Circle Theorems Calculator — Inscribed Angle, Thales, Cyclic Quadrilateral, Chords, Secants & Tangent

The circle theorems calculator is the perfect tool to understand the multiple geometric relationships between the parameters of a circumference and external parameters, such as tangent or secant lines 📐. We've grouped all of the most important circle theorems in this simple calculator for you to better understand and practice this subject.

We also paired this calculator with a short text describing, but not limited to, the following:

• What the circle theorems are;
• What the inscribed angle theorem formula is;
• The exterior angle of a circle theorem, also called the intersecting secants theorem;
• How to find the angles in a cyclic quadrilateral;
• The tangent to a circle formula; and
• More!

What are the circle theorems?

The circle theorems are a way to explain many mathematical properties and relationships between circles and all kinds of angles and line segments you can form with them. These theorems are usually taught to high school students as a way to simplify more complex geometry theorems.

In this circle theorems calculator, we've gathered six of them to aid in your studies. Depending on who you ask, there might be a couple more theorems, but these are the ones that are easier to explain with a calculator. In this tool, you will find:

1. The inscribed angle theorem;
2. The Thales theorem;
3. The cyclic quadrilateral theorem;
4. The equidistant chords theorem;
5. The intersecting secants theorem; and
6. The tangent to a circle theorem.

Inscribed angle theorem ─ what is the inscribed angle theorem?

The inscribed angle theorem states:

• The inscribed angle is half the central angle which subtends the same arc; and
• Angles on the circumference subtended by the same arc are equal.

For example, if θ_i is an angle subtended by an arc (both within the circumference of the circle), and we measure the angle subtended by the same arc from the center of the circle (θ_c), it will be twice the first angle. Also, if another angle on the circumference were to subtend the same arc as θ_i, according to the theorem, they would be equal.

You can either manually use the inscribed angle theorem formula or input the angle in our circle theorems calculator using any unit. Give it a try!

Thales theorem

Thales theorem states that if A, B, and C are three points on the circumference of a circle, and AC lies on its diameter, then the angle ∠ABC is a right angle.

This is especially useful if we pair it with the Pythagorean theorem, because it allows us to find the missing angle or side. Since we already know the value of one angle (90°) and the hypotenuse (the diameter), we can easily find the rest if we know the value of one additional side or angle by doing a bit of arithmetic:

where d is the diameter (hypotenuse) and a and b are the two legs of the right triangle.

Cyclic quadrilateral theorem ─ how to find angles in a cyclic quadrilateral

What is a cyclic quadrilateral? A cyclic quadrilateral is simply any quadrilateral whose vertices lie on the circumference of a circle.

The cyclic quadrilateral theorem says that any two opposite angles in a cyclic quadrilateral add up to 180°.

So if you know two adjacent angles, you can immediately find the remaining two: each opposite angle equals 180° minus the given one.

Equidistant chords theorem

This theorem is straightforward: chords equidistant from the center of a circle are equal in length. This means that if you take any two chords of a circle that are equally separated from its center, they will also be equal in length. The reverse also applies: if two chords are equal, they will be equidistant from the circle's center.

The length of a chord at a distance d from the center of a circle of radius r is:

Intersecting secants theorem ─ exterior angle of a circle theorem

When two secant lines are drawn from a single external point, the angle formed between them is related to the two arcs they cut out of the circle. The exterior angle is equal to half the difference of the intercepted arcs:

Here, arc_far is the larger (farther) intercepted arc and arc_near is the smaller (nearer) intercepted arc.

Tangent to a circle theorem ─ tangent to a circle formula

A tangent is a line that touches a circle at exactly one point and is perpendicular to the radius at that point of contact. Because of this right angle, the radius, the tangent, and the line from the external point to the center form a right triangle, so the tangent length from a point a distance d from the center is:

For a real tangent to exist, the point must lie on or outside the circle (d ≥ r).

How to use this calculator

1. Choose a theorem from the drop-down list — inscribed angle, Thales, cyclic quadrilateral, equidistant chords, intersecting secants, or tangent to a circle.
2. Pick your unit system — metric (cm) or US/Imperial (in). This applies to the length-based theorems (Thales, chords, tangent). The angle-based theorems work purely in degrees.
3. Enter the required values for your chosen theorem.
4. The calculator instantly returns the result, the related quantities, and the formula used.

Unit Systems

Metric System

Enter lengths in centimeters (cm). Angles are always given in degrees (°).

US / Imperial System

Enter lengths in inches (in). Angles are always given in degrees (°).

FAQs

What is the inscribed angle theorem formula?

The inscribed angle equals half of the central angle subtending the same arc: θ_i = θ_c / 2. Equivalently, the central angle is twice any inscribed angle on the same arc.

Why is the angle in a semicircle always 90°?

This is Thales's theorem. Any triangle whose hypotenuse is a diameter of a circle and whose third vertex lies on the circle is a right triangle, with the right angle at that third vertex.

How do you find a missing angle in a cyclic quadrilateral?

Opposite angles of a cyclic quadrilateral sum to 180°. So a missing angle equals 180° minus the angle opposite to it.

Are two chords equidistant from the center always equal?

Yes. Chords equidistant from the center of a circle are equal in length, and equal chords are always equidistant from the center.

What is the exterior angle of a circle?

When two secants meet at a point outside the circle, the angle between them equals half the positive difference of the two intercepted arcs: (arc_far − arc_near) / 2.

How many tangents can be drawn from an external point?

Exactly two tangents of equal length can be drawn from a point outside the circle, one from a point on the circle, and none from a point inside it.