Circle Theorems Calculator
Apply any of the four major circle theorems to calculate unknown angles from arc and angle measurements.
⭕ What are Circle Theorems?
Circle theorems are a set of geometric results that describe the precise relationships between the angles, chords, tangents, arcs, and secants of a circle. They allow you to calculate unknown angles purely from arc measurements or from other known angles — no radius required. Mastering circle theorems is a key part of secondary school geometry and standardised test preparation worldwide.
The four theorems supported by this calculator cover the most widely examined cases. The Central and Inscribed Angle Theorem (also called the inscribed angle theorem) states that an inscribed angle — whose vertex lies on the circle — is always exactly half the central angle that subtends the same arc. This means if you know the central angle is 120°, the inscribed angle intercepting the same arc is 60°, and any other inscribed angle intercepting that same arc is also 60°, regardless of where on the remaining arc the vertex is placed.
The Chord-Chord Angle Theorem applies when two chords cross inside a circle. The angle at the intersection equals half the sum of the two intercepted arcs. For example, if the two chords carve out arcs of 80° and 60° on opposite sides of the intersection, the angle is (80° + 60°) ÷ 2 = 70°. Engineers and architects encounter this when designing circular structures with intersecting supports. The Tangent-Chord Angle Theorem states that when a tangent meets a chord at the point of tangency, the angle formed equals half the arc cut off by the chord on that side. A tangent never enters the circle, so it creates a unique angle relationship not found with secants.
The Secant-Secant Theorem governs the angle formed outside a circle when two secants (lines that pass through the circle) are drawn from the same external point. The angle equals half the positive difference of the two intercepted arcs: (far arc − near arc) ÷ 2. This theorem unifies with the tangent version when one or both secants become tangents, since a tangent intersects the circle at exactly one point. All four theorems are consequences of the fundamental inscribed angle theorem and can be proved by drawing radii and applying triangle angle sums.
This calculator supports all four theorems with dedicated input modes. Select the mode that matches your configuration, enter the known arc or central angle, and the calculator applies the correct formula, shows the result, and displays the theorem used — ideal for checking homework, preparing for exams, or verifying geometric constructions.
📐 Circle Theorem Formulas
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Inscribed Angle Theorem
A central angle of 140° intercepts an arc of 140°. The inscribed angle intercepting the same arc = 140° ÷ 2 = 70°. Any other inscribed angle intercepting that same 140° arc is also 70°, no matter where on the remaining arc the vertex is placed.
Example 2 — Chord-Chord Angle
Two chords intersect inside a circle. The two intercepted arcs on opposite sides of the intersection measure 90° and 50°. The angle at the intersection = (90° + 50°) ÷ 2 = 70°. The vertical angle across from it is also 70°. The other two vertical angles each equal (360° − 90° − 50°) ÷ 2 = 110°.
Example 3 — Tangent-Chord Angle
A tangent and a chord meet at a point on a circle. The chord intercepts an arc of 150°. The tangent-chord angle = 150° ÷ 2 = 75°. The supplementary angle on the other side intercepts the remaining arc (360° − 150° = 210°), so that angle = 210° ÷ 2 = 105°. Note that 75° + 105° = 180° ✓ (straight line).
Example 4 — Secant-Secant External Angle
Two secants from an outside point intercept arcs of 200° (far) and 60° (near). The external angle = (200° − 60°) ÷ 2 = 70°. If the near arc were 0° (one secant becomes a tangent), the formula would give 200° ÷ 2 = 100°, consistent with the tangent-secant theorem.