Segment Area Calculator

A circular segment is the region between a chord and the arc it cuts off.

How the Math Works

The Segment Area Calculator uses the formula Area = ½r²(θ - sin θ) to find the area of a circular segment. Here, r is the radius of the circle and θ is the central angle in radians. This formula subtracts the area of triangle formed by the two radii and the chord from the area of the circular sector, leaving only the segment area between the chord and the arc. The sine function accounts for the triangular portion that gets removed, making this a precise way to calculate curved segment areas.

Practical Applications

To use this calculator, simply enter the circle's radius and the central angle that defines your segment. The angle must be in radians - if you have degrees, convert them first by multiplying by π/180. For example, if you need the area of a circular arch with radius 10 feet and central angle 45 degrees (π/4 radians), plug these values in to get the exact segment area. This is particularly useful in engineering, architecture, and manufacturing when working with curved surfaces or partial circular shapes.

Day-to-Day Use

This calculation appears in everyday situations like determining how much material you need for curved architectural features, calculating the area of circular garden beds with arc-shaped boundaries, or figuring out the surface area of partially filled circular containers. Whether you're installing a circular window with curved trim, designing a running track with curved sections, or even estimating the area of a pizza slice with a specific angle, understanding segment areas helps you make accurate measurements and material estimates for practical projects.

Worked example

Radius 5, angle 90° → ½·25·(π/2 − 1) ≈ 7.13.

FAQ