Arc Length Calculator
Calculate arc length, sector area, chord length, and arc-to-chord ratio from radius and central angle. Includes preset common angles, a properties table, and an arc diagram.
Central Angle Calculator
Calculate the central angle from arc length and radius. Find inscribed angle relationships, sector area, and segment properties with presets, properties table, and circle visual.
Central Angle Calculator
Central Angle⧉
90.0000° / 1.5708 rad
Angle at the center subtended by the arc
Inscribed Angle⧉
45.0000°
Half the central angle (Inscribed Angle Theorem)
Arc Length⧉
15.7080
s = rθ = 10.00 × 1.5708
Sector Area⧉
78.5398
Area of the pizza-slice region = ½r²θ
Chord Length⧉
14.1421
Straight line between arc endpoints = 2r sin(θ/2)
Segment Area⧉
28.5398
Region between chord and arc = Sector − Triangle
Sagitta⧉
2.9289
Max distance from chord to arc = r − r cos(θ/2)
Circle Fraction⧉
25.00%
This arc is 25.00% of the full circumference
Circle Diagram
Blue = arc & central angle, Red = chord, Green = inscribed angle
Arc as Fraction of Circle
0%50%100%
Properties for Common Angles (r = 10.00)
| Central (°) | Inscribed (°) | Arc Length | Chord | Sector Area | Segment Area |
|---|---|---|---|---|---|
| 15° | 7.5° | 2.618 | 2.611 | 13.090 | 0.149 |
| 30° | 15° | 5.236 | 5.176 | 26.180 | 1.180 |
| 45° | 22.5° | 7.854 | 7.654 | 39.270 | 3.915 |
| 60° | 30° | 10.472 | 10.000 | 52.360 | 9.059 |
| 90° | 45° | 15.708 | 14.142 | 78.540 | 28.540 |
| 120° | 60° | 20.944 | 17.321 | 104.720 | 61.418 |
| 135° | 67.5° | 23.562 | 18.478 | 117.810 | 82.454 |
| 150° | 75° | 26.180 | 19.319 | 130.900 | 105.900 |
| 180° | 90° | 31.416 | 20.000 | 157.080 | 157.080 |
| 210° | 105° | 36.652 | 19.319 | 183.260 | 208.260 |
| 240° | 120° | 41.888 | 17.321 | 209.440 | 252.741 |
| 270° | 135° | 47.124 | 14.142 | 235.619 | 285.619 |
| 300° | 150° | 52.360 | 10.000 | 261.799 | 305.101 |
| 330° | 165° | 57.596 | 5.176 | 287.979 | 312.979 |
| 360° | 180° | 62.832 | 0.000 | 314.159 | 314.159 |
Planning notes, formulas, and examples
About the Central Angle Calculator
The central angle calculator determines the angle at the center of a circle given the arc length and radius. The central angle theorem states that the central angle is twice any inscribed angle that subtends the same arc, a cornerstone of circle geometry. The relationship is elegantly simple: θ = s / r, where s is the arc length and r is the radius. This calculator goes further by computing the inscribed angle, sector area, segment area, chord length, sagitta, and the fraction of the full circle the angle represents. Whether you are an engineer designing a curved road section, a student working through circle theorems, or a carpenter cutting a circular arc, the page keeps those related properties together. Choose between computing the angle from arc length + radius, or enter the angle directly to find the corresponding arc length. Preset examples cover common geometry problems, and the interactive circle diagram highlights the central angle, inscribed angle, and arc. A reference table shows properties for standard angles from 15° to 360°, so you can quickly compare or verify your work. Understanding central and inscribed angles is essential for coordinate geometry, trigonometry proofs, and the geometry of circles on standardized tests.
When This Page Helps
Central angle problems require computing multiple related properties — inscribed angle, sector area, segment area, chord length, and sagitta — which is time-consuming when done by hand. This calculator solves in both directions (angle-to-arc and arc-to-angle), computes all circle properties simultaneously, and visualizes the relationship with an interactive diagram. It is ideal for geometry students working through circle theorems, engineers designing curved structures, and anyone who needs to quickly convert between arc measurements and angles.
How to Use the Inputs
- Choose the solving mode: "Angle → Arc Length" if you know the angle, or "Arc Length → Angle" if you know the arc.
- Enter the radius of the circle in the Radius field.
- If in Angle mode, enter the central angle in degrees. If in Arc mode, enter the arc length.
- Click a preset like "90°, r=8" or "Arc=5π, r=10" to load a common example.
- Review the central angle, inscribed angle, sector area, chord, sagitta, and segment area in the output cards.
- Examine the circle diagram showing the central and inscribed angle relationship.
- Check the reference table for properties of standard angles from 15° to 360°.
Formula used
θ = s / r (radians)
Inscribed Angle = θ / 2
Sector Area = ½ r² θ
Chord = 2r sin(θ/2)Example Calculation
Result: 90° (π/2 rad)
For arc length s = 5π and radius r = 10: θ = 5π/10 = π/2 rad = 90°. Inscribed angle = 45°. Sector area = ½×100×π/2 ≈ 78.54.
Tips & Best Practices
- Keep angle units consistent; mixing degrees and radians is the most common source of wrong results.
- Use a simple known case or diagram to confirm the sign and scale of the answer.
Central and Inscribed Angle Theorem
The Inscribed Angle Theorem states that an inscribed angle (vertex on the circle) is always exactly half the central angle that subtends the same arc. If a central angle is 80°, every inscribed angle subtending the same arc is 40°, regardless of where the vertex sits on the circle. This theorem is the basis for many geometry proofs and appears frequently on standardized tests like the SAT and ACT.
Applications in Engineering and Design
Central angles are essential in civil engineering for designing road curves and railroad bends. The degree of curvature (central angle per unit chord) determines the sharpness of a turn and the safe speed limit. Architects use central angles when designing arched windows, circular colonnades, and amphitheater seating. In surveying, central angles help calculate property boundaries along curved roads. GPS navigation systems use central angles on great circles to compute shortest-path distances on Earth's surface.
Sectors, Segments, and Chord Geometry
A sector is the "pizza slice" region bounded by two radii and an arc; its area is ½r²θ. A segment is the region between a chord and its arc; its area equals the sector area minus the triangle formed by the two radii and the chord. The sagitta (height of the arc above the chord) is h = r − r cos(θ/2), crucial for arch and bridge design. Understanding the relationships between these elements allows you to solve complex geometric problems by breaking them into simpler sector and triangle components.
Sources & Methodology
Last updated:
Frequently Asked Questions
A central angle is an angle whose vertex is at the center of a circle and whose sides pass through two points on the circle.
The inscribed angle is always exactly half the central angle when both subtend the same arc (Inscribed Angle Theorem).
Divide the arc length by the radius: θ = s/r. This gives the angle in radians; multiply by 180/π for degrees.
A sector is the "pizza slice" region bounded by two radii and the arc between them.
A segment is the region between a chord and the arc it subtends. Its area equals the sector area minus the triangle area.
Yes. A central angle greater than 180° corresponds to a major arc (more than half the circle).
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