Circle Sector Calculator

What is a Circle Sector?

A circle sector is the region of a circle bounded by two radii and the arc between them. Often described as a "pie slice," a sector's shape is defined by its radius and central angle. The larger the central angle, the bigger the sector. A full circle is a sector with a central angle of 360 degrees, while a semicircle is a sector with a central angle of 180 degrees. Sectors with a central angle less than 180 degrees are called minor sectors, and those greater than 180 degrees are called major sectors.

The sector area formula is A = (θ/360) × πr² when the angle is measured in degrees, or A = ½r²θ when measured in radians. This formula derives from the proportional relationship between the sector and the full circle. Since the sector is a fraction of the entire circle, its area is the same fraction of the circle's area. The key measurements that describe a sector include the radius, central angle, arc length (the curved outer edge), chord length (the straight line between the arc endpoints), and sector perimeter (the total boundary length).

Circle sector calculations appear frequently in real-world applications. Engineers use them to design gears with sector-shaped teeth. Architects calculate sector areas for curved windows, fan-shaped layouts, and circular amphitheaters. Bakers and chefs use sector measurements for slicing pies, cakes, and pizzas into equal portions. Urban planners apply sector geometry to design roundabouts, traffic circles, and radial road networks.

How to Use the Circle Sector Calculator

This calculator solves for any of the three primary variables in the sector equation. Select which variable you want to calculate, enter the known values, and all related sector properties are computed instantly.

• Solve for Sector Area: Enter the radius and central angle (in degrees or radians) to find the sector area, along with the arc length, chord length, and perimeter.
• Solve for Radius: Enter the sector area and central angle to determine the circle's radius and all other sector properties.
• Solve for Central Angle: Enter the radius and sector area to find the central angle in both degrees and radians, along with arc length, chord length, and perimeter.

Sector Area Formula Derivation

The sector area formula is derived from the proportional relationship between the sector and the full circle. A full circle has an area of πr² and a central angle of 2π radians (360 degrees). Since the sector's area is proportional to its central angle, the sector area A equals the full circle area multiplied by the ratio of the sector angle to the full circle angle: A = πr² × (θ/360) in degrees, or A = πr² × (θ/2π) = ½r²θ in radians.

The arc length of a sector follows the same proportional relationship. The full circumference is 2πr, so the arc length s = 2πr × (θ/360) = (πrθ)/180 in degrees, or simply s = rθ in radians. The chord length, which is the straight-line distance between the endpoints of the arc, is calculated using c = 2r × sin(θ/2). The chord is always shorter than the arc for any angle greater than zero.