Circle Theorems Calculator

The inscribed angle theorem states that an inscribed angle (formed by two chords meeting on the circle) is exactly half the central angle that subtends the same arc. If the central angle is θ, the inscribed angle is θ/2.

The central angle is the angle formed at the center of the circle by two radii. It equals the measure of its intercepted arc in degrees. A central angle is always twice the inscribed angle that subtends the same arc.

Any angle inscribed in a semicircle (with the diameter as its base) is always exactly 90°. This is Thales' theorem, one of the oldest results in geometry. It follows from the inscribed angle theorem since the semicircle arc is 180°.

The angle between a tangent line and a chord at the point of tangency equals half the measure of the intercepted arc. If the arc is θ, the tangent-chord angle is θ/2.

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 applies to any quadrilateral inscribed in a circle.

A tangent line to a circle is perpendicular to the radius at the point of tangency. If the tangent touches the circle at point T and the center is O, then OT is perpendicular to the tangent line.

Angles subtended by the same arc (or chord) at the circumference are equal. If two inscribed angles both subtend the same arc, they are equal regardless of where on the circle they are located.

The angle between two intersecting chords equals half the sum of the intercepted arcs: angle = (arc₁ + arc₂) / 2. This applies when the chords intersect inside the circle.

The angle between two secants meeting outside the circle equals half the difference of the intercepted arcs: angle = (far arc - near arc) / 2. This is the external angle theorem for circles.

The angle between a tangent and a chord equals the angle in the alternate segment. If the tangent touches at T and the chord goes to A and B, then angle between tangent and TA equals angle TBA.

For a point P outside a circle, the power is PA × PB = PC × PD, where A, B and C, D are the intersection points of two secants from P. For a tangent PT: PT² = PA × PB.

The central angle of a semicircle is 180°. By the inscribed angle theorem, any inscribed angle in a semicircle is 180°/2 = 90°. This is why the angle in a semicircle is always a right angle.

Arc length = (θ/360°) × 2πr, where θ is the central angle in degrees and r is the radius. For θ = 90° and r = 5: arc = (90/360) × 10π ≈ 7.85.

Equal chords subtend equal central angles. Longer chords subtend larger central angles. The chord length is c = 2r sin(θ/2), where θ is the central angle and r is the radius.

Parallel chords intercept equal arcs between them. If two chords are parallel, the arcs between their endpoints are equal. This creates symmetric configurations useful in geometric proofs.

For a tangent at T and a secant through A and B (with A closer to T): angle = (arc TB - arc TA) / 2. If the secant passes through the center, the angle equals half the intercepted arc.

If M is the midpoint of a chord PQ, and AB and CD are chords through M, then M is also the midpoint of the chord connecting the intersections of AC with PQ and BD with PQ. It is a beautiful result in circle geometry.

Circle theorems are used in navigation (great circle routes), architecture (dome and arch design), engineering (gear and pulley systems), astronomy (celestial mechanics), and computer graphics (circular interpolation).

The angle at the center is twice the angle at the circumference when both subtend the same arc. Equivalently, the inscribed angle is half the central angle. This is the most fundamental circle theorem.

Yes. If you know a chord length and the inscribed angle it subtends, you can find the radius using c = 2r sin(θ/2). Solve for r: r = c / (2 sin(θ/2)).

The Circle Theorems Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Circle Theorems Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.