Tangent-Secant Angles Calculator
Angles formed by tangents, secants, and chords
Select Configuration
Two Secants from External Point
Angle = ½ × |Far Arc - Near Arc|
Diagram
Angles Formed by Tangents and Secants
When tangents and secants intersect a circle, they form angles related to the intercepted arcs. The formulas depend on whether the intersection is inside, on, or outside the circle.
The Four Cases
Two Secants (External)
Angle = ½ × |Far Arc - Near Arc|
Lines intersect outside the circle
Two Tangents (External)
Angle = ½ × |Major Arc - Minor Arc|
Both lines tangent from same point
Secant-Tangent (External)
Angle = ½ × |Far Arc - Near Arc|
One tangent, one secant from external point
Tangent-Chord (On Circle)
Angle = ½ × Intercepted Arc
Tangent meets chord at point on circle
Summary of Angle Formulas
| Location | Formula |
|---|---|
| Inside circle (chords) | θ = ½(arc₁ + arc₂) |
| On circle (inscribed) | θ = ½ × arc |
| Outside circle | θ = ½|arc₁ - arc₂| |
Frequently Asked Questions
What's the difference between a secant and a chord?
A chord is a segment with both endpoints on the circle. A secant is a line that intersects the circle at two points and extends beyond.
Why do we subtract arcs for external angles?
When the vertex is outside the circle, the angle is formed by the "gap" between the arcs, hence the difference formula.
What is the far arc vs near arc?
The far arc is further from the external point (between the far intersection points). The near arc is between the closer intersection points.
How is tangent-chord different?
The tangent-chord angle has its vertex on the circle, so it's similar to an inscribed angle. The formula is simply half the arc (no subtraction).
Can the angle be greater than 90°?
For external angles, the maximum is less than 90° (when arcs differ by 180°). For tangent-chord, the angle can be up to 90°.