Central Angle Calculator
Choose a solve mode, enter two known values, and the calculator instantly finds the unknown. Mode 1 finds the central angle from arc length and radius. Mode 2 finds arc length from a central angle and radius. Mode 3 converts between an inscribed angle and the central angle that subtends the same arc, using the Central Angle Theorem. Switch between degrees and radians, or between metres and feet, at any time.
Formula
Worked example
A circle of radius 10 m has an arc of 5 m. Central angle = 5 / 10 = 0.5 rad = 28.648°. Chord = 2 x 10 x sin(0.25) = 4.9479 m. Sector area = 0.5 x 100 x 0.5 = 25 m². The arc spans 0.5 / (2pi) = 7.96% of the full circle.
What is a central angle?
A central angle is an angle whose vertex sits at the centre of a circle and whose two sides (radii) extend outward to meet the circumference. The arc that lies between the two radii is called the intercepted arc. The central angle and its intercepted arc share the same degree measure, so a 60° central angle intercepts exactly one-sixth of the circle (since 60 / 360 = 1/6). Central angles appear throughout geometry, trigonometry, navigation, engineering and astronomy whenever a portion of a circle needs to be measured or described precisely.
The central angle formula
The core relationship is theta = s / r, where theta is the central angle in radians, s is the arc length and r is the radius. Rearranged, s = r * theta gives the arc length when the angle is known. To convert between degrees and radians use theta_deg = theta_rad x (180 / pi), or theta_rad = theta_deg x (pi / 180). Once the central angle in radians is known, the chord (straight-line distance between the arc endpoints) is 2r sin(theta / 2), and the area of the sector (pie-slice region) is (1/2) r^2 theta. All four quantities are computed automatically by this calculator.
The Central Angle Theorem and inscribed angles
The Central Angle Theorem (also called the Inscribed Angle Theorem) states that a central angle is exactly twice the inscribed angle that subtends the same arc. An inscribed angle is one whose vertex lies on the circle itself rather than at the centre. For example, if an inscribed angle measuring 35° subtends a given arc, the central angle subtending that same arc is 70°. Conversely, every central angle equals twice the corresponding inscribed angle. This relationship is fundamental in circle theorems and is used in proofs, coordinate geometry, and problems involving angles in semicircles (where the inscribed angle is always 90° because the central angle is 180°).
Degrees, radians and unit conversions
Degrees divide a full rotation into 360 equal parts, a convention inherited from ancient Babylonian astronomy. Radians measure angles as arc-to-radius ratios and are the natural unit for calculus and physics: a full circle is exactly 2pi radians. To switch from degrees to radians, multiply by pi/180 (about 0.01745 per degree). To switch from radians to degrees, multiply by 180/pi (about 57.296). Common benchmarks: 90° = pi/2 rad, 180° = pi rad, 360° = 2pi rad. Both units are fully supported here.
Common central angles, arc fractions and radians
| Central angle | Radians | Arc fraction | Sector (% of circle) |
|---|---|---|---|
| 30° | pi/6 (0.5236) | 1/12 | 8.33% |
| 45° | pi/4 (0.7854) | 1/8 | 12.5% |
| 60° | pi/3 (1.0472) | 1/6 | 16.67% |
| 90° | pi/2 (1.5708) | 1/4 | 25% |
| 120° | 2pi/3 (2.0944) | 1/3 | 33.33% |
| 135° | 3pi/4 (2.3562) | 3/8 | 37.5% |
| 150° | 5pi/6 (2.6180) | 5/12 | 41.67% |
| 180° | pi (3.1416) | 1/2 | 50% |
| 270° | 3pi/2 (4.7124) | 3/4 | 75% |
| 360° | 2pi (6.2832) | 1/1 | 100% |
These standard values appear frequently in geometry, trigonometry and engineering.
Frequently asked questions
What is the formula for a central angle?
The central angle in radians equals arc length divided by radius: theta = s / r. In degrees, multiply the result by 180 / pi. If you know the central angle and radius but want the arc length, rearrange to s = r * theta (theta must be in radians).
How do I convert between inscribed angles and central angles?
The Central Angle Theorem says that the central angle is twice the inscribed angle subtending the same arc. So multiply the inscribed angle by 2 to get the central angle, or divide the central angle by 2 to get the inscribed angle. For example, a 45° inscribed angle corresponds to a 90° central angle.
What is the difference between a central angle and an arc length?
A central angle is a measure of rotation (in degrees or radians) at the centre of the circle. Arc length is a measure of distance along the curved circumference. They are related by s = r * theta, where theta is in radians. A larger radius produces a longer arc for the same central angle.
How do I find the chord length from a central angle?
Chord length = 2 * radius * sin(theta / 2), where theta is the central angle in radians. For example, with radius 10 and central angle 60° (pi/3 rad), chord = 2 * 10 * sin(pi/6) = 20 * 0.5 = 10. This calculator computes chord length automatically alongside the central angle.
What is the sector area formula?
The area of a circular sector (the pie-slice region bounded by two radii and the arc) is A = (1/2) * r^2 * theta, where theta is in radians. Equivalently, A = (theta_deg / 360) * pi * r^2. Both forms give the same answer and the calculator shows all four related quantities (central angle, arc length, chord and sector area) for every mode.
Why does this calculator show the result in both degrees and radians?
Degrees are intuitive for everyday geometry while radians are required by most formulas in calculus and physics. Showing both simultaneously removes the need for a separate conversion step and prevents errors caused by mixing units. You can also switch the input unit between degrees and radians at any time.