Free Circle Theorems Calculator

The Circle Theorems Calculator is an interactive geometry tool that brings together six essential circle relationships in one simple interface. Whether you are studying the Inscribed Angle Theorem, working with Thales’ Theorem (angle in a semicircle), analyzing a Cyclic Quadrilateral, determining chord lengths, applying the Intersecting Secants Theorem, or finding a tangent line, this calculator helps you move quickly from known values to missing angles and segment lengths. It is designed as a free online geometry calculator that accepts different angular units and instantly returns results based on the chosen theorem.

Inscribed Angle Theorem

The inscribed angle theorem states two important facts about angles that sit on the circumference of a circle. First, an inscribed angle measures exactly half the measure of the central angle that subtends the same arc. Second, any two inscribed angles that subtend the same arc are equal. In formal terms, if $\theta_i$ is an inscribed angle and $\theta_c$ is the corresponding central angle, then:

$$\theta_i = \frac{1}{2} \theta_c$$

Additionally, if $\theta_{i1}$ and $\theta_{i2}$ subtend the same arc, then $\theta_{i1} = \theta_{i2}$. Using the calculator, you enter either the inscribed or central angle and the tool applies this relation automatically.

Thales Theorem – Angle in a Semicircle

Thales theorem is a special case of the inscribed angle theorem. If three points $A$, $B$, and $C$ lie on the circumference of a circle and the segment $AC$ is a diameter, then the angle at $B$ ($\angle ABC$) is a right angle:

$$\angle ABC = 90^\circ$$

Because the diameter forms the hypotenuse of a right triangle, you can combine this result with the Pythagorean Theorem to find missing sides or angles when only one side or one acute angle is known. The calculator lets you input the known components and solves for the rest automatically.

Cyclic Quadrilateral Theorem

A cyclic quadrilateral is any four‑sided polygon whose vertices all lie on a single circle. The main property of such a quadrilateral is that its opposite angles are supplementary – they add up to $180^\circ$. If a quadrilateral has vertices $A, B, C, D$ on the circle, then:

$$\angle A + \angle C = 180^\circ, \quad \angle B + \angle D = 180^\circ$$

Given any three angles, the fourth can be found directly. The calculator uses this rule to compute the missing angle in a cyclic quadrilateral after you enter the known ones.

Chord Length Theorem (Equidistant Chords)

Chords that are at the same distance from the center of the circle have equal lengths. Conversely, equal chords are equidistant from the center. The relationship between chord length $c$, distance from center $d$, and radius $r$ is derived from the right triangle formed by half the chord, the distance to the center, and the radius:

$$c = 2\sqrt{r^{2} - d^{2}}$$

If you know any two of these three variables, you can rearrange the equation to solve for the third. The calculator performs this rearrangement for you, making it easy to find chord length or its distance from the center.

Intersecting Secants Theorem (Exterior Angle of a Circle)

When two secant lines intersect outside a circle, the angle formed at the intersection equals half the difference of the intercepted arcs. If the far arc (larger) measures $a$ and the near arc (smaller) measures $b$, then:

$$\theta = \frac{1}{2} \left(a - b\right)$$

The same configuration also yields a proportional relationship among the segment lengths. If the secants intersect at point $P$ and meet the circle at points $A$, $D$ (on one secant) and $B$, $C$ (on the other), then:

$$PA \cdot PD = PB \cdot PC$$

Entering any three of the four segment lengths (or the arc measures) into the calculator gives you the missing value immediately.

Tangent to a Circle Theorem

A line that touches a circle at exactly one point – a tangent – is perpendicular to the radius drawn to that point of tangency. If the circle has center $(h,k)$ and radius $r$, then the tangent line at a point $(x_1, y_1)$ on the circle satisfies:

$$(x_1 - h)(x - h) + (y_1 - k)(y - k) = r^{2}$$

In slope‑intercept form, the tangent can be expressed as $y = m x + b$ where the slope $m$ is determined by the radius’s perpendicular direction. The calculator accepts any point on the circumference and outputs the corresponding tangent line equation, allowing you to skip the manual derivation.

Each of these six theorem areas is integrated into a single tool, so you can switch between them without leaving the page. By entering the values you have, the Circle Theorems Calculator applies the correct formula and returns the missing angle, length, or line equation – supporting study, homework, and quick geometric exploration.