Circle Tangent and Secant Calculator
Circle theorems with numbers: the length of a tangent from an outside point, the angle between two tangents, the tangent–secant theorem, intersecting chords and a chord's length from its distance to the centre.
About the Circle Tangent and Secant Calculator
A tangent touches a circle at one point and is perpendicular to the radius there, so a tangent from a point at distance d from the centre has length √(d² − r²). The two tangents from one point are equal.
If a secant from P cuts the circle at A and B, then PT² = PA × PB; if two chords cross at P, AP × PB = CP × PD.
The formula
PT = √(d² − r²); PT² = PA × PB; AP × PB = CP × PD
Worked example
Example: Length of each tangent = 8
- Inputs: I want: Length of a tangent from an outside point; Radius r: 6; Distance of the point from the centre: 10.
- Formula: PT = √(d² − r²); PT² = PA × PB; AP × PB = CP × PD
- Tangent² = d² − r²: (10)^2-(6)^2 = (64)
- Result: Length of each tangent = 8
- Tangent²: 64
- Angle between the tangents: 73.7398°
Questions people ask
Why are the two tangents equal?
The two right triangles OTP and OT′P share the hypotenuse OP and have equal radii, so they're congruent (RHS).
How long is the tangent from a point on the circle?
There's only one tangent from a point on the circle, and its length from the point is 0.