Last updated October 13, 2026 · Report an error
A tangent line touches a circle at exactly one point, never crossing into its interior — a concept that shows up constantly in geometry, engineering (like belt-and-pulley systems), and design work involving circular shapes. This calculator handles two related but distinct problems: finding the length of a tangent line segment drawn from an external point to a circle, and checking whether a given line is tangent to a circle, a secant (crossing through it at two points), or entirely non-intersecting. Both problems rely on the same core geometric fact — that a tangent line is always perpendicular to the circle's radius at the exact point of tangency — which is what makes both calculations solvable with straightforward distance formulas rather than needing calculus or iterative approximation.
How it works
Choose which calculation you need. For tangent length, enter the circle's center coordinates and radius, along with the external point you're measuring from. Because the tangent line, the radius to the point of tangency, and the line from the external point to the center form a right triangle (with the tangent segment and radius as legs, and the distance to the center as the hypotenuse), the calculator applies the Pythagorean-theorem-derived formula L = √(d² − r²), where d is the distance from the external point to the circle's center and r is the radius. For the tangency check, enter the circle and a line in the standard form Ax + By + C = 0, and the calculator computes the perpendicular distance from the circle's center to that line, then compares it against the radius to classify the line.
- Enter what to calculate.
- Enter center x.
- Enter center y.
- Enter radius (r).
- Enter external point x.
- Enter external point y.
- Enter line A (Ax + By + C = 0).
- Enter line B.
- Enter line C.
- Click Calculate to see your results.
Examples
Circle at origin, r=5, point (13,0)
Distance from (13,0) to the center (0,0) is 13, so the tangent length is √(13² − 5²) = √(169 − 25) = √144 = 12.
A closer external point
For a circle centered at (0,0) with radius 5 and an external point at (6,8), the distance to center is √(6²+8²) = √100 = 10, giving a tangent length of √(10² − 5²) = √75 ≈ 8.66.
Checking a line's relationship to a circle
For a circle centered at (0,0) with radius 5, a line at perpendicular distance exactly 5 from the center is tangent (touching at one point); a line at distance 3 is a secant (crossing at two points); a line at distance 8 doesn't intersect the circle at all.
Who should use it
- Geometry homework involving tangent lines and circles.
- Engineering problems involving belts, pulleys, or circular clearances.
- Verifying whether a designed line or edge just touches a circular boundary.
Industry applications
- Geometry education
- Mechanical engineering (belt and pulley design)
- CAD and technical drawing
Advantages
- Solves both tangent-length and tangency-classification problems in one tool.
- Uses exact, formula-based geometry rather than approximation.
- Clearly reports whether a line is tangent, secant, or non-intersecting.
Limitations
- Requires the point to be outside the circle for a valid tangent-length result.
Common mistakes to avoid
- Trying to compute a tangent length from a point that's actually inside the circle.
- Confusing a secant line (crossing through the circle at two points) with a tangent line (touching at exactly one point).
- Forgetting that two tangent lines (not one) exist from any external point, both of equal length.
Best practices
- Confirm the point is genuinely outside the circle (distance to center greater than the radius) before computing tangent length.
- Use the perpendicular-distance-to-line formula carefully, matching the line's Ax + By + C = 0 coefficients precisely.
- Sanity-check a tangent length result using the Pythagorean relationship L² + r² = d².
Tips
- Remember there are always two tangent lines (of equal length) from any external point, not just one.
- Use the Pythagorean relationship L² + r² = d² as a quick way to verify your result.