What is the inscribed angle theorem?

It states that an inscribed angle is always half the central angle that intercepts the same arc. If the central angle is θ, the inscribed angle is θ/2.

What is Thales' theorem?

A special case of the inscribed angle theorem: any angle inscribed in a semicircle (subtending a diameter) is exactly 90°. This forms a right triangle.

Can the inscribed angle be larger than 90°?

Yes. If the central angle exceeds 180°, the inscribed angle exceeds 90°. For example, a 240° central angle gives a 120° inscribed angle.

How do I find the chord length from the inscribed angle?

First double the inscribed angle to get the central angle θ, then compute chord = 2r × sin(θ/2).

The sagitta (or versine) is the distance from the midpoint of a chord to the midpoint of its arc. It equals r(1 − cos(θ/2)).

Do inscribed angles work for any circle?

Yes. The inscribed angle theorem holds for all circles, regardless of size. The radius affects lengths (arc, chord, sagitta) but not the angle relationship.

Inscribed Angle Calculator — Central Angle, Arc & Chord

Calculate inscribed angles, central angles, arc length, chord length, sagitta, sector area, and segment area for a circle. Includes Thales' theorem detection and a reference table.

Inscribed Angle Calculator

Input Mode

Central Angle (°)

Circle Radius

Decimal Places

Common Inscribed & Central Angle Pairs

Central Angle	Inscribed Angle	Arc Fraction	Chord / Diameter
30°	15°	1/12	0.2588
60°	30°	1/6	0.5000
90°	45°	1/4	0.7071
120°	60°	1/3	0.8660
150°	75°	5/12	0.9659
180°	90°	1/2	1.0000
240°	120°	2/3	0.8660
270°	135°	3/4	0.7071
300°	150°	5/6	0.5000
360°	180°	1	0.0000

Thales’ Theorem (special case)

When the two endpoints of the arc form a diameter (central angle = 180°), any inscribed angle subtending that arc is exactly 90°. This is Thales’ theorem, one of the oldest known geometry theorems. Use the “Central 180°” preset above to verify.

Planning notes, formulas, and examples

About the Inscribed Angle Calculator — Central Angle, Arc & Chord

An inscribed angle is an angle formed by two chords that share an endpoint on a circle. The Inscribed Angle Theorem states that an inscribed angle is always exactly half the central angle that subtends the same arc. This elegant relationship is one of the cornerstones of circle geometry.

Formally, if a central angle measures θ degrees, the inscribed angle on the same arc is θ/2 degrees. Conversely, if you know the inscribed angle α, the central angle is 2α. The intercepted arc has a degree measure equal to the central angle.

A famous special case is Thales' theorem: when the inscribed angle subtends a diameter (central angle = 180°), the inscribed angle is exactly 90°. This means any triangle inscribed in a semicircle with the diameter as one side is a right triangle.

Beyond angles, the arc, chord, and sagitta are all related. The arc length is rθ (in radians). The chord length is 2r sin(θ/2). The sagitta (the height of the circular segment) is r(1 − cos(θ/2)). Sector area is ½r²θ and segment area adds the triangle subtraction.

This calculator accepts a central angle, inscribed angle, or arc length with radius, and computes all related quantities. It flags Thales' theorem when it applies, provides a reference table of common angle pairs, and visualizes the proportional relationships with bars.

When This Page Helps

The Inscribed Angle Calculator — Central Angle, Arc & Chord is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Central Angle, Inscribed Angle, Arc Length in one pass, with conversions and derived values shown together.

How to Use the Inputs

Choose input mode: from central angle, inscribed angle, or arc length & radius.
Enter the angle (in degrees) or arc length.
Enter the circle radius for dimensional quantities.
Set decimal precision.
Click a preset (60°, 90°, 120°, 180° Thales, etc.) to explore.
Read the central angle, inscribed angle, arc, chord, sagitta, sector area, and segment area.
Check if Thales' theorem applies (central angle = 180°).

Formula used

Inscribed Angle = ½ × Central Angle
Central Angle = 2 × Inscribed Angle
Arc Length = r × θ (θ in radians)
Chord Length = 2r × sin(θ/2)
Sagitta = r × (1 − cos(θ/2))
Sector Area = ½r²θ
Segment Area = ½r²(θ − sin θ)
Thales' Theorem: inscribed angle = 90° when subtending a diameter

Example Calculation

Result: Inscribed Angle = 45°, Arc ≈ 15.708, Chord ≈ 14.142, Sagitta ≈ 2.929

For central angle 90° and r = 10: Inscribed = 90/2 = 45°. Arc = 10 × π/2 ≈ 15.708. Chord = 2 × 10 × sin(45°) = 20 × 0.7071 ≈ 14.142. Sagitta = 10 × (1 − cos 45°) = 10 × 0.2929 ≈ 2.929.

Tips & Best Practices

All inscribed angles that subtend the same arc are equal — no matter where on the circle the vertex sits.
Thales' theorem is the basis for constructing right angles with just a compass and straightedge: place the diameter, pick any point on the semicircle, and the angle is 90°.
An inscribed angle of 0° means the two chords overlap (degenerate case). An inscribed angle of 180° means the vertex is at the minor arc side of a full chord.
The chord/diameter ratio equals sin(central/2), providing a quick lookup for common angles.

How This Inscribed Angle Calculator — Central Angle, Arc & Chord Works

Where It Helps In Practice

Inscribed Angle Calculator — Central Angle, Arc & Chord calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.

Accuracy And Setup Tips

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

It states that an inscribed angle is always half the central angle that intercepts the same arc. If the central angle is θ, the inscribed angle is θ/2.