Circle Chord and Arc Calculator

Introduction

A circle is one of the most familiar shapes in geometry, but it contains several different measurements that describe very different ideas. A chord is the straight-line distance between two points on the circle. An arc is the curved portion of the circumference between those same points. When that chord cuts off part of the circle, the enclosed region between the chord and the arc is called a segment. This calculator brings those ideas together by taking a radius and a central angle, then computing the corresponding chord length, arc length, and segment area.

That combination is useful because many real design and classroom problems start with the radius of a circle and the angle at its center. If you know those two values, you can estimate the straight span of a curved opening, the length of the curved edge itself, and the area trapped between them. Those measurements show up in window design, arch layouts, wheel parts, circular tracks, surveying, machining, and trigonometry lessons. Instead of switching between several formulas by hand, you can enter the inputs once and compare all three outputs side by side.

How to Use

Enter the circle's radius in the first field and the central angle in degrees in the second field. The radius can be in any positive unit such as inches, centimeters, meters, or feet. The calculator keeps that unit consistent: if the radius is entered in centimeters, then the chord length and arc length are also reported in centimeters, while the segment area is reported in square centimeters. The angle field is set up for values greater than 0 and up to 360 degrees.

After you submit the form, the page shows a plain-language result summary and a compact table with the three computed values. The copy button lets you place the summary on your clipboard for notes, homework, or design reports. The calculations happen in your browser, so the tool responds immediately and does not require sending data anywhere else. For the most typical geometry problems involving a single visible segment, users often work with angles from 0° to 180°, but the calculator will still evaluate larger central angles as entered.

Formula

The calculator uses the radius $r$ and the central angle $\theta$. Because JavaScript trigonometric functions use radians, the entered angle in degrees is first converted to radians. From there, the three outputs follow from standard circle geometry.

The chord length comes from the isosceles triangle formed by two radii and the chord. If that triangle is split down the middle, each half becomes a right triangle with hypotenuse $r$ and half-chord $\frac{c}{2}$. The angle at the center becomes $\frac{\theta }{2}$. Using sine gives the half-chord, so the full chord length is:

Formula: c = 2 r sin θ / 2

$c=2r\sin \frac{\theta }{2}$

The arc length is more direct. A full circle measures $2\pi$ radians and has circumference $2\pi r$, so an angle of $\theta$ radians cuts off an arc length of:

Formula: s = r θ

$s=r\theta$

The segment area is found by subtracting the area of the triangle from the area of the sector. The sector area is $\frac{1}{2}rr\theta$ and the triangle area is $\frac{1}{2}rr\sin \theta$. That leads to the segment formula:

Formula: A = 1 / 2 r^2(θ - sin θ)

$A=\frac{1}{2}{r}^2(\theta -\sin \theta )$

Two quick interpretations help these formulas feel intuitive. First, arc length grows directly with the angle when the radius is fixed, which is why arcs and radians are so closely connected. Second, the chord does not grow linearly in the same way because it depends on the sine of half the angle. For small angles, the arc and the chord are nearly the same length. As the angle gets larger, the curved path pulls away from the straight-line shortcut.

Worked Example

Suppose a circle has radius 10 units and central angle 60°. Converting the angle to radians gives $\theta =60\times \frac{\pi }{180}=\frac{\pi }{3}$. The chord length becomes $2\times 10\times \sin \frac{\pi }{6}=10$ units. The arc length is $10\times \frac{\pi }{3}\approx 10.47$ units.

For the segment area, the sector area equals $\frac{1}{2}10\times 10\times \frac{\pi }{3}\approx 52.36$, while the triangle area is $\frac{1}{2}10\times 10\times \sin \frac{\pi }{3}=43.30$. Subtracting those values gives a segment area of about $9.06$ square units. This example shows why the result set is useful: the same input pair describes a straight span, a curved span, and an enclosed curved area.

Interpreting the Results

If the chord length is close to the arc length, your angle is relatively small and the curved edge is not bending very far away from a straight line. If the chord is much shorter than the arc, the arc covers a larger portion of the circle. The segment area tends to stay small for tiny angles because the chord sits close to the circumference, leaving only a thin sliver of area between the straight line and the arc. As the angle increases, that segment region becomes more substantial.

This is especially helpful in applied settings. A builder may care about the chord because it gives the straight opening between two endpoints. A fabricator may care about the arc because it describes the length of material needed along a curved edge. A designer or engineer may care about segment area when estimating paint, glass, metal, or concrete within a curved boundary.

Limitations and Assumptions

This tool assumes standard plane geometry and treats the radius as a positive real number. It also assumes the central angle is entered in degrees, even though the formulas are evaluated in radians internally. If you are working from a diagram that already labels the angle in radians, convert it to degrees before typing it into the form or adjust it mentally so the input matches the interface.

The most common textbook meaning of a circular segment refers to the region cut off by a chord and the corresponding minor arc. For that reason, many lessons focus on angles from 0° to 180°. The calculator accepts larger angles because the underlying formulas can still be evaluated, but the geometric interpretation becomes less conventional. When you want the familiar minor-segment picture, keep the angle at or below 180°.

Finally, the displayed results are rounded to four decimal places for readability. That is ideal for most homework checks and quick design estimates, but highly sensitive engineering work may require carrying more digits, tracking measurement tolerances, or using a CAD workflow that preserves full precision throughout a project.

Sample Values

The table below summarizes chord length and arc length for several common angles when the radius is 1 unit. Because both formulas scale with the radius, you can multiply the listed values by any radius to estimate results for a larger or smaller circle.

These sample values reveal an important pattern. At small angles, the chord and arc are nearly equal because a very short piece of a circle looks almost straight. As the angle approaches 180°, the chord approaches the diameter while the arc approaches half the circumference. That contrast is exactly why chord length and arc length should not be treated as interchangeable.

History and Applications

The study of chords and arcs goes back to ancient astronomy. Before sine and cosine tables were standardized in their modern form, mathematicians often used chord tables to connect angles with measurable distances in a circle. Ptolemy's Almagest is a famous historical example. Those early tables helped scholars estimate the positions of stars and planets and eventually influenced the development of trigonometry itself.

Modern applications are just as practical. In civil engineering, chord length and segment area matter when designing arches, domes, and curved retaining features. In manufacturing, a chord can describe the straight width across a circular part while the arc captures the length along its edge. Surveyors encounter arc lengths on curved boundaries, and designers use segment areas when estimating materials for curved panels, windows, or decorative cutouts. Even everyday examples, such as a pizza slice edge or a curved garden border, can be understood with the same geometry.

Connections to Trigonometry

Circle geometry is one of the clearest places to see trigonometry in action. The chord formula uses the sine of half the central angle, which ties a circle problem directly to right-triangle reasoning. The arc formula, meanwhile, shows why radians are natural: one radian is defined as the angle that cuts off an arc equal in length to the radius. In that sense, the relationship $s=r\theta$ is not just a formula to memorize; it is almost the definition of radian measure.

The segment area formula also points ahead to more advanced work. Students often derive it by subtracting triangle area from sector area, and later they may revisit the same geometry in calculus, numerical modeling, and computer graphics. Understanding how a curve differs from its straight-line chord is useful when approximating circles on a screen, analyzing curved motion, or interpreting graphs built from periodic functions.

Further Exploration

Once you are comfortable with these outputs, there are several natural extensions. Equal chords in the same circle subtend equal arcs and equal central angles. The perpendicular bisector of any chord passes through the center. Chords equidistant from the center are congruent. Each of those statements can be explored visually with a dynamic geometry sketch and then checked numerically with this calculator.

You can also study how changing the radius rescales everything. If the angle stays fixed and the radius doubles, the chord length and arc length double, while the segment area grows by a factor of four. That scaling behavior is a useful reminder that lengths depend linearly on size, but areas depend on the square of size. Seeing all three outputs together makes that contrast much easier to remember.

For a more intuitive feel, try entering several angles with the same radius and watch how the outputs move. Then keep the angle fixed and change only the radius. That simple experiment builds strong geometric intuition: the angle controls the shape of the slice, while the radius controls the overall scale of the entire figure. The optional mini-game below turns that idea into a quick timing challenge so you can practice matching central angles and see how they drive chord and arc measurements in real time.