What this chord calculator does
This tool calculates measurements for a chord of a circle. A chord is a line segment with both endpoints on the circle. Depending on what values you already know, the calculator can solve for chord length, central angle, arc length, sagitta, and the area of the circular segment.
It is useful in geometry homework, construction layout, machining, architecture, and any project where circle segments appear.
Core circle-chord formulas
1) From radius and central angle
d = r cos(θ/2)
h = r - d
arc length = rθ (θ in radians)
If you know the circle radius and angle at the center, this is the most direct route to chord length and all related values.
2) From radius and distance to chord
θ = 2 arccos(d/r)
Here, d is the perpendicular distance from the circle center to the chord. This model is common in engineering drawings.
3) From radius and sagitta
d = r - h
θ = 2 arccos((r - h)/r)
Sagitta is the height from the midpoint of the chord up to the arc. This is common in road, bridge, and arch geometry.
4) From radius and chord length
d = √(r² - (c/2)²)
h = r - d
If you measured chord length physically, this mode helps recover the angle and segment properties.
How to use the calculator (quick steps)
- Select a method from the dropdown.
- Enter the required values (radius plus one other measure).
- Optionally enter units such as cm, m, or inches.
- Click Calculate to view all results.
Practical examples
Example A: Radius + angle
Suppose radius = 10 and central angle = 60°. The calculator returns:
- Chord length ≈ 10
- Distance from center to chord ≈ 8.6603
- Sagitta ≈ 1.3397
Example B: Radius + sagitta
If radius = 15 and sagitta = 3, the chord is:
- c = 2√(2·15·3 - 3²) = 2√81 = 18
Common mistakes to avoid
- Using a central angle in degrees inside formulas that require radians.
- Entering impossible values (for example, chord length greater than diameter).
- Confusing sagitta with the distance from center to chord.
- Mixing units in one problem (e.g., radius in meters and chord in centimeters).
When a chord calculator is especially useful
- Designing curved components and supports.
- Estimating arc spans in construction and fabrication.
- Converting between angle-based and distance-based circle measurements.
- Checking hand calculations quickly during study or work.
Final note
The calculator reports geometric values for a circle segment based on valid circle relationships. For best results, keep consistent units, use precise inputs, and round only at the end of your calculation workflow.