# Circle Theorems Calculator

URL: https://speedycalculator.com/calculators/math/circle-theorems/
Last updated: 2026-10-30

> The circle theorems calculator applies six classic circle theorems: the inscribed angle theorem, Thales' theorem, cyclic quadrilaterals, intersecting chords, two secants and the tangent–secant theorem.

## Inputs

- 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

- Inscribed angle
- Other acute angle
- Angle opposite the diameter
- Angle C (opposite A)
- Angle D (opposite B)
- ED
- PD (whole secant 2)
- Tangent length PT

## 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
```

## Explanation

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

FieldWhat to enterTheoremChoose one: Inscribed angle theorem, Thales' theorem, Cyclic quadrilateral, Intersecting chords …Central angle (°)A numberAngle at one end of the diameter (°)A numberAngle A (°)A numberAngle B (°)A numberAEA numberEBA numberCEA numberPA (outer part of secant 1)A numberPB (whole secant 1)A numberPC (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 (°)50257537.51005012562.515075200100

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [Wolfram MathWorld – Inscribed angle](https://mathworld.wolfram.com/InscribedAngle.html)
- [Wolfram MathWorld – Circle power](https://mathworld.wolfram.com/CirclePower.html)