The Circular Segment Calculator for Arcs, Chords, and More

Welcome to the world of curves and calculations with the Circular Segment Calculator! This tool is your key to unlocking the mysteries of circular segments, allowing you to effortlessly compute arc length, chord length, height, perimeter, and area.

A circular segment is the region enclosed by a chord and the arc subtended by that chord within a circle (The yellow part in the picture above).

The arc length is the measure of the curve along the circle’s circumference formed by a central angle (b in the picture).

The chord length is the straight-line distance between the endpoints of a circular segment (s).

The height is the perpendicular distance between the chord and the center of the circle (h).

The perimeter is the total length of the arc and the chord (b + s).

The area is the space enclosed by the circular segment, including the region under the arc and above the chord (A).

Let’s embark on a journey through the step-by-step instructions for calculating various properties of a circular segment:

Identify the radius (R) and the central angle (θ).

Use the formula

Arc Length = θ/360° * 2πR

Example: For a circle with a radius of 10 units and a central angle of 60 degrees, the arc length is

Arc Length = θ/360° * 2πR = 60°/360° * 2π * 10 = 10.472 units

Identify the radius (R) and the central angle (θ).

Use the formula

Chord Length = 2R * sin(θ/2)

Example: With a radius of 12 units and a central angle of 45 degrees, the chord length is

Chord Length = 2R * sin(θ/2) = 2 * 12 * sin(45°/2) = 9.184 units

Identify the radius (R) and the central angle (θ).

Use the formula

Height = R – R * cos(θ/2)

Example: If the radius is 15 units and a central angle of 90 degrees, the height is

Height = R – R * cos(θ/2) = 15 – 15 * cos(90°/2) = 4.393 units

Identify the radius (R) and the central angle (θ).

Use the formula

Perimeter = Arc Length + Chord Length = θ/360° * 2πR + 2R * sin(θ/2)

Example: If a circle with a radius of 10 units and a central angle of 60 degrees, the perimeter is

Perimeter =  θ/360° * 2πR + 2R * sin(θ/2)

= 60°/360° * 2π * 10 + 2 * 10 * sin(60°/2)

= 20.472 units

Identify the radius (R) and the central angle (θ).

Use the formula

Area= πR² * θ/360° – 1/2 * R² * sin(θ)

Example: With a radius of 15 units and a central angle of 60 degrees, the area is

Area = πR² * θ/360° – 1/2 * R² * sin(θ)

=π * 15² * 60°/360° – 1/2 * 15² * sin(60°)

= 20.382 square units

Navigate the curves of circular segments with confidence using the Circular Segment Calculator. Calculate arcs, chords, heights, perimeters, and areas effortlessly, and embrace the elegance of circular geometry. Happy calculating!