Math July 21, 2026 · 12 min read

Circle Theorems Calculator: Inscribed Angles, Central Angles & More (2026)

Calculate angles using circle theorems — inscribed angles, central angles, tangent-chord angles, and cyclic quadrilateral properties.

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Put these formulas into practice with our instant, step-by-step Circle Theorems Calculator.

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Circle theorems are the geometric rules that describe the relationships between angles, arcs, chords, tangents, and secants in circles. From the inscribed angle theorem to the tangent-chord theorem, these results form the foundation of circle geometry. Our circle theorems calculator applies each theorem to find unknown angles with step-by-step geometric proofs, making it an essential tool for students, teachers, and anyone working with circular geometry.

Key Takeaway

The inscribed angle is half the central angle subtending the same arc. Any angle in a semicircle is 90°. Opposite angles in a cyclic quadrilateral sum to 180°.

The Inscribed Angle Theorem

Inscribed angle = ½ × Central angle

The inscribed angle theorem states that an angle formed by two chords meeting at a point on the circle is exactly half the central angle that subtends the same arc. This is the most fundamental circle theorem and the basis for many others. If the central angle is 120°, the inscribed angle is 60°.

The Angle in a Semicircle (Thales' Theorem)

Any angle inscribed in a semicircle — with the diameter as its base — is always exactly 90°. This is a direct consequence of the inscribed angle theorem: the semicircle arc is 180°, so the inscribed angle is 180°/2 = 90°. Thales' theorem is one of the oldest results in mathematics, dating back to around 600 BC.

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a circle. The key property is that opposite angles are supplementary (add to 180°). This property is used extensively in advanced geometry and competition mathematics.

How to Use the Calculator

  1. Select the circle theorem type from the dropdown.
  2. Enter the known angle or arc measurement.
  3. The calculator applies the theorem and shows the result with a geometric proof.

Frequently Asked Questions

What is the inscribed angle theorem?
An inscribed angle is half the central angle subtending the same arc.
What is the angle in a semicircle?
Always 90°. This is Thales' theorem.
What is a cyclic quadrilateral?
A four-sided polygon inscribed in a circle. Opposite angles sum to 180°.
What is the tangent-chord angle?
The angle between a tangent and a chord equals half the intercepted arc.

Conclusion

Circle theorems are essential for understanding the geometry of circles. Use our free circle theorems calculator to apply these theorems and find unknown angles with step-by-step proofs.