How to Use Circle Theorem Calculator
The Circle Theorem Calculator lets you apply a chosen circle theorem to compute an unknown angle. Select the theorem that matches your problem, enter the known angle, and the result is shown immediately with the formula and reasoning. The Circle Theorem Calculator runs entirely in your browser.
- Select a circle theorem — Choose from the six supported theorems using the dropdown.
- Enter the known angle — Type the angle (in degrees) that corresponds to the theorem’s input. Some theorems have a fixed answer and require no input.
- Review the result — The Circle Theorem Calculator shows the computed angle, the formula used, and a brief explanation.
For theorems with fixed results (Angle in a Semicircle = 90°, Tangent–Radius = 90°), the result is shown automatically without any angle input.
Formula & Theory - Circle Theorem Calculator
The Circle Theorem Calculator implements six classic theorems from Euclidean geometry:
1. Central Angle Theorem:
Central angle = 2 × Inscribed angle
→ Inscribed angle = Central angle / 2
2. Angle in a Semicircle:
Inscribed angle = 90° (always)
3. Inscribed Angles on the Same Arc:
All inscribed angles subtending the same arc are equal.
→ Other inscribed angle = Given inscribed angle
4. Tangent–Radius:
Angle between tangent and radius at point of tangency = 90° (always)
5. Tangent–Chord Angle:
Tangent-chord angle = Intercepted arc / 2
6. Cyclic Quadrilateral:
Opposite angles sum to 180°
→ Opposite angle = 180° − Known angle| Theorem | Input | Output |
|---|---|---|
| Central Angle | Central angle (°) | Inscribed angle (°) |
| Semicircle | — | 90° |
| Same Arc | One inscribed angle (°) | Other inscribed angle (°) |
| Tangent–Radius | — | 90° |
| Tangent–Chord | Intercepted arc (°) | Tangent-chord angle (°) |
| Cyclic Quad | One angle (°) | Opposite angle (°) |
Assumptions and Limits
All theorems assume a standard Euclidean circle with the points, chords, and tangents drawn in the conventional configuration. Angles must be positive and within their valid ranges (0°–360° for arcs, 0°–180° for most inscribed angles).
Use Cases for Circle Theorem Calculator
The Circle Theorem Calculator is a valuable resource for geometry students, teachers, and anyone working with circular shapes. Common uses include:
- Geometry homework — Instantly verify angle calculations for circle theorem problems.
- Exam preparation — Practice applying each theorem and check reasoning against the calculator.
- Teaching aid — Demonstrate the relationship between central and inscribed angles, or prove that the angle in a semicircle is always 90°.
- Engineering and design — Quickly check angular relationships in circular mechanisms, gear layouts, or architectural arches.
The Circle Theorem Calculator shows the theorem, the substitution, and the computed angle together, making it easy to understand the reasoning behind each result.