Circle Theorems Calculator - Inscribed Angles, Cyclic Quadrilaterals
Apply circle theorems to calculate inscribed angles, central angles, arc measures, cyclic quadrilateral angles, and tangent-chord angles instantly.
Select a theorem, enter the known angle or arc measure, and get the unknown value with an explanation of the theorem used.
Circle Theorems Calculator - Inscribed Angles, Cyclic Quadrilaterals
Apply circle theorems to calculate inscribed angles, central angles, arc measures, cyclic quadrilateral angles, and tangent-chord angles instantly.
An inscribed angle is half the central angle that subtends the same arc. Enter the central angle to find the inscribed angle, or vice versa.
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About the circle theorems calculator
Circle theorems are a collection of fundamental results in Euclidean geometry that describe the relationships between angles, arcs, and line segments associated with circles. They provide powerful tools for solving geometric problems without requiring coordinate geometry or trigonometry, making them a staple of secondary school mathematics curricula worldwide.
The inscribed angle theorem is the most widely used circle theorem. It states that an inscribed angle — an angle whose vertex lies on the circle and whose sides are two chords — is exactly half the central angle that subtends the same arc. Equivalently, all inscribed angles that subtend the same arc are equal. This theorem transforms problems about angles inside circles into straightforward halving or doubling operations.
Thales' theorem is the oldest and most elegant special case: when the chord subtended by an inscribed angle is the diameter of the circle, the inscribed angle is always 90°. This means that if you know two endpoints of a diameter, every point on the circle forms a right angle with those two endpoints. Thales' theorem is used in practical geometry to find the centre of a circle: any two right angles inscribed on a chord will locate the diameter.
The cyclic quadrilateral theorem extends the inscribed angle idea to four-sided figures. A quadrilateral is cyclic (i.e., all four vertices lie on a circle) if and only if its opposite angles sum to 180°. This property is used to test whether four points are concyclic and to solve for unknown angles in geometric figures.
The tangent-chord angle theorem states that the angle between a tangent to a circle and a chord from the point of tangency equals half the intercepted arc. This parallels the inscribed angle theorem but involves a tangent line instead of a second chord. It is particularly useful in problems involving circles touching each other or touching a straight line.
This calculator implements five theorem types: Inscribed Angle, Central Angle (the reverse of inscribed), Angle in Semicircle (Thales), Cyclic Quadrilateral, and Tangent-Chord Angle. For each type, you enter the known value and the calculator applies the appropriate theorem to find the unknown. The results include a brief statement of the theorem used, helping you learn the geometry alongside the computation.
All angle inputs and outputs are in degrees. The calculator validates that input angles fall within physically meaningful ranges — for example, a central angle must be between 0° and 360°, and a known angle in a cyclic quadrilateral must be between 0° and 180°. Results outside these ranges indicate an input error rather than a valid geometric configuration.
Circle theorems examples
Three worked examples applying different circle theorems to realistic geometry problems.
| Theorem & Input | Result | Explanation |
|---|---|---|
| Inscribed angle theorem: central angle = 80° | Inscribed angle = 40° | By the inscribed angle theorem, the inscribed angle is always half the central angle subtending the same arc. So 80° ÷ 2 = 40°. |
| Cyclic quadrilateral: known angle = 110° | Opposite angle = 70° | Opposite angles in a cyclic quadrilateral are supplementary: they sum to 180°. So 180° − 110° = 70°. |
| Tangent-chord angle: arc measure = 120° | Tangent-chord angle = 60° | The angle between a tangent and a chord equals half the intercepted arc. So 120° ÷ 2 = 60°. |
| Angle in semicircle (no input needed) | 90° | By Thales' theorem, any angle inscribed in a semicircle — with its vertex on the circle and its two sides passing through the diameter endpoints — is always a right angle (90°). |
How to use the circle theorems calculator
- Select the theorem type that matches your problem: Inscribed Angle, Central Angle, Angle in Semicircle, Cyclic Quadrilateral, or Tangent-Chord Angle.
- If more than one calculation mode is available for the selected theorem, choose which quantity you want to find.
- Enter the known angle or arc measure in degrees in the input field. For the Angle in Semicircle theorem, no input is needed.
- Click 'Calculate' to see the result along with a brief explanation of the theorem applied.
- Use the example buttons to load pre-set scenarios and verify that you understand how each theorem works before entering your own values.
Circle theorems FAQ
What is the inscribed angle theorem?
The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. If a central angle measures 80°, the inscribed angle subtending the same arc measures 40°. This theorem holds regardless of where on the major arc the vertex of the inscribed angle is placed.
What is Thales' theorem?
Thales' theorem is a special case of the inscribed angle theorem: any angle inscribed in a semicircle — that is, an angle whose two rays pass through the endpoints of the diameter — is always a right angle (90°). Historically it is one of the oldest recorded geometric theorems, attributed to Thales of Miletus around 600 BCE.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided polygon whose four vertices all lie on a single circle. The key property of a cyclic quadrilateral is that each pair of opposite angles sums to 180°. Not every quadrilateral is cyclic; a rectangle is always cyclic, but a general parallelogram is cyclic only if it is a rectangle.
What is the tangent-chord angle theorem?
The tangent-chord angle theorem states that the angle formed between a tangent to a circle and a chord drawn from the point of tangency equals half the arc intercepted by the chord. This is analogous to the inscribed angle theorem but applies when one side of the angle is a tangent rather than a chord.
How are circle theorems used in real life?
Circle theorems are used in engineering and architecture when designing arches, domes, and curved structures. In navigation, they help compute angles between lines of sight. In computer graphics, they are applied in curve fitting and circular arc generation. In astronomy, Thales' theorem is used to determine distances when a triangle is inscribed in a circle whose diameter is a known baseline.
Can inscribed angles exceed 90°?
Yes. If the central angle is between 180° and 360° (i.e., the inscribed angle subtends the minor arc), the central angle exceeds 180°, giving an inscribed angle greater than 90°. However, when the problem refers to the minor arc, the central angle is between 0° and 180°, so the inscribed angle is between 0° and 90°. This calculator handles the full 0°–360° range for the central angle.