Circular Segment Area Calculator: Find the Area of a Circular Segment
ByMuhammad Ali•Founder of KruskalCode
22:04
6 min read
Understanding the area of a circular segment is a common task in geometry, whether you're working on homework or solving real-world problems. A circular segment is a specific part of a circle, defined by a chord and the arc it cuts off. It's often confused with a circular sector, but they are distinct concepts. This guide will walk you through what a circular segment is, its formula, and how to use our free online calculator to find its area quickly.
Explanation
A circular segment is the region of a circle that is bounded by a chord and the arc subtended by that chord. Imagine a full circle, and then draw a straight line (a chord) across it. The area enclosed by this chord and the curved edge (the arc) is the circular segment. It's essentially what's left if you take a 'pizza slice' (a sector) and remove the triangular part formed by the two radii and the chord. Calculating the area of a circular segment involves knowing the circle's radius and the central angle that corresponds to the arc of the segment. The key is to remember that the central angle must be expressed in radians for the standard formula to work correctly. Our calculator simplifies this by allowing you to input the angle in degrees, handling the conversion for you.
Formula
The formula to calculate the area of a circular segment is: Area = r² * (θ - sin(θ)) / 2 Where: * 'r' is the radius of the circle. * 'θ' (theta) is the central angle in radians.
Example
Let's work through an example to solidify your understanding. Suppose you have a circle with a radius of 8 meters, and a chord creates a segment with a central angle of 120 degrees. 1. **Convert the angle to radians:** θ = 120° * (π / 180°) = 2π/3 radians (approximately 2.0944 radians). 2. **Apply the formula:** Area = 8² * (2π/3 - sin(2π/3)) / 2 Area = 64 * (2.0944 - 0.8660) / 2 Area = 64 * (1.2284) / 2 Area = 39.3088 square meters This calculation shows the area of the circular segment. Our calculator performs these steps instantly, saving you time and ensuring accuracy.
How to use the related calculator
Using the ProMathTools Circular Segment Area Calculator is straightforward. Simply enter the 'Radius (r)' of your circle into the first input field. Then, input the 'Central Angle (θ in degrees)' into the second field. Make sure your radius is a positive number and your angle is between 0 and 360 degrees. Once you've entered both values, the calculator will instantly display the calculated area of the circular segment, along with the input values for easy reference.
Try the related calculator
Open toolFAQ
What is a circular segment?
A circular segment is the area of a circle that is 'cut off' from the rest of the circle by a chord. It's the region bounded by the chord and the arc it subtends. Think of it as a slice of pizza (a sector) with the triangle part removed.
How is a circular segment different from a circular sector?
A circular sector is a region bounded by two radii and the arc between them, like a slice of pizza. A circular segment, on the other hand, is the region bounded by a chord and the arc it subtends. You can find the area of a segment by subtracting the area of the triangle formed by the two radii and the chord from the area of the corresponding sector.
Why does the formula use radians for the angle?
The standard formula for the area of a circular segment, Area = r² * (θ - sin(θ)) / 2, requires the angle θ to be in radians for the 'θ - sin(θ)' part to work correctly. If you use degrees directly, the result will be incorrect. Our calculator handles the conversion for you, so you can input degrees directly.
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About the author
Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.
Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.
