Circle Theorems Calculator
Select a circle theorem, enter the known values, and the calculator finds every missing angle or length instantly. Six theorem modes are covered: the inscribed angle theorem, Thales theorem (angle in a semicircle), the cyclic quadrilateral theorem, the equidistant chords theorem, the intersecting secants theorem, and the tangent-radius theorem. Each result panel shows the formula, a step-by-step worked solution, and a circle diagram.
Formula
Worked example
Inscribed angle example: central angle = 80 deg, radius = 5 units. Inscribed angle = 80/2 = 40 deg. Arc length = 5 * (80 * pi/180) = 5 * 1.3963 = 6.9813 units.
What are circle theorems?
Circle theorems are a set of geometric rules that describe exact relationships between angles, arcs, chords, tangents, and secants inside and around circles. They are foundational results in Euclidean geometry, proven from a few basic axioms, and appear in everything from engineering design to navigation. The six theorems covered here are the most commonly needed: the inscribed angle theorem, Thales theorem, the cyclic quadrilateral theorem, the equidistant chords theorem, the intersecting secants theorem, and the tangent-radius theorem.
Inscribed angle theorem and Thales theorem
The inscribed angle theorem states that an angle formed by two chords that meet on the circumference is exactly half the central angle that subtends the same arc. If the central angle is 80 degrees, the inscribed angle is 40 degrees, no matter where on the arc the vertex is placed. Thales theorem is the special case where the chord is the diameter: the central angle becomes 180 degrees, so the inscribed angle becomes 90 degrees. Any triangle drawn inside a circle with one side as the diameter will always have a right angle at the third vertex. This is a powerful result that appears throughout geometry and trigonometry.
Cyclic quadrilateral, chord, secant, and tangent theorems
A cyclic quadrilateral has all four vertices on the circumference. Its opposite angles always sum to 180 degrees, because both pairs of opposite angles are inscribed angles that together subtend the full 360-degree circle. The equidistant chords theorem uses Pythagoras: the perpendicular from the centre bisects any chord, so if you know the chord length and radius you can find the distance from the centre with d = sqrt(r^2 - (chord/2)^2). Equal chords in the same circle are always the same distance from the centre. For intersecting secants drawn from an external point, the angle at that external point equals half the difference of the two intercepted arcs. Finally, the tangent-radius theorem says the radius drawn to the point of tangency is always perpendicular to the tangent line, giving a 90-degree angle, from which you can find the distance to the centre and the tangent length using Pythagoras.
How to use this calculator
Select the theorem you need from the dropdown. Only the inputs relevant to that theorem are shown. Enter the values you know and the calculator computes all the missing angles and lengths immediately, along with a step-by-step worked solution. For the inscribed angle theorem, enter the central angle and radius to get the inscribed angle and arc length. For Thales theorem, enter the diameter and one leg to get the other leg and the two non-right angles. For cyclic quadrilaterals, enter two adjacent angles to find the opposite pair. For the chord theorem, enter the chord length and radius to find the distance from the centre. For intersecting secants, enter both intercepted arcs to find the external angle. For the tangent-radius theorem, enter the radius and tangent length to find the distance to the centre and the angle at the external point.
Circle Theorems Summary
| Theorem | Key relationship | Formula |
|---|---|---|
| Inscribed Angle | Inscribed angle = half the central angle subtending the same arc | theta_i = theta_c / 2 |
| Thales' Theorem | Any angle inscribed in a semicircle is a right angle | theta = 90 deg |
| Cyclic Quadrilateral | Opposite angles in a cyclic quadrilateral are supplementary | A + C = 180 deg; B + D = 180 deg |
| Equidistant Chords | Distance from centre to chord: apply Pythagoras with half-chord and radius | d = sqrt(r^2 - (AB/2)^2) |
| Intersecting Secants | External angle = half the difference of intercepted arcs | theta = (far arc - near arc) / 2 |
| Tangent-Radius | Radius to point of tangency is perpendicular to the tangent | theta = 90 deg at tangency point |
Key relationships between angles, arcs, chords, and tangents in circle geometry.
Frequently asked questions
What is the inscribed angle theorem?
The inscribed angle theorem states that an angle formed at the circumference of a circle by two chords is exactly half the central angle that intercepts the same arc. If the central angle is 100 degrees, the inscribed angle is 50 degrees. Crucially, the inscribed angle stays the same regardless of where on the arc the vertex is placed, as long as it stays on the same side of the chord.
What does Thales' theorem say?
Thales' theorem is a specific case of the inscribed angle theorem. It says that any angle inscribed in a semicircle (that is, the angle at a point on the circumference where the two chords span the full diameter) is always exactly 90 degrees. This is because the diameter subtends a central angle of 180 degrees, and half of 180 is 90. In practice this means any triangle inscribed in a circle with the hypotenuse as the diameter is a right triangle.
What is special about angles in a cyclic quadrilateral?
In a cyclic quadrilateral (a four-sided polygon with all vertices on a circle), each pair of opposite angles adds up to exactly 180 degrees. This property is also called supplementary opposite angles. If one angle is 75 degrees, the opposite angle must be 105 degrees. The sum of all four angles is still 360 degrees, which is true of any quadrilateral, but the 180-degree pairing is unique to cyclic quadrilaterals.
How do I find the distance from the centre to a chord?
The perpendicular from the centre of a circle to a chord always bisects the chord (cuts it into two equal halves). This creates a right triangle with the radius as the hypotenuse, half the chord as one leg, and the distance from the centre as the other leg. You can then use Pythagoras: distance = sqrt(radius^2 - (chord/2)^2). For example, a chord of length 8 units in a circle of radius 5 units: distance = sqrt(25 - 16) = sqrt(9) = 3 units.
Why is a tangent always perpendicular to the radius at the point of tangency?
A tangent touches the circle at exactly one point. If it were not perpendicular to the radius at that point, you could draw a shorter line from the centre to the tangent, which would be closer to the centre than the radius, making the centre inside the tangent line and allowing a second intersection point, contradicting the definition of a tangent. The perpendicularity can also be proved formally from the definition of a circle as the set of points equidistant from the centre.
What is the intersecting secants theorem?
When two secants (lines that cross a circle at two points) are drawn from a single external point, they intercept two arcs on the circle. The angle at the external point equals half the difference between the two intercepted arc measures: external angle = (far arc - near arc) / 2. For example, if the far arc is 120 degrees and the near arc is 40 degrees, the external angle is (120 - 40) / 2 = 40 degrees.