Circle and Tangent Calculator

Tangent from an outside point to a circle

Two tangents can be drawn to a circle from a point outside it. A tangent is perpendicular to the radius at the touch point, so the centre, the outside point and the touch point form a right triangle. Its hypotenuse is the distance d from the point to the centre, one leg is the radius r and the other leg is the tangent length: L = √(d² − r²).

Example: from a point 5 units from the centre of a circle of radius 3, the tangent length is √(25 − 9) = 4 units. The angle between the two tangents is 2 × arcsin(3 / 5) = 73.74 degrees.

Does a line cut the circle?

Compare the distance from the centre to the line with the radius. If it is smaller the line cuts the circle at two points, if equal the line is tangent, and if larger they do not meet. When it cuts, the chord length is k = 2 × √(r² − d²).

Example: the circle of radius 5 centred at the origin meets the line y = x + 1 at (3, 4) and (−4, −3). The chord is 9.899 units long.

Position of two circles

Compare the distance d between the centres with the sum and the difference of the radii. If d exceeds the sum the circles are separate, if it equals the sum they touch externally, between the difference and the sum they cut at two points, equal to the difference they touch internally, and below the difference one lies inside the other.

For two separate circles the external common tangent has length √(d² − (r₁ − r₂)²) and the internal one √(d² − (r₁ + r₂)²).

Where is it used?

The straight belt span between two pulleys, the point where a corner fillet meets the straight edge, the clearance of a pipe passing a tank or a column, and whether two circular areas overlap all come from these calculations.

Limits

Enter all dimensions in one unit. No tangent exists from a point inside the circle. Do not expect exact tangency from measured values: the calculator uses a very tight numerical threshold.