Tangent Circle Calculator

Background.

From a point outside a circle you can draw exactly two straight lines that just graze it, one on each side. Each touches the circle at a single point, and the stretch of line from your point to that touch point is the tangent length. The two are always equal, which is why a cord pulled taut from a peg around both sides of a round post leaves the post at matching distances.

The length follows from one fact. A radius drawn to the point of contact is perpendicular to the tangent there. So the circle's centre, the point of contact and your outside point form a right-angled triangle, with the line from the centre to your point as its hypotenuse. Pythagoras finishes the job: the tangent length squared equals the distance to the centre squared, minus the radius squared.

The best-known use of that triangle is the horizon. Your eye is the outside point, the Earth is the circle, and the tangent length is how far you can see across a calm sea before the surface curves away. With the Earth's mean radius of 6,371 km and eyes 1.7 m above the water, the geometric horizon is about 4.65 km off, and atmospheric refraction stretches the visible horizon a little beyond that.

You supply two numbers in the same unit: the distance from the point to the centre of the circle, and the radius. The answer comes back in that unit. The point has to lie outside the circle, so the distance must be larger than the radius; a point on the circle has a tangent length of zero, and from a point inside, every line cuts the circle twice.

What is tangent circle calculator?

The tangent length is the distance along a tangent line from an external point to the point where that line touches the circle. Every external point has two such tangents, mirror images of each other across the line to the centre, and their lengths are equal. Squared, the tangent length is also the power of the point with respect to the circle, d² − r², which equals the product of the two distances along any secant drawn from the point through the circle.

How to use this calculator.

1. 01Work out the distance from the outside point to the circle's centre, not to its edge. If you know the gap to the nearest point of the circle instead, add the radius: a point 8 units from the edge of a radius-5 circle is 13 from the centre.
2. 02Enter the radius in the same unit. If you were given a diameter, halve it first.
3. 03Keep the distance larger than the radius. A value equal to or below it is refused, because the point would sit on the circle or inside it.
4. 04Read the tangent length in the unit you used. It is the length of each of the two tangents from the point.

The formula.

t = √(d² − r²). Here d is the distance from the outside point P to the centre O, r is the radius and t is the distance from P to the point of contact T. The radius OT meets the tangent at T at a right angle; if it did not, the line would dip into the circle instead of touching it. Triangle OTP therefore has its right angle at T and its hypotenuse along OP, so Pythagoras gives d² = r² + t², and rearranging gives the formula. Requiring d to exceed r keeps the quantity under the square root positive. As P moves away, t approaches d, because the radius becomes a small part of the triangle; as P closes in on the circle, t shrinks quickly toward zero. The same quantity d² − r² equals PA × PB for any line from P that cuts the circle at A and B.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.