Circle theorems connect angles and lengths in circles. They are a staple of school geometry and are used in navigation (danger circles), optics and architecture.
How to use the Circle Theorems Calculator
- Theorem
- Central angle (°)
- Angle at one end of the diameter (°)
- Angle A (°)
- Angle B (°)
- AE
- EB
- CE
- PA (outer part of secant 1)
- PB (whole secant 1)
- PC (outer part of secant 2)
Results update as you type, and everything is calculated in your browser.
How it works
Choose the theorem; only the fields it needs appear. Lengths can be in any single unit; angles are in degrees.
Inputs explained
| Field | What to enter |
|---|---|
| Theorem | Choose one: Inscribed angle theorem, Thales’ theorem, Cyclic quadrilateral, Intersecting chords … |
| Central angle (°) | A number |
| Angle at one end of the diameter (°) | A number |
| Angle A (°) | A number |
| Angle B (°) | A number |
| AE | A number |
| EB | A number |
| CE | A number |
| PA (outer part of secant 1) | A number |
| PB (whole secant 1) | A number |
| PC (outer part of secant 2) | A number |
What each result means
- Inscribed angle = [Central angle (°)]÷2
- Other acute angle = 90 – [Angle at one end of the diameter (°)]
- Angle opposite the diameter = 90
- Angle C (opposite A) = 180 – [Angle A (°)]
- Angle D (opposite B) = 180 – [Angle B (°)]
- ED = [AE]×[EB]÷[CE]
- PD (whole secant 2) = [PA (outer part of secant 1)]×[PB (whole secant 1)]÷[PC (outer part of secant 2)]
- Tangent length PT = sqrt([PA (outer part of secant 1)]×[PB (whole secant 1)])
Example values
How the main result changes when Central angle changes and everything else stays at the example values above:
| Central angle (°) | Inscribed angle (°) |
|---|---|
| 50 | 25 |
| 75 | 37.5 |
| 100 | 50 |
| 125 | 62.5 |
| 150 | 75 |
| 200 | 100 |
Formula
Inscribed angle = central angle / 2 · Angle in a semicircle = 90°
Cyclic quadrilateral: opposite angles add to 180° · Chords: AE·EB = CE·ED
Secants: PA·PB = PC·PD · Tangent–secant: PT² = PA·PB
Worked example
Inputs — Theorem: Inscribed angle theorem; Central angle (°): 100.
Results — Inscribed angle = 50 °.
Key terms
- Inscribed angle
- An angle with its vertex on the circle.
- Secant
- A line cutting the circle at two points.
Frequently asked questions
What does the inscribed angle theorem say?
An inscribed angle is half the central angle on the same arc.
What is Thales' theorem?
Any angle inscribed in a semicircle is a right angle.
When is a quadrilateral cyclic?
When its opposite angles add up to 180°.
What is the power of a point?
The constant product PA·PB for every line through P meeting the circle.