principles. The primary formula for the area of a circular segment based on the central angle (measured in radians) is: Segment Area = 0.5 × r^2 × (θ - sin(θ)), where r is the circle's radius and θ is the central angle. When dealing with degrees rather than radians, the angle is converted using the relation θ_radians = θ_degrees × (π / 180). Depending on your inputs, the calculator also leverages related formulas such as Arc Length = r × θ and Chord Length = 2 × r × sin(θ / 2) to solve for missing parameters before determining the final area.

Let us walk through a practical calculation. Suppose you need to find the area of a circular segment where the circle's radius is 5 cm, and the central angle is 90 degrees (which equals π/2 or approximately 1.5708 radians). First, substitute the radius and angle into the segment area formula: Area = 0.5 × (5)^2 × (1.5708 - sin(90°)). Since sin(90°) equals 1, the equation simplifies to: Area = 0.5 × 25 × (1.5708 - 1). Next, calculate the difference inside the parentheses: 1.5708 - 1 = 0.5708. Finally, multiply the components together: 12.5 × 0.5708 ≈ 7.14 square centimeters. Thus, the segment area is approximately 7.14 cm².

Always verify whether your angle input is set to degrees or radians before running calculations, as mixing them up will yield incorrect results. If you are measuring a physical object, take multiple measurements of the chord length and height (sagitta) to average out any manufacturing or measurement imperfections. Keep in mind that if your central angle exceeds 180 degrees, you are calculating a major segment; for angles under 180 degrees, you are working with a minor segment.