Chord Calculator
Calculate circular segment properties (chord length, arc length, sagitta) using known dimensions.
Note: Reverse calculations inherently compute the properties of the minor circular segment (angle ≤ 180°).
A chord calculator helps you calculate important properties of a circular segment from known circle measurements. Using values such as the radius and central angle, the calculator can determine the chord length, arc length, and sagitta.
These measurements describe different parts of the same circular segment. The chord is the straight line between two points on the circle, the arc follows the curved circumference between those points, and the sagitta measures the height of the arc above the chord.
What Is a Chord Calculator?
A chord is a straight line segment whose two endpoints lie on the circumference of a circle. Chords can have many different lengths depending on their position and the angle they subtend at the center.
This circle chord calculator uses known dimensions to calculate related circular-segment properties. For example, when the radius (r) and central angle (θ) are known, the calculator can determine:
- Chord length
- Arc length
- Sagitta
- Other related circular-segment measurements supported by the selected input mode
The diameter is a special type of chord. It passes directly through the center of the circle and is the longest possible chord.
How to Use the Chord Calculator
Using the calculator requires only a few steps:
- Select the available pair of known variables under Given Variables.
- For the Radius & Central Angle option, enter the circle’s radius.
- Enter the central angle in degrees.
- Click Calculate Properties.
- Review the calculated chord, arc, and segment measurements.
- Click Reset when you want to perform another calculation.
Keep all length measurements in the same unit system. If the radius is entered in centimeters, for example, calculated chord length, arc length, and sagitta will also be expressed in centimeters.
Understanding the Chord Calculator Results
The calculated values describe different parts of the circular segment.
| Result | What It Measures | How to Interpret It |
|---|---|---|
| Chord length | Straight distance between two points on the circle | Useful when you need the direct span |
| Arc length | Curved distance along the circumference | Useful when measuring material or travel along the curve |
| Sagitta | Perpendicular height from the chord midpoint to the arc | Describes the rise or depth of the circular segment |
Chord length and arc length should not be confused. They share the same endpoints, but the chord travels directly between them while the arc follows the curve of the circle.
How Chord Length Is Calculated
When the radius and central angle are known, chord length is calculated using:
Chord length = 2r × sin(θ ÷ 2)
where:
- r is the radius
- θ is the central angle
The calculation works because a line from the center of the circle to the midpoint of the chord divides the geometry into two equal right triangles.
For example, if a circle has a radius of 10 units and a central angle of 90°, the chord length is:
2 × 10 × sin(45°) ≈ 14.14 units
The calculator performs this trigonometric calculation automatically.
Arc length is calculated from the portion of the complete 360° circle covered by the central angle:
Arc length = 2πr × (θ ÷ 360)
Sagitta can be calculated from:
Sagitta = r × [1 − cos(θ ÷ 2)]
Together, these calculations describe the straight span, curved distance, and rise of the same circular segment.
Chord Length vs Arc Length vs Sagitta
These three measurements are related but serve different purposes.
Chord length is usually the measurement you need when determining a straight span across part of a circle.
Arc length measures the actual curved path. For any non-zero minor arc, the arc is longer than the corresponding chord because it follows the curvature instead of taking the direct route.
Sagitta, also called arc height or segment height, measures how far the arc rises above the midpoint of the chord.
For fabrication, you might use the chord for a straight opening, arc length for the amount of curved material required, and sagitta for determining how much the curve rises.
How Radius and Central Angle Affect Chord Length
Both radius and central angle directly affect the result.
If the central angle remains unchanged, increasing the radius increases the chord length proportionally. Doubling the radius doubles the chord length.
For a minor circular segment, increasing the central angle from near 0° toward 180° also increases the chord length. The two chord endpoints move farther apart around the circle.
At a central angle of 180°, the chord passes through the center and becomes the diameter:
Chord length = 2 × radius
This is the maximum possible chord length in a circle.
Angles greater than 180° require careful interpretation because the same chord can correspond to both a minor arc and a major arc. Make sure the selected segment matches the geometry you are trying to measure.
Where Circle Chord Measurements Are Used
Chord calculations appear in many practical applications beyond classroom geometry.
Common uses include:
- Designing arches and curved architectural features
- Measuring circular tanks and pipes
- Fabricating curved metal or wood components
- Checking CAD drawings
- Laying out circular construction elements
- Surveying curved boundaries
- Calculating straight spans across circular structures
- Determining arc rise for templates and curved surfaces
The correct measurement depends on the task. Use chord length for a direct span, arc length for distance along a curve, and sagitta when the height of the arc matters.
Common Chord Calculation Mistakes
Several input and interpretation mistakes can produce misleading results:
- Entering the diameter when the calculator expects the radius
- Confusing arc length with chord length
- Measuring the angle somewhere other than the center of the circle
- Mixing units such as centimeters and inches
- Assuming every chord passes through the center
- Treating sagitta as another name for radius
- Ignoring whether the intended measurement refers to the minor or major arc
Check the meaning and units of every input before calculating.
Chord Calculator FAQ
What is a chord in a circle?
A chord is a straight line segment connecting two points on a circle’s circumference. A diameter is a chord that passes through the center.
What is the difference between chord length and arc length?
Chord length measures the straight distance between two points on a circle. Arc length measures the curved distance along the circumference between the same points.
What is sagitta?
Sagitta is the perpendicular distance from the midpoint of a chord to the corresponding arc. It is also commonly called arc height or segment height.
When does a chord equal the diameter?
A chord becomes a diameter when it passes through the center of the circle. For radius-and-angle calculations, this occurs at a central angle of 180°.
Can a chord be longer than the diameter?
No. The diameter is the longest possible chord of a circle. Every other chord is shorter than the diameter.
Does chord length use the same unit as the radius?
Yes. If the radius is entered in meters, centimeters, inches, or another length unit, the resulting chord length uses the same unit.