Free Segment Area Calculator

Circular Segment Area Calculator Overview

The Segment Area Calculator is a free online tool designed to compute the area of a circular segment. Beyond that, it also derives the chord length, arc length, and sagitta (segment height) from just two known variables. Whether you are dealing with minor or major segments, this calculator handles them all with ease.

Understanding Circular Segments and Chords

A circular segment is formed when a straight line (the chord) cuts across a circle, separating the area into two parts. The smaller portion is known as a minor segment, while the larger one is called a major segment. When the chord passes through the center, the segment becomes a semicircle (central angle of 180°). It's helpful not to confuse a segment with a sector, which is bounded by two radii and an arc.

In geometric terms, a chord is any line segment whose endpoints lie on the circle. Its extension beyond the circle is called a secant. The longest possible chord in a circle passes through the center and is referred to as the diameter.

Two Essential Formulas for Circular Segment Area

The area of a circular segment can be calculated through two primary formulas, depending on the known parameters.

Formula 1: Given Radius (r) and Central Angle (α in radians)

$$A_{\text{segment}} = \frac{1}{2} r^{2} \left( \alpha - \sin \alpha \right)$$

This expression is derived by subtracting the area of the isosceles triangle (formed by the two radii and the chord) from the area of the sector:

$$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2} r^{2} \alpha - \frac{1}{2} r^{2} \sin \alpha$$

Formula 2: Given Radius (r) and Segment Height or Sagitta (h)

$$A_{\text{segment}} = r^{2} \arccos\left( \frac{r-h}{r} \right) - (r-h) \sqrt{2rh - h^{2}}$$

This version is especially useful when you know the height of the liquid in a partially filled pipe or tank.

How to Use This Circle Segment Calculator

Using the tool is straightforward. Suppose you need the cross‑sectional area of a partially filled pipe:

1. Enter the circle radius. For example, set the pipe radius to 5 inches.
2. Provide a second variable. In this scenario, the liquid height is 3 inches, so input that value into the “height” (sagitta) field.
3. Read the results. The calculator instantly returns the segment area (19.8 in²), chord length (9.17 in), arc length (11.6 in), and central angle (132.84°).

This same process works if you instead start with the central angle or the chord length. The tool thus functions as a chord length calculator, an arc length calculator, and a sagitta calculator all in one, making it a versatile resource for circular segment area problems.

Application in Real‑World Tasks

The formulas and the calculator find frequent use in engineering (e.g., designing arches), plumbing (flow‑area calculations), architecture (curved structures), and geometry education. By providing the circle segment area formula in a ready‑to‑use form, this free online tool saves time and reduces manual errors.