Last updated: July 31, 2026
Inscribed Angle Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
inscribed angle = central angle / 2, central angle radians = central angle degrees × π/180, arc length = radius × central angle radians.
Where:
- θcentral=Central angle in degrees(degrees)
- θinscribed=Inscribed angle in degrees(degrees)
- θrad=Central angle in radians(radians)
- r=Circle radius
- s=Arc length
Worked Examples
Quarter-circle central angle
Use a 90° central angle.
- 1Half of 90° is 45°.
- 2Convert 90° to radians: π/2 ≈ 1.570796.
- 3No radius was entered, so arc length stays 0 in this calculator output.
Semicircle with radius
Use a 180° central angle and radius 5.
- 1Half of 180° is 90°.
- 2Convert 180° to radians: π ≈ 3.141593.
- 3Arc length = 5 × π ≈ 15.707963.
Small arc example
Use a 60° central angle.
- 1Half of 60° is 30°.
- 2Convert 60° to radians: π/3 ≈ 1.047198.
- 3Without radius, only the angle conversion outputs are used.
Large arc with radius
Use a 270° central angle and radius 2.5.
- 1Half of 270° is 135°.
- 2Convert 270° to radians: 3π/2 ≈ 4.712389.
- 3Arc length = 2.5 × 4.712389 ≈ 11.780972.
Introduction
The inscribed angle theorem is a foundational circle-geometry rule: an inscribed angle intercepting the same arc measures half the corresponding central angle. This calculator packages that theorem with a radians conversion and optional arc-length computation so one input can support several circle-geometry checks.
The Inscribed Angle Theorem
When two angles intercept the same arc, the central angle is twice the inscribed angle. Equivalently, the inscribed angle is one-half of the central angle.
Central angle sits at the center of the circle.
Inscribed angle has its vertex on the circle.
Both angles must intercept the same arc.
The ratio central : inscribed is always 2 : 1.
A semicircle creates a 180° central angle and a 90° inscribed angle, which is the classic right-angle case.
Why the Calculator Also Converts to Radians
Arc length formulas use radians naturally, so the tool converts the central angle from degrees into radians as a supporting output.
Multiply degrees by π/180.
90° becomes π/2.
180° becomes π.
360° becomes 2π.
Radians are especially important in calculus, trigonometry, and physics formulas involving circles.
How Arc Length Fits In
If the radius is known, the same central angle also determines the intercepted arc length using s = rθ, where θ is in radians.
Enter a positive radius to unlock arc length.
A larger radius produces a longer arc for the same angle.
A larger angle produces a longer arc for the same radius.
If no radius is provided, this tool reports arc length as 0.
Allowed Input Range
The central angle must be greater than 0° and at most 360°. A radius is optional, but if you provide one it must be positive.
0° is rejected because it does not create a meaningful intercepted arc.
360° is accepted as a full-circle case.
Negative radii are invalid.
Blank radius is allowed when only angle relationships are needed.
How to Use the Tool
Start with the central angle in degrees. If you also know the circle radius, add it to obtain arc length in the same run.
Enter the central angle.
Optionally enter a positive radius.
Read the inscribed angle first.
Use the radian and arc-length outputs for deeper geometry work.
Common Geometry Applications
These relationships appear in proofs, diagram labeling, and practical design work involving curved paths or circular sectors.
High-school circle theorems.
Arc and chord problems.
Wheel and pulley measurements.
CAD sketches involving circular arcs.
Special Cases to Remember
Certain central angles are worth recognizing instantly because they show up often in textbooks and exams.
90° central → 45° inscribed.
180° central → 90° inscribed.
60° central → 30° inscribed.
270° central → 135° inscribed.
Memorizing these benchmark cases can help you spot diagram errors immediately.
FAQs
What is an inscribed angle?
It is an angle whose vertex lies on the circle and whose sides intercept an arc of the circle.
How is an inscribed angle related to a central angle?
For the same intercepted arc, the inscribed angle equals one-half of the central angle.
Why is arc length optional here?
Arc length needs a radius. If you only want the inscribed angle relationship, the radius is not necessary.
Why does the calculator convert degrees to radians?
The arc-length formula s = rθ requires θ in radians, so the conversion is included automatically.
Can I use a central angle greater than 360°?
No. This calculator restricts the input to angles above 0° and up to 360° inclusive.
What happens if I leave the radius blank?
The inscribed angle and radian conversion are still computed, and arc length is reported as 0.
Why is a 180° central angle important?
It creates a 90° inscribed angle, which explains why an angle subtending a diameter is a right angle.
How precise are the outputs?
The displayed numeric outputs are rounded to six decimal places for consistent geometry calculations.