Tangent of a Circle Calculator

The tangent length L from an external point to a circle equals the square root of (d squared minus r squared), where d is the distance from the external point to the center and r is the radius. This comes from the Pythagorean theorem applied to the right triangle formed by the radius, the tangent segment, and the line to the center.

A tangent line touches a circle at exactly one point. At that point of tangency, the radius drawn to that point is perpendicular to the tangent line. This is a fundamental theorem of circle geometry. If the angle were anything other than 90 degrees, the line would either miss the circle or cut through it as a secant.

Yes. From any external point, both tangent segments drawn to the same circle have equal length. This is the equal tangent theorem. The proof uses congruent right triangles: both triangles share the same hypotenuse (distance to center) and the same leg (radius), so the third sides (tangent lengths) must be equal.

The full angle between two tangents from an external point equals 2 times arcsin(r/d), where r is the radius and d is the distance to the center. For example, with r equal to 5 and d equal to 13, the angle is 2 times arcsin(5/13), which equals approximately 45.24 degrees.

For a circle with center (h, k) and radius r, a tangent line with slope m has equation y = mx + c where c equals (k minus mh) plus or minus r times the square root of (1 plus m squared). The two values of c give the two parallel tangent lines with that slope, one on each side of the circle.

A tangent line touches a circle at exactly one point and lies entirely outside the circle except at that contact point. A secant line intersects a circle at two points, passing through the interior. The discriminant of the quadratic system (circle plus line) equals zero for a tangent and is positive for a secant.

No. A tangent from an internal point is impossible. For a tangent to exist, the formula L equals the square root of (d squared minus r squared) requires d to be greater than r. If d is less than r, the expression under the square root is negative, meaning no real tangent length exists. A tangent is only possible from an external point.

The central angle between the two points of tangency equals 180 degrees minus the full angle at the external point. Equivalently, it equals 2 times arccos(r/d). For r equal to 5 and d equal to 13, this is 2 times arccos(5/13), approximately 134.76 degrees.

The area of the quadrilateral formed by the two tangent segments and the two radii to the tangent points equals the radius times the tangent length (r times L). This quadrilateral has two right angles at the tangent points, so its area is twice the area of the right triangle formed by one radius, one tangent, and the line to the center.

Vertical tangent lines have undefined slope and cannot be represented as y equals mx plus c. For a circle with center (h, k) and radius r, the two vertical tangents are the lines x equals h plus r and x equals h minus r. These are not handled in the slope mode but can be identified directly from the center and radius.

Circle tangents appear in road design (transition curves where straight roads meet circular roundabouts), belt-and-pulley engineering (finding the length of a belt running over two circular pulleys), satellite orbit geometry, and optics (finding tangent rays from a light source to a circular lens or mirror).

The tangent length depends only on the distance from the external point to the center, not on the direction. Any two points at the same distance d from the center have identical tangent lengths. Moving the external point along a circle of radius d centered on the circle center keeps the tangent length constant.