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Tangent Circle Calculator

Calculate tangent line lengths from an external point to a circle, or the external and internal common tangent lengths between two circles.

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How the Tangent Circle Calculator Works

The tangent circle calculator determines the length of tangent line segments using the fundamental geometric principle that a tangent to a circle is always perpendicular to the radius at the point of tangency. This right-angle relationship transforms every tangent-length problem into a straightforward application of the Pythagorean theorem. The calculator supports three modes: tangent length from an external point to a single circle, external common tangent length between two circles, and internal common tangent length between two circles.

The General Formula

All three modes share the same structural form: L = √(d² − k²), where d is a center-to-center or point-to-center distance and k is a radius term that changes by mode. The distance d is always computed from the coordinate distance formula: d = √((x2 − x1)2 + (y2 − y1)2).

Mode 1: Tangent Length from an External Point

In this mode k = r, the single circle's radius. Given an external point P(x1, y1) and a circle centered at C(x2, y2) with radius r, the tangent length is L = √(d² − r²). The radius, the tangent segment, and line PC form a right triangle with hypotenuse d, so the Pythagorean theorem isolates L directly. Example: A point at (0, 0) and a circle centered at (5, 12) with radius 5 give d = √(25 + 144) = √169 = 13, so L = √(169 − 25) = √144 = 12 units.

Mode 2: External Common Tangent Between Two Circles

An external common tangent runs alongside both circles without crossing between them. Setting k = |r1 − r2| gives Lext = √(d² − (r1 − r2)2). This formula is valid when d > |r1 − r2|, meaning one circle does not contain the other. Example: Two circles with radii 7 and 3 whose centers are 10 units apart yield L = √(100 − 16) = √84 ≈ 9.165 units.

Mode 3: Internal Common Tangent Between Two Circles

An internal common tangent crosses through the gap between the two circles. Setting k = r1 + r2 gives Lint = √(d² − (r1 + r2)2), valid only when d > r1 + r2. Extending the example to centers 15 units apart with radii 7 and 3: L = √(225 − 100) = √125 ≈ 11.180 units.

Derivation and Theoretical Basis

The tangent-radius perpendicularity theorem is a cornerstone of Euclidean circle geometry. As detailed in the Baruch College Precalculus tutorial on equations of tangents to circles, the tangent point is the unique location on the circle where the radius and the tangent line meet at exactly 90°. Applying the Pythagorean theorem to the resulting right triangle — hypotenuse d, one leg r — isolates L = √(d² − r²) directly.

For the two-circle cases, the geometry reduces to an equivalent single-right-triangle problem by projecting onto the line connecting the two centers. Dr. Steven Gubkin's research on tangent circles at Cleveland State University demonstrates how the relative positions of mutually tangent circles constrain possible tangent configurations, providing the rigorous geometric foundation for both external and internal formulas.

Practical Applications

  • Mechanical engineering: Computing open and crossed belt lengths over pulleys of different radii, where tangent segments determine the straight-run sections between contact arcs.
  • Civil engineering: Designing horizontal road alignments where straight tangent sections meet circular curve segments at transition points.
  • Robotics: Building Dubins paths — the shortest curves between two directional poses — which consist of circular arcs joined by straight tangent segments.
  • Computer graphics: Blending circular arcs smoothly in vector font outlines and CAD spline curves by computing tangent-point transitions.
  • Surveying: Measuring unobstructed sight-line distances from an observation station past cylindrical structures such as storage tanks or grain silos.

Validity Conditions

The tangent length is real only when the expression under the square root is non-negative. Mode 1 requires d ≥ r. Mode 2 requires d ≥ |r1 − r2|. Mode 3 requires d ≥ r1 + r2. When equality holds exactly, the tangent length is zero, corresponding to a geometric tangency condition — the external point sits on the circle, or the two circles touch at a single point.

Reference

Frequently asked questions

What is the formula for tangent length from an external point to a circle?
The tangent length from an external point to a circle is L = √(d² − r²), where d is the straight-line distance from the external point to the circle's center and r is the radius. For example, a point 13 units from the center with radius 5 gives L = √(169 − 25) = √144 = 12 units. This result follows from the Pythagorean theorem applied to the right triangle formed by the tangent segment, the radius to the tangent point, and the line connecting the external point to the circle's center.
What is the difference between an external and internal common tangent between two circles?
An external common tangent lies outside both circles and does not pass between them; its length is √(d² − (r1 − r2)²). An internal common tangent crosses through the space between the two circles; its length is √(d² − (r1 + r2)²). External tangents exist whenever one circle does not contain the other (d > |r1 − r2|). Internal tangents exist only when the circles are completely separate (d > r1 + r2). Overlapping or internally nested circles have no internal common tangent and the formula produces no real result.
Why must the external point lie outside the circle for the tangent length formula to be valid?
The formula L = √(d² − r²) requires d > r, which holds only when the point is strictly outside the circle. If d equals r, the point is on the circle and the tangent length is zero. If d is less than r, the point is inside the circle, the expression under the square root becomes negative, and no real tangent line can be drawn. The entire derivation depends on the perpendicular right triangle construction, which only exists when a tangent segment can physically reach the circle from an exterior location.
How does the calculator handle two circles with equal radii?
When r1 equals r2, the external common tangent formula simplifies to L = √(d² − 0) = d, so the tangent length equals the full center-to-center distance. The internal common tangent formula becomes L = √(d² − (2r)²). For example, two circles each with radius 1, whose centers are 4 units apart, produce an internal tangent length of √(16 − 4) = √12 ≈ 3.464 units. Enter equal radii in both fields, input the center coordinates, and select the appropriate two-circle mode to compute either result.
What are the most common real-world applications of tangent circle length calculations?
Tangent circle calculations appear across multiple engineering and design disciplines. Mechanical engineers use them to size open and crossed belt drives by computing the straight tangent runs between pulley contact arcs. Civil engineers apply the geometry when laying out horizontal road curves where tangent sections meet circular arcs. Roboticists compute Dubins shortest paths using tangent segments joining circular arcs between waypoints. Computer graphics developers blend arcs in font outlines and CAD profiles using tangent-point geometry. Surveyors also apply tangent calculations to estimate clearance distances from observation points to circular-cross-section structures like storage tanks.
What happens to tangent length when two circles are externally or internally tangent to each other?
When two circles are externally tangent, d = r1 + r2 exactly, so the internal common tangent formula evaluates to √(0) = 0 — the circles touch at one point and no crossing tangent exists as a segment of positive length. When two circles are internally tangent, d = |r1 − r2|, and the external common tangent length equals zero for the same reason. In both boundary cases the calculator returns zero. Any configuration where d falls below these thresholds produces no real tangent of the corresponding type because the geometry becomes impossible.