Circle Chord, Arc Length, and Segment Area Calculator

Introduction to circle chords, arcs, and segment area

This circle chord and arc calculator focuses on three measurements that come from the same two points on a circle but describe different geometric ideas. A chord is the straight-line distance between the points, an arc is the curved part of the circumference between them, and a segment is the region trapped between that chord and that arc. When you know a circle's radius and the central angle that subtends those endpoints, you can compute all three measurements from one consistent setup.

That matters because real problems rarely stop at just one number. In a classroom diagram, you may need the chord to check a triangle step, the arc to understand radian measure, and the segment area to finish a geometry proof. In a design or fabrication setting, the same pair of inputs can describe the width across a curved opening, the length of trim along the edge, and the amount of material inside the curved slice. This page gathers those outputs in one place so you do not have to switch among separate formulas or recalculate the same angle conversion more than once.

The calculator is especially useful when a problem gives a radius and a central angle directly, which is one of the most common ways circle geometry is introduced. Once those values are entered, the tool returns a quick summary along with a small result table. The explanation below also connects the formulas to the underlying picture, so the calculator works as both a practical tool and a study aid for geometry and trigonometry.

How to Use the circle chord and arc calculator

This circle chord and arc calculator expects two inputs: the radius of the circle and the central angle in degrees. Enter the radius as any positive number. You may use inches, centimeters, meters, feet, or another unit, as long as you stay consistent. The calculator does not ask for the unit name, so it treats the radius as a pure number and carries that same unit into the chord length and arc length results. Segment area is then reported in the corresponding square unit.

Enter the angle in degrees in the second field. The form is designed for values greater than 0 and up to 360. In many textbook problems, the most familiar segment picture uses an angle between 0° and 180°, because that creates the minor segment most students visualize first. Still, the calculator can evaluate larger central angles too, which is useful when you want to compare a major arc or explore how the formulas behave beyond the standard classroom sketch.

After you press Calculate, the page shows a sentence-length result summary followed by a compact table listing chord length, arc length, and segment area separately. If you want to save the output for homework notes, a lab sheet, or a design memo, the copy button places the summary and table values on your clipboard. All calculations run in your browser, so the response is immediate and no data needs to be sent elsewhere.

Formula for chord length, arc length, and segment area

The circle chord and arc formulas on this page begin with radius $r$ and central angle $\theta$. Because JavaScript trigonometric functions use radians, the angle entered in degrees is converted before the main calculations are performed. Once that conversion happens, the three outputs follow from standard geometry relationships that are closely connected to triangles, sectors, and radian measure.

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 follows from the definition of radian measure. 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 comes from subtracting the triangle under the chord from the sector formed by the two radii. 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)

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

These formulas reveal an important pattern. Arc length grows directly with the angle when the radius stays fixed, which is why radians are such a natural unit for circular motion and geometry. Chord length does not grow in a perfectly linear way because it depends on the sine of half the angle. For very small angles, the chord and arc are close in value because a tiny arc looks almost straight. As the angle increases, the curved path becomes noticeably longer than the straight-line connection.

Segment area adds one more layer of intuition. A very small angle creates only a thin sliver of area between the arc and the chord. A larger angle opens that region wider, so the sector grows faster than the triangle being subtracted. That is why the segment area starts near zero and becomes much more substantial as the central angle expands.

Worked Example: a 10-unit radius with a 60° central angle

This worked example for circle chord and arc calculations uses a radius of 10 units and a central angle of 60°. Converting the angle to radians gives $\theta =60\times \frac{\pi }{180}=\frac{\pi }{3}$. The chord length then 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. The example is a good reminder that one radius-and-angle pair can produce a straight span, a curved boundary, and an enclosed area all at once.

If you compare the two lengths, you can already see the geometry story. The 60° chord is exactly 10 units, while the arc is slightly longer at about 10.47 units. That difference is small because 60° is a moderate angle, not a very large slice of the circle. If you repeated the example with 150° or 180°, the gap between chord length and arc length would become much larger.

Interpreting the Results for chord length, arc length, and segment area

The results from this circle chord and arc calculator are easiest to understand when you think about what each output measures physically. Chord length is the direct straight-line distance between the arc endpoints. Arc length is the distance along the curve itself. Segment area is the amount of space enclosed between that curved edge and the chord. If the chord length is close to the arc length, the chosen angle is relatively small and the curve has not bent far away from a straight line.

If the arc length is much larger than the chord length, the circle slice covers a more dramatic portion of the circumference. The segment area usually follows that same visual trend. Small angles create a very thin segment, while larger angles cut off a wider region. This makes the tool helpful in applied work: a builder may need the chord as the straight opening across an arch, a fabricator may need the arc as the length along a curved edge, and a designer may need the segment area to estimate material, coating, or glazing inside that curved boundary.

It also helps to interpret the numbers as a consistency check. For a fixed radius, the arc should always be at least as long as the chord for the same endpoints, with equality approached only for extremely small angles. Likewise, a larger radius with the same angle should increase both lengths proportionally, while the segment area should grow more quickly because area depends on the square of the radius. If your manual calculation does not behave that way, the issue is often a degree-versus-radian mix-up or a copied value entered in the wrong field.

Limitations and Assumptions for circle chord and arc calculations

This circle chord and arc calculator assumes ordinary plane geometry, a positive real radius, and a central angle entered in degrees. Internally, the code converts degrees to radians before using the trigonometric functions, because that is how the browser's math functions are defined. If your source material already gives $\theta$ in radians, convert that value to degrees before entering it here so the interface and the formulas stay aligned.

The segment-area formula on the page is the standard sector-minus-triangle relationship. That interpretation is most familiar for minor segments, which is why many geometry classes focus on angles from 0° to 180°. The calculator still accepts larger angles because the numerical formulas can be evaluated, but the phrase “segment area” becomes less conventional in casual classroom use when the angle describes a major arc instead of a minor one. If you want the standard textbook picture of a chord cutting off a smaller cap-shaped region, keep the angle at or below 180°.

Another practical limitation is rounding. The displayed results are rounded to four decimal places for readability, which is usually ideal for homework checks, shop estimates, and general-purpose geometry work. Highly sensitive engineering, surveying, or CAD workflows may require more precision, explicit unit tracking, and tolerance analysis beyond what a simple browser calculator is meant to provide.

Sample Values for a unit circle

The sample values below show how circle chord length and arc length behave when the radius is 1 unit. Because both formulas scale directly with the radius, you can multiply these numbers by any positive radius to estimate the corresponding values for a larger or smaller circle. Looking at the unit-circle case first makes the angle-driven pattern easier to spot.

These unit-circle values highlight a reliable trend in circle geometry. At small angles, the chord and the arc are nearly equal because a short part of a circle looks almost straight. As the angle approaches 180°, the chord approaches the diameter, while the arc approaches half the circumference. That widening gap is exactly why a straight span across a curve and the curve itself should never be treated as interchangeable measurements.

History and Applications of circle chords and arcs

The history of circle chords and arcs reaches far back into astronomy and early trigonometry. Before modern sine and cosine tables became standard, mathematicians often used chord tables to connect angles with measurable distances in a circle. Ptolemy's Almagest is one of the best-known examples. Those historical tables were not just theoretical curiosities; they supported calculations involving celestial positions, angular measurement, and the eventual development of more familiar trigonometric functions.

Today, the same circle relationships appear in much more everyday contexts. Architects and builders use chords when they need the straight width across a curved opening. Fabricators and machinists may use arc length to estimate material along an edge or to lay out curved cuts. Engineers and designers use segment areas when estimating glass, metal, concrete, paint, or other materials inside curved boundaries. Surveying, roadway design, wheel geometry, and decorative pattern work all rely on the same basic idea: a radius and an angle can describe both straight and curved dimensions of the same feature.

Connections to Trigonometry and radian measure

Circle chords and arcs provide one of the clearest bridges between geometry and trigonometry. The chord formula uses the sine of half the central angle, so a circle problem turns into a right-triangle relationship almost immediately. The arc formula shows why radians are so useful: one radian is the angle that cuts off an arc equal in length to the radius. In that sense, the equation $s=r\theta$ is more than a convenient rule. It expresses the basic meaning of radian measure itself.

The segment-area formula also previews more advanced mathematical thinking. Students often learn it as a sector-minus-triangle subtraction in geometry, but the same idea appears later in calculus, numerical modeling, and computer graphics. Whenever you compare a curve with the straight chord connecting its endpoints, you are stepping into the same geometric intuition used in approximation methods, screen rendering, and circular-motion analysis.

Further Exploration with radius and angle changes

Further exploration of circle chords becomes more intuitive when you vary one input at a time. Hold the radius fixed and change only the angle: the arc grows linearly, the chord grows more slowly according to the sine rule, and the segment area thickens from a narrow sliver into a substantial curved region. Then reverse the experiment and hold the angle fixed while changing the radius. The shape stays similar, but every linear measurement scales with the radius and every area scales with the square of the radius.

You can also use the calculator to test familiar theorems about circles. Equal chords in the same circle subtend equal central angles and equal arcs. The perpendicular bisector of any chord passes through the center. Chords that sit the same distance from the center are congruent. Those statements become easier to remember when you plug in several examples and watch how the chord and arc values move together.

If you want a more playful way to build that intuition, the optional mini-game below turns the same geometry into a timing challenge. The target angle changes, the sweep speed changes, and each round reinforces the same central idea behind the calculator: once the radius and central angle are known, the chord and the arc are determined, and their relationship becomes easier to see with repeated practice.