Tangent of a Circle Calculator

When a line is tangent to a circle, it intersects the circle at exactly one point. A fundamental geometric theorem states that the radius drawn to the point of tangency is always perpendicular to the tangent line. This forms a right-angled triangle where the hypotenuse is the distance (d) from the external point to the circle's center, one leg is the radius (r), and the other leg is the length of the tangent segment (tan). Using the Pythagorean theorem, the relationship is expressed as:

length_tan = sqrt(length_d^2 - radius^2)

Where length_d is the distance from the external point to the center, and radius is the distance from the center to the outer edge of the circle.

Imagine you are solving a geometric design problem where you need to find the length of a tangent segment. Let us assume the distance from an external point to the circle's center (length d) is 13 units, and the circle's radius is 5 units. First, square the distance: 13 squared equals 169. Next, square the radius: 5 squared equals 25. Subtract the squared radius from the squared distance: 169 minus 25 equals 144. Finally, take the square root of 144, which gives you a tangent length of 12 units.

Always ensure that your distance measurement (d) is taken from the center of the circle to the external point, not from the circle's outer edge. Double-check that your units of measurement match consistently for both the radius and the distance d. If you encounter a negative value under the square root, it indicates an impossible geometric configuration where the external point lies inside the circle rather than outside.