Welcome to the Circle Theorems Calculator. This tool helps you solve geometry problems involving circles, angles, arcs, chords, tangents, and cyclic quadrilaterals. Select the theorem you want to use, enter the values you know, and the calculator will find the missing measurements instantly.
Circle Theorems Calculator
Result:
0°
Related Tools:
Understanding Circle Theorems
Circle theorems are rules that describe the relationships between angles, lines, and arcs in circles. They are essential for solving geometry problems in exams, construction, and design.
1. Angle at the Center Theorem
The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.
Formula: Angle at center = 2 × Angle at circumference
2. Angle in a Semicircle Theorem
The angle inscribed in a semicircle is always a right angle (90 degrees).
Formula: If the diameter is the base, the angle at the circumference = 90°
3. Angles in the Same Segment Theorem
Angles in the same segment of a circle are equal.
Formula: Angle A = Angle B (if both stand on the same arc)
4. Cyclic Quadrilateral Theorem
The opposite angles of a cyclic quadrilateral sum to 180 degrees.
Formula: Angle A + Angle C = 180°, Angle B + Angle D = 180°
5. Tangent and Radius Theorem
The tangent to a circle is perpendicular to the radius at the point of contact.
Formula: Angle between radius and tangent = 90°
6. Alternate Segment Theorem
The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
Formula: Angle between tangent and chord = Angle in opposite segment
7. Chord Theorem (Perpendicular from Center)
The perpendicular from the center of a circle to a chord bisects the chord.
Formula: If a line from center is perpendicular to a chord, it cuts the chord into two equal parts.
8. Intersecting Chords Theorem
When two chords intersect inside a circle, the products of the segments of each chord are equal.
Formula: AE × EB = CE × ED
9. Tangent Secant Theorem
When a tangent and a secant intersect outside a circle, the square of the tangent length equals the product of the whole secant and its external segment.
Formula: (Tangent)² = Whole Secant × External Segment
Quick Reference Table
| Theorem | Rule | Formula |
|---|---|---|
| Angle at Center | Center angle = 2 × circumference angle | θ_center = 2 × θ_circumference |
| Angle in Semicircle | Angle is always 90° | θ = 90° |
| Same Segment | Angles on same arc are equal | θ₁ = θ₂ |
| Cyclic Quadrilateral | Opposite angles sum to 180° | A+C=180°, B+D=180° |
| Tangent & Radius | Tangent ⟂ radius | Angle = 90° |
| Alternate Segment | Angle = angle in opposite segment | θ_tangent_chord = θ_alternate |
| Chord Perpendicular | Perpendicular bisects chord | AM = MB |
| Intersecting Chords | Product of segments equal | a×b = c×d |
| Tangent Secant | Tangent² = secant × external | t² = s × e |
Frequently Asked Questions
A circle theorem is a mathematical rule that describes how angles, lines, and arcs behave inside and around circles.
There are eight main circle theorems taught in school geometry. This calculator covers nine including variations.
Circle theorems help solve real world problems in engineering, architecture, navigation, and design.
This calculator handles the most common circle theorem problems. For complex cases, you may need additional steps or multiple theorems.
Yes, completely free. No registration or payment required