Tangent to Circle Calculator

Circle: (x − h)² + (y − k)² = r². Let the point be P(x1, y1). Define dx = x1 − h, dy = y1 − k, and distance d = √(dx² + dy²).

• If d < r, the point is inside, so no real tangents exist.
• If d = r, one tangent exists at the point on the circle.
• If d > r, two tangents exist, with tangent length ℓ = √(d² − r²).

Touchpoints are computed in translated coordinates, then shifted back by (h, k). The angle between tangents is θ = 2·asin(r/d) in degrees.

1. Enter the circle center values h and k.
2. Enter the radius r (must be positive).
3. Enter the point coordinates x1 and y1.
4. Click Calculate to view tangents and touchpoints.
5. Use Download CSV or Download PDF to export results.

Circle and external point setup

This calculator models a circle using center (h, k) and radius r, then evaluates a point P(x1, y1). It first computes dx = x1 − h and dy = y1 − k, so the distance d = √(dx² + dy²) is immediate. Comparing d with r classifies the geometry as inside, on, or outside the circle. These inputs determine the tangent outcome very reliably.

When tangents exist and how many

If d < r, the point is internal and no real tangents exist. If d = r, the point lies on the boundary and a single tangent is defined at that location. If d > r, exactly two tangents exist, symmetric about the line from the center to P.

Tangent length and angular separation

For an external point, the tangent length is ℓ = √(d² − r²). This value is useful for quick feasibility checks in design and construction problems where clearance matters. The angle between tangents is θ = 2·asin(r/d), reported in degrees for interpretation and plotting. As d grows larger than r, ℓ approaches d and θ becomes small, matching intuition.

Touchpoint coordinates from stable vectors

Touchpoints are computed without iterative solvers. In translated coordinates, the point vector is (dx, dy). The calculator forms a weighted combination of that vector and its perpendicular (−dy, dx). Two touchpoints result from adding and subtracting the perpendicular component, then shifting back by (h, k). This approach is numerically stable even when the point is far from the circle.

Line equations and slope reporting

Each tangent is returned in Ax + By + C = 0 form, which avoids division by zero and supports vertical lines naturally. The slope is reported as −A/B when B ≠ 0; otherwise the slope is marked undefined. This representation also supports accurate PDF and CSV exports. Coefficients are rounded for display, yet computations use full precision internally.

How to interpret the graph and exports

The plot shows the circle, the external point, the touchpoints, and tangent segments extended across the view window. Use the CSV to store inputs and computed metrics, and the PDF to share a clean summary in reports. Recheck units, especially when r is not in meters. If you need a longer visual line, zoom the plot; the equation remains unchanged.

1) What inputs does the calculator require?

Enter the circle center (h, k), radius r, and the point (x1, y1). The tool classifies the point and computes tangents when they exist.

2) Why do I sometimes see no tangents?

If the point is inside the circle, real tangents do not exist because every line through the point intersects the circle at two points.

3) What does “undefined slope” mean?

An undefined slope indicates a vertical tangent line. The equation is still valid in Ax + By + C = 0 form.

4) How is tangent length calculated?

For an external point, tangent length is √(d² − r²), where d is the distance from the point to the circle’s center.

5) Are the touchpoints always accurate?

Yes, touchpoints are computed using a closed-form vector method. It avoids iterative solving and remains stable for large coordinates.

6) What should I export: CSV or PDF?

Use CSV for spreadsheets and further calculations. Use PDF when you need a clean, shareable summary of equations, touchpoints, and key metrics.