Tangent of a Circle Calculator (tangent length, tangent points & line equations)

Understanding the tangent to a circle calculator — concept and principle

This calculator finds the tangent lines from a point to a circle. Given the circle's centre and radius and a point P, it gives the tangent length, the distance from P to the centre, both tangent points, the equations of both tangent lines and the angle between them. Use it for coordinate geometry homework, CAD, belt and pulley layouts and line-of-sight problems.

The radius meets each tangent at a right angle, so the tangent length is √(d² − r²) = 10.

How it is calculated

With centre C = (h, k), radius r and P at distance d from C:

Tangent length PT = √(d² − r²).
Tangent points: T = (h + r·cos(φ ± α), k + r·sin(φ ± α)), where φ is the direction of P from C and α = arccos(r ÷ d).
Tangent line at T: (Tx − h)(x − h) + (Ty − k)(y − k) = r², shown as y = mx + b (or x = c if vertical).
Angle between the two tangents = 2·arcsin(r ÷ d).

A point on the circle has one tangent; a point inside has none.

The principle

A tangent touches the circle at exactly one point and is perpendicular to the radius there. From an outside point, the centre, the tangent point and P form a right triangle with the right angle at T. The hypotenuse is CP = d, one leg is the radius r, so the other leg, the tangent length, is √(d² − r²) by Pythagoras. There are two such triangles, mirror images across the line CP, which is why an outside point always has two tangents of equal length.

Worked example

Default inputs: centre (0, 0), r = 5, P = (10, 5).

d = √(10² + 5²) = √125 ≈ 11.1803.
Tangent length = √(125 − 25) = √100 = 10.
Tangent point 1 = (0, 5); its tangent line is y = 5.
Tangent point 2 = (4, −3); its tangent line is 4x − 3y = 25, or y = 1.33333x − 8.33333.
Angle between the tangents = 2·arcsin(5 ÷ 11.1803) ≈ 53.13°.

Each point is the calculator re-run with only “Point P: x” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Point P: x	Tangent length from the point
2.5	2.5
5	5
7.5	7.5
10	10
12.5	12.5
15	15
20	20

How to read the result

Both tangents from the same point have the same length. Check each tangent point: it should be exactly r from the centre, and the tangent line should pass through P. A small angle between the tangents means P is far from the circle; the angle approaches 180° as P moves close to the circle.

Common mistakes

Using d + r or d − r instead of √(d² − r²).
Forgetting the second tangent.
Applying the formula to a point inside the circle, which has no tangents.
Writing the tangent line through the centre rather than perpendicular to the radius.

Keep in mind: For belts, pulleys and mechanical layouts, include clearances and tolerances; the ideal geometry assumes perfectly round parts and zero-thickness lines.

Frequently asked questions

Why is a tangent perpendicular to the radius?

The radius to T is the shortest distance from the centre to the line; the shortest segment from a point to a line is perpendicular to it.

Are the two tangent lengths from a point always equal?

Yes. The two right triangles are congruent.

What is the tangent line at a point on the circle?

The line through that point perpendicular to the radius: (x₀ − h)(x − h) + (y₀ − k)(y − k) = r².

Sources: OpenStax, Prealgebra 2e, §9.3 (Pythagorean theorem); circle tangent–radius theorem (Euclid, Elements, Book III, Proposition 18)

Checked 9 Nov 2026. Standard theorem of circle geometry: a tangent is perpendicular to the radius at the point of tangency; with the Pythagorean theorem this gives the tangent length √(d² − r²) from an external point. Tangent points and line equations derived here from that right triangle (standard analytic geometry).

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