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Tangent Length Calculator

Find the length of the tangent from an external point to a circle.

Result

9.1652

How it is calculated

The tangent touches the circle at a right angle to the radius, so the centre, the point of contact and the external point form a right triangle whose hypotenuse is the distance from the external point to the centre.

t = √(d² − r²)

Here d is the distance from the external point to the circle's centre and r is the radius, both in the same length unit. The result is in that unit too and is shown to 4 decimal places. The tool returns no result when the radius is zero or negative, or when the distance is not strictly larger than the radius, because a point on or inside the circle has no tangent line.

Worked example

Take a point 10 units from the centre of a circle with radius 4. Square both: 10² = 100 and 4² = 16. Subtract: 100 − 16 = 84. Take the square root of 84. The tangent length is 9.1652 units.

Try this example →

Common mistakes

  • Entering the distance to the nearest point on the circle instead of the distance to the centre. The distance must be measured to the centre.
  • Entering the diameter in the radius field, which makes the radius twice too large and the tangent length wrong.
  • Mixing units, for example a distance in centimetres with a radius in millimetres.
  • Entering a distance equal to or smaller than the radius. The point is then on or inside the circle, so there is no tangent and no result.

About this calculator

When a straight line is drawn from a point outside a circle so that it touches the circle at exactly one point, the segment from the external point to that point of tangency is the tangent length, and it follows directly from the fact that a tangent line is always perpendicular to the circle’s radius at the point of contact. That perpendicularity creates a right triangle with the radius, the tangent segment, and the line from the external point to the center, so by the Pythagorean theorem the tangent length equals the square root of (distance to center squared minus radius squared).

This relationship is used in geometry classrooms, in CAD and technical drafting when constructing tangent lines to circular features, and in surveying and machining where tangent lengths determine clearances or transition curves between straight and circular paths.

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