Calculator
Example Data Table
| Case | Known Values | Expected Main Output |
|---|---|---|
| Radius and angle | Radius 10, angle 60 degrees | Chord 10, arc 10.472 |
| Radius and chord | Radius 12, chord 14 | Angle 71.166 degrees |
| Chord and sagitta | Chord 16, sagitta 4 | Radius 10 |
| Sector and angle | Area 78.54, angle 90 degrees | Radius about 10 |
Formula Used
The calculator uses standard circle relationships. The central angle is shown as theta.
- Arc length: s = r × theta
- Chord length: c = 2r × sin(theta / 2)
- Sagitta: h = r × (1 - cos(theta / 2))
- Sector area: A = 0.5 × r² × theta
- Segment area: A = 0.5 × r² × (theta - sin(theta))
- Radius from chord and sagitta: r = c² / (8h) + h / 2
- Central angle from radius and chord: theta = 2 × asin(c / 2r)
Radians are used inside the calculations. Degree input is converted before solving.
How to Use This Calculator
- Select the pair of values you already know.
- Enter only the fields needed for that selected case.
- Choose degrees or radians when using an angle.
- Set decimal places for the final report.
- Press Calculate to show results below the header.
- Use CSV or PDF buttons to export the same report.
Chords and Arcs Guide
Why Chords and Arcs Matter
A circle looks simple, yet its parts carry useful measurements. Chords connect two points on the curve. Arcs trace the curved path between those points. Builders, designers, students, and planners often need both values. They may know a radius, an angle, or a sagitta. This calculator turns that known pair into a complete circle report.
What the Calculator Solves
The tool can solve several common geometry cases. You can enter radius and angle, diameter and angle, radius and chord, radius and arc, chord and sagitta, radius and sagitta, arc and chord, or sector area and angle. The result shows radius, diameter, central angle, chord length, arc length, sagitta, sector area, and segment area. It also gives complementary arc values for the same chord.
Practical Uses
Chords and arcs appear in layout work, furniture curves, arch planning, circular tanks, road bends, pattern drafting, and learning tasks. A small input change can shift the final chord or segment area. That is why a full result table is helpful. It lets you compare values before cutting, drawing, or checking a design.
Accuracy Notes
The calculator uses standard circle formulas. It assumes a flat circle and consistent units. If you enter feet, the length results are feet. Area results become square feet. If you enter meters, area becomes square meters. Decimal precision can be adjusted before calculation. Very large or very small values may still be limited by normal rounding.
Better Workflow
Use the example table first when learning the layout. Then enter your own known values. Review the warning messages if a result is impossible. A chord cannot be longer than the circle diameter. An arc should be longer than its chord for the arc and chord mode. Export the report when you need a saved record.
Reading the Results
Start with the selected arc length and central angle. These values describe the curved path you chose. The complementary arc describes the rest of the circle. Segment area is the space between the chord and arc. Sector area is the wedge from the center. Sagitta shows the arc height above the chord. Together, these outputs make curved geometry easier to verify. They also support quick independent field checks.
FAQs
What is a chord?
A chord is a straight line segment joining two points on a circle. It does not need to pass through the center.
What is an arc?
An arc is the curved part of a circle between two points. Its length depends on radius and central angle.
What is sagitta?
Sagitta is the height from the chord midpoint to the arc. It helps define shallow or deep circular curves.
Can I use any unit?
Yes. Use one length unit consistently. If radius is in meters, chord and arc results are also in meters.
Why must the chord be less than diameter?
The longest possible chord is the diameter. A larger chord cannot fit inside the selected circle.
Are area results unit based?
Yes. If lengths are in feet, area results are square feet. If lengths are meters, areas are square meters.
What does complementary arc mean?
It is the remaining arc around the circle. The selected arc and complementary arc form the full circumference.
Why use radians in formulas?
Circle formulas for arc, sector, and segment area use radians. Degree input is converted automatically before calculation.