Circle Chord Length

Chord, arc, and central angle are different

The chord is a straight line segment inside the circle, while the arc is the curved portion of the circumference between the same endpoints. A central angle is formed by two radii drawn from the center to those endpoints. One angle and one radius therefore determine both a chord and an arc, but the two lengths are not the same except at the zero limit. The calculator returns only the chord. It does not silently substitute the curved distance along the circle or the area swept by the radii.

The page uses the minor central angle, restricted to 0 through 180 degrees. At 0 degrees the two endpoints coincide and the chord is zero. At 180 degrees the endpoints are opposite and the chord is a diameter, equal to 2r. Between those endpoints the chord is shorter than the corresponding semicircular or smaller arc. Specifying the angle branch prevents an ambiguous supplementary or reflex angle from producing a result that belongs to a different geometric question.

• A chord is straight; an arc is curved.
• The input angle is the minor central angle.
• Zero degrees gives a coincident endpoint pair.
• 180 degrees gives the diameter.

Read the radius and angle fields

Enter the circle radius in the radius field and the included central angle in degrees in centralAngle. The radius must be positive, because a zero-radius object does not provide the intended circle scale and would make every chord collapse. The angle may be fractional, so 60.5 degrees is a valid input within the range. Negative angles, reflex angles above 180 degrees, and values outside the displayed finite limits are not accepted by the handler.

The unit attached to the radius is carried to the chord. If the radius is 5 centimetres, the result is in centimetres; if it is 5 inches, the result is in inches. The calculator has no length-unit selector and cannot convert a mixed entry. The angle field is explicitly in degrees even though the internal sine call uses radians. That distinction is displayed in the steps so a degree value is not accidentally passed to a radian formula by a downstream implementation.

• Radius is a positive length.
• centralAngle is entered in degrees.
• Fractional angles are allowed within 0 through 180.
• Length units are preserved, not converted.

Deriving the chord formula

Join the circle center to both chord endpoints. The two radii have equal length r and form an isosceles triangle whose vertex angle is theta. Bisect that triangle through the center and the midpoint of the chord. Each half is a right triangle with hypotenuse r, angle theta/2 at the center, and opposite side c/2. The sine ratio gives sin(theta/2) = (c/2)/r. Rearranging produces c = 2r sin(theta/2).

The trigonometric function expects theta in radians, so the entered degree value is multiplied by pi/180 before half-angle evaluation. Keeping that conversion explicit is essential: sin(30) in a programming language usually means 30 radians, not 30 degrees. The engine handles the two exact endpoints directly, returning zero at zero degrees and 2r at 180 degrees. Interior angles use the same formula and receive a finite-result check before being returned.

• The radii and chord form an isosceles triangle.
• Bisecting gives a right triangle with opposite side c/2.
• Use sine of half the central angle.
• Convert degrees to radians before calling sine.

Worked example with a sixty-degree angle

Take radius r = 5 length units and centralAngle = 60 degrees. Convert the angle: theta = 60 pi/180 = pi/3 radians. Half the angle is pi/6, whose sine is 1/2. Substitution gives c = 2 x 5 x 1/2 = 5 length units. The result is equal to the radius in this particular example because a 60-degree central angle creates an equilateral triangle from the two radii and the chord.

The same example can be checked geometrically without relying on the final decimal. A radius from the center to each endpoint and the chord make three equal sides when the included angle is 60 degrees. If the radius is changed to 10 while the angle stays 60 degrees, the chord becomes 10. If the angle is changed while the radius stays 5, the chord changes in proportion to the sine of half the new angle. Those checks test both scale and angle handling.

• r = 5 and theta = 60 degrees.
• theta becomes pi/3 radians.
• sin(theta/2) is 1/2.
• The chord is 5 length units.

Endpoint and monotonic behavior

At centralAngle = 0, the formula's limiting value is zero and the handler returns exactly zero. At centralAngle = 180, the sine of 90 degrees is one, so the chord is exactly the diameter 2r. For a fixed positive radius, the chord increases as the minor angle increases from zero to 180 degrees. For a fixed angle, doubling the radius doubles the chord. These relationships are useful reasonableness checks for a result and do not require a diagram.

The angle range excludes a reflex central angle because a reflex angle would identify the major arc and can make the phrase chord length ambiguous without an endpoint convention. The straight chord associated with a pair of endpoints is still the same segment, but the intended intercepted arc and segment formula would need an explicit branch. Keeping this page on the minor branch allows the result and its related segment-area page to use one clear domain.

• Chord length rises with angle on the minor branch.
• Chord length scales linearly with radius.
• Zero angle returns zero exactly.
• A 180-degree angle returns the diameter exactly.

Numerical validation and safe bounds

The catalog bounds the radius from 0.000001 through 1,000,000 length units and the angle from 0 through 180 degrees. The pure handler repeats those checks and rejects non-number values, NaN, infinities, and out-of-range entries. A radius just below the lower boundary is not silently raised to the boundary, because doing so would make the displayed input and the calculated geometry disagree. The same principle applies to an angle just above 180 degrees.

The degree-to-radian conversion and the chord result both receive finite checks. The numeric result is returned as a renderer-safe number with an explicit same-length-units label and a presentation precision. The handler does not round the chord before returning it. At very small angles, the true chord can be small and a display rounded to a few places may show zero; retain the underlying numeric value when small differences matter. Display precision is not a claim about measurement precision.

• Radius and angle bounds are inclusive.
• Invalid finite shape and nonfinite values are rejected.
• The result is checked before rendering.
• Small displayed values may need unrounded downstream use.

Chord versus arc length

For the same radius and minor angle, arc length is r theta in radians, while chord length is 2r sin(theta/2). The arc follows the circumference and the chord cuts across the interior. For positive angles below 180 degrees, the arc is longer than the chord. Near zero the difference becomes small, which can make the quantities look interchangeable in a rough drawing, but they answer different questions. Use the chord formula when the straight segment itself is the requested dimension.

A sector-area formula also uses the central angle but includes a squared radius and an angular area term. A circular-segment area combines a sector with a triangle and is not obtained by multiplying the chord by an arbitrary height. The calculator's title and result label deliberately say chord length so a downstream reader can distinguish the output from the arc and area pages. If a drawing supplies a curved boundary, identify which boundary is being measured before entering the angle.

• Arc length uses r theta.
• Chord length uses 2r sin(theta/2).
• The arc is longer than the chord on the interior minor branch.
• Area formulas need additional terms and outputs.

Common mistakes in circle problems

A frequent error is using the full central angle rather than half of it inside sine. Another is entering a diameter in the radius field, which doubles the scale and doubles the answer. Degree-radian confusion is especially common when a hand calculation and a programming library are compared. Check the step that displays the converted angle. Also make sure the stated angle is central rather than an inscribed angle; an inscribed angle subtending the same arc is half the central angle and must be converted before using this page.

Do not use the chord formula for a straight distance from the center to an endpoint; that distance is the radius. Do not use it for the length around the rim; that is an arc. If the endpoints are known but the angle is not, a separate distance or inverse-trigonometric setup may be needed before this calculator can be used. The page requires the radius and central angle directly and does not reconstruct hidden geometry from a diagram or external coordinates.

• Use half the central angle inside sine.
• Enter radius, not diameter.
• Convert inscribed angles before using the central-angle formula.
• Distinguish chord, radius, and arc measurements.

Interpretation, use cases, and limits

The result is useful for circle diagrams, circular segment dimensions, cable or panel layouts, geometric construction, and software tests of a known trigonometric relation. It can be passed to a segment-area calculation or compared with a diameter bound. In each use, preserve the radius unit and the minor-angle convention. The calculator does not infer a physical material, thickness, tension, tolerance, or manufacturing process from a chord value.

For surveying, structural fabrication, navigation, or safety-critical design, validate the geometry, angle measurement, coordinate frame, and tolerance with domain-specific tools. A theoretical chord assumes an ideal circle and exact endpoint placement. Real edges may be elliptical, rough, offset, or measured with uncertainty. The page is an auditable formula check, not a guarantee that a physical part fits or that a path follows the circular boundary.

• Use the output for an ideal circular chord model.
• Retain the minor-angle convention with the value.
• Physical dimensions need measurement and tolerance review.
• Do not interpret a chord as a path along the circumference.

Reproducible review of a chord result

To reproduce the result, record r, the central angle in degrees, the converted radian value, and the expression 2r sin(theta/2). Check the two endpoints as limits: angle zero must give zero and angle 180 must give 2r. For an interior value, compare the result with both zero and the diameter; it should lie between them. If the radius is positive and the angle is on the minor branch, a negative chord indicates a sign or unit error rather than a geometric result.

When a result is handed to another page, state whether it will be used as a straight segment, a triangle side, or an input to a segment-area model. Preserve the unrounded number for later arithmetic and keep the visible unit attached. The calculator intentionally stops at one length. It does not choose an arc branch, calculate a sector, or decide whether an engineered or measured circle satisfies a tolerance. Those are separate, reviewable questions.

• Record radius, degree angle, radians, and formula.
• Check zero and diameter endpoint behavior.
• Confirm the result lies between zero and 2r.
• State the downstream geometric meaning explicitly.

The midpoint and sag connection

The perpendicular from the circle center to a chord bisects the chord. If the chord has length c and radius r, its half-length is c/2, and the distance from the center to the chord follows a right triangle. That construction gives useful geometric context for the chord formula: a small central angle produces a short chord near the circle's rim, while a 180-degree angle places the chord through the center. The calculator does not return the midpoint distance or sag, but those quantities can be derived separately when needed.

A circular segment page may use the same radius and central angle to calculate the cap area. The chord is one boundary of that cap, so the two pages are related without being duplicates. If a problem supplies a chord and a perpendicular height instead of an angle, use the right-triangle relationships to derive the missing angle or radius with an explicit algebraic step. Do not add an unrequested sag input to this handler, because it would create a different contract and different validation rules.

• The center-to-chord perpendicular bisects the chord.
• Chord half-length is c/2 in the central triangle.
• Sag and midpoint distance are related but not returned.
• Segment area can reuse the same radius and angle separately.

Sensitivity to radius and angle

The chord is linear in radius, so a one-percent change in radius produces a one-percent change in chord at a fixed angle. Its response to angle is trigonometric: differentiating with respect to the radian angle gives r cos(theta/2). Near zero, a small angle change changes the chord at approximately r times the angle change. Near 180 degrees, the local slope approaches zero, so the chord is close to its diameter and changes more slowly as the angle increases.

These sensitivity statements describe the ideal formula, not the uncertainty of a measuring instrument. If radius or angle has a tolerance, propagate that tolerance with a separate analysis and preserve the angle unit. A degree uncertainty must be converted before it is combined with a radian derivative. The calculator returns a nominal chord from nominal inputs and does not claim a confidence interval or manufacturing tolerance.

• Chord scales linearly with radius.
• Angle sensitivity is highest near the small-angle end.
• Sensitivity decreases near the diameter limit.
• Input uncertainty needs separate propagation.

A clear handoff for circle geometry

When passing a chord result to another person or tool, include the radius, central angle in degrees, converted radians, and whether the chord belongs to the minor branch. Also state whether the number is a straight segment or an input to an area construction. This small amount of context prevents a recipient from interpreting the value as an arc or a diameter. The unrounded numeric value is preferable when the chord will be used in later trigonometric arithmetic.

For a final review, test a 60-degree example, a zero-angle endpoint, and a 180-degree endpoint. Exchange a degree input for its radian equivalent only after converting deliberately, and verify that the resulting chord agrees. If a result is outside zero through 2r for a valid positive radius and minor angle, investigate the input convention or implementation. The page's narrow output makes that audit straightforward.

When a chord is used in a larger drawing, mark the two endpoints and the circle center before assigning the number to an edge. This makes clear which radius and central angle generated it. Preserve any later rotation or translation as a geometric transformation that does not alter the length, and preserve any scale conversion as a separate unit operation. A reviewer can then distinguish a changed picture from a changed chord contract.

• Attach radius, angle unit, branch, and output meaning.
• Keep the unrounded chord for downstream formulas.
• Test 60, zero, and 180-degree cases.
• A valid chord lies from zero through the diameter.

Choosing the right chord interpretation

In a construction, the two endpoints may be specified by a central angle, an arc, a sag, or coordinates. This page is designed for the first case: radius plus included central angle. If the supplied angle is measured at the circumference, identify it as an inscribed angle and double it when it subtends the same arc before using the central-angle formula. If the supplied length is a diameter, halve it for the radius. Every conversion should be visible in the record.

A chord can be a side of an isosceles triangle, a straight cut across a circular panel, or a geometric reference in a diagram. The same numeric length can acquire a different practical meaning depending on the surrounding construction. Keep the ideal circle assumption, endpoint pair, and unit attached. The calculator does not know whether the chord is an edge, a brace, a screen distance, or an abstract segment.

When comparing two chords, hold the intended variable fixed. Equal radii with larger minor angles give longer chords, while equal angles with larger radii scale the chord directly. Chords from different circles cannot be compared by angle alone because radius supplies the length scale. Chords from the same circle cannot be compared by radius because that radius is fixed; their angle or endpoint placement controls the difference.

A robust implementation test should include a right-angle central example, a 60-degree exact example, and the diameter endpoint. A 90-degree central angle gives c = r sqrt(2), while a 60-degree angle gives c = r. These identities are independent of the unit size and expose errors in half-angle handling. The current engine returns finite values for each valid test.

If a later workflow needs a tolerance around the chord, calculate the nominal value here and attach tolerance analysis separately. The formula is sensitive to the radius and angle in different ways, and degree rounding can matter near a specified fit. A readable formatted result is useful for a person, but the unrounded number and input precision should be retained for a technical comparison.

• Convert inscribed angle or diameter inputs explicitly.
• Keep endpoint, circle, unit, and ideal-model context.
• Compare chords with radius and angle variables identified.
• Use 60, 90, and 180-degree checks for implementation review.
• Perform tolerance analysis outside the nominal formula.