Chord Length Calculator - Radius, Angle, Sagitta

The chord length calculator finds the straight-line distance between two points on a circle when you know the radius and the central angle that the chord subtends. Use it for geometry homework, structural layout work, lens and mirror checks, or any drawing where you need the chord, the arc, and the sagitta in the same step.

• • Geometry problems: Solve textbook questions when the central angle and radius are given.
• • Layout and cutting: Translate a measured arc into a straight chord, a sagitta, and a segment area.
• • Lens and dish design: Check the chord and depth of a circular dish or curved panel from one radius and angle.
• • Quick cross-checks: Run a textbook theta with the same radius to confirm the chord, sagitta, arc length, and segment area match the formula.

The result is a length, not an angle. Keep the radius and the chord outputs in the same unit. If you enter the radius in inches, the chord, sagitta, arc, and area come out in inches and square inches. The angle input accepts degrees or radians, so you do not need to convert the angle yourself.

When the same circle also needs area, circumference, or diameter, the Circle Calculator gives those values from the same radius input.

The calculator applies the standard circle formula that links a chord, the radius, and the central angle. It then derives the sagitta, arc length, and circular segment area from the same angle.

• r: radius of the circle
• theta: central angle in radians (convert degrees to radians first if needed)
• chord length c: straight-line distance between the two points on the circle
• sagitta h: r * (1 - cos(theta / 2)), the depth of the segment from chord to arc
• arc length s: r * theta, the curved distance between the same two points

The chord formula c = 2r sin(theta/2) comes from dropping a perpendicular from the center to the chord. That perpendicular splits the chord in half and splits the central angle into two equal angles, so the right triangle has hypotenuse r, opposite side c/2, and half-angle theta/2. The sagitta is the remaining piece of the radius after that perpendicular, which is r cos(theta/2). The arc length uses the same theta in radians, and the segment area is the sector area minus the area of the isosceles triangle formed by the two radii and the chord.

When you switch the angle unit from degrees to radians, the chord length calculator does the conversion in the same step, so the result is identical for theta = 60 degrees and theta = pi/3 radians.

According to Wolfram MathWorld, a chord of a circle of radius r subtended by a central angle theta has length 2r sin(theta/2).

The arc length row here uses the same r and theta as the standalone Arc Length Calculator, so cross-checking between the two is direct.

These four terms describe the same pair of points on a circle, and they each have a different role in geometry problems and physical layouts.

The sagitta shows how far the arc bows away from the chord. A 60 degree central angle has sagitta r(1 - sqrt(3)/2), about 0.134r. A 120 degree angle has sagitta r/2 = 0.5r. A 180 degree angle has sagitta r, the full radius.

If the segment is part of a larger regular polygon instead of a circle, the Polygon Area Calculator helps with the polygon-area version of the same problem.

Use the chord length calculator with a clear radius, a single central angle, and the matching unit so every result stays in one consistent system.

1. 1 Enter the radius: Use the distance from the center of the circle to any point on the circle. Pick a unit and stay with it for the chord, sagitta, and arc outputs.
2. 2 Enter the central angle: Type the angle at the center between the two radii that touch the chord endpoints. Use 0 to 360 for degrees or 0 to 2*pi for radians.
3. 3 Choose the angle unit: Match the unit to the value you just typed. Switching the unit does not change the angle, only how the calculator reads it.
4. 4 Read the chord length: Use this output as the straight-line distance between the two points on the circle for layout, cutting, or a geometry answer.
5. 5 Check the sagitta and arc length: Use the sagitta as the depth of the curved piece and the arc length as the curved distance between the same two points.
6. 6 Use the segment area when needed: Use the segment area for area estimates of a cutout, a dish, or a curved panel that matches the same radius and central angle.

Suppose you are cutting a 60 degree wedge from a 10 inch circle. The chord length is 10.00 inches, which is the straight edge of the wedge. The sagitta is 1.34 inches, the arc length is 10.47 inches, and the segment area is 9.06 square inches. Use the chord value first, then add any waste or overlap.

For the right triangle that drops from the center to the chord midpoint, the Right Triangle Calculator gives the hypotenuse and side lengths when you only know an angle.

Reading all four results in one step turns a single radius and angle into a complete picture of a circular segment.

• • Four results from two inputs: Get chord length, sagitta, arc length, and segment area at the same time.
• • Degrees or radians: Use the angle unit you already have. The calculator does the radian conversion internally.
• • Layout and cutting support: Pair the chord with the sagitta to lay out a curved cutout, a dish, or a panel that needs a flat back.
• • Decimal and unit friendly: Decimals, fractions, and large or small radii all work for both classroom problems and measured drawings.
• • Single-formula audit trail: Every result is derived from the same chord formula and is easy to check against a textbook.

Because the calculator returns the full set, it also works as a formula checker. Enter the angle and radius, then read off the sagitta or segment area to see if it matches the expected value. The output rows also help pick the number for the next step, whether the chord for a cutting list, the sagitta for a drawing, or the segment area for a material estimate.

If the angle is given in turns, gradians, or another non-degree unit, the Angle Converter can convert it to degrees or radians before entering it here.

The chord formula is compact, but the choice of input unit and the way you measure the angle and the radius can change which result you trust most.

• • The calculator assumes a flat 2D circle. For a sphere, the chord is the same as the 2D chord, but segment area and arc length are different curves on the surface and need spherical geometry formulas.
• • The segment area output is the segment on the side of the chord the central angle sweeps. For central angles up to 180 degrees (pi radians) it is the minor segment, and for larger angles it is the major segment. To get the opposite piece, subtract the returned area from the full circle area pi r^2.
• • Rounded output can differ by a few hundredths from a hand calculation that rounds after every intermediate step. Keep full precision in the input and read the rounded result with the same tolerance.

The chord formula is exact for an ideal circle, so treat the result as a working value rather than a certified measurement. The calculator is for the open interval (0, 2*pi) radians or (0, 360) degrees; values at 0 or 360 degenerate to a point and have no meaningful chord.

For classroom work, the most common cause of a wrong answer is the angle unit. A 60 degree angle and a 60 radian angle look similar in a worksheet, but the radian version wraps more than nine full circles and the chord formula will return a meaningless value or fail the upper-bound check. Convert to degrees or radians first if the input is in turns, gradians, or other unusual units.

According to Wolfram MathWorld, a circular segment's height (sagitta) is r(1 - cos(theta/2)) and its area is (1/2)r^2(theta - sin theta) with theta in radians.

According to Wolfram MathWorld, trigonometric formulas such as sin(theta) are direct only when theta is in radians.

For the right triangle at the heart of the chord formula, the Pythagorean Triples Calculator can verify whether a given radius, half-chord, and perpendicular form a Pythagorean triple.