Length of a Tangent from a Point

Finds the distance from a point outside a circle to where its tangent touches. A tangent meets the radius at a right angle, so the Pythagorean theorem is all that is needed. The angle between the two tangents and the distance between the two touching points are also shown.

From any point outside a circle, exactly two tangents can be drawn. The distance from that point to where a tangent touches follows from the Pythagorean theorem alone.

L = \sqrt{d^2 - r^2}

$L$ is the tangent length, from the point to the touching point
$r$ is the radius of the circle
$d$ is the distance from the centre to the point

The formula is this simple because a tangent meets the radius at a right angle. Joining the centre, the touching point and the original point forms a right triangle whose hypotenuse is $d$ and whose legs are $r$ and $L$ .

It is worth seeing why that angle must be right. A tangent meets the circle at exactly one point. If the angle were not right, the foot of the perpendicular from the centre would fall somewhere else on the line, closer to the centre than $r$ . The line would then pass through the interior and cut the circle twice. Only a right angle avoids that.

Example

Take the defaults: a circle of radius 3, and a point 5 away from its centre.

$\sqrt{5^2 - 3^2}$ is $\sqrt{16}$ , so the tangent length is exactly 4. The familiar 3-4-5 right triangle has appeared.

The distance between the two touching points is $\dfrac{2rL}{d}$ , which is $2 \times 3 \times 4 \div 5$ , or 4.8. The angle between the tangents is $2\arcsin\dfrac{r}{d}$ , and since $\arcsin(0.6)$ is about 36.87 degrees, that angle is about 73.74 degrees.

Connection with the power of a point

Rearranging gives $L^2 = d^2 - r^2 = (d + r)(d - r)$ . The right-hand side is the product of the distances to the two intersection points of the line drawn from the point through the centre.

The power of a point theorem states that whatever direction a line is drawn in, the product of the distances to its two intersections with the circle is the same. A tangent is the limiting case where those two intersections merge into one, so the product becomes $L \times L$ . That is where the two results meet.

Points to watch

No tangent exists from a point inside the circle. From a point exactly on the circle, the point is itself the touching point and the tangent length is 0. This calculator reports both cases rather than returning a number.

When $d$ is very large compared with $r$ , the tangent length is almost equal to $d$ . Seen from far enough away, a circle is indistinguishable from a point.