Circle Tangent Length Calculator – From External Point to Circle
A tangent line from an external point to a circle is a straight line that touches the circle at exactly one point, called the tangency point. From any point located outside a circle, you can always draw exactly two such tangent lines, and a remarkable property of Euclidean geometry guarantees that both segments are of equal length. This calculator computes that tangent length instantly using the Pythagorean theorem.
The Tangent Length Formula
When you connect the external point P, the circle's center O, and the tangency point T, you form a right triangle. The radius OT is always perpendicular to the tangent line PT at the point of tangency. This right-angle relationship is the cornerstone of the calculation.
Using the Pythagorean theorem on triangle OPT:
d² = t² + r²
Solving for the tangent length t:
t = √(d² − r²)
where d is the distance from the external point to the circle's center and r is the circle's radius. The constraint d > r must hold — if the point is on or inside the circle, no real tangent line can be drawn.
Tangent Angle and the Full Opening Angle
In addition to the tangent length, this calculator computes the tangent angle θ— the angle at the external point between the line PO and each tangent line:
θ = arcsin(r / d)
Because there are two equal tangent lines from P, they form a symmetric "V" shape opening toward the circle. The full angle between both tangent lines is simply 2θ. As the external point moves farther from the circle (d increases), this angle narrows; as it approaches the circle surface, the angle widens toward 180°.
Diameter Input Mode
Many practical problems specify the diameter of a pipe, wheel, or column rather than its radius. The Diameter input mode lets you enter the full width of the circle directly. The calculator halves it internally to derive the radius before computing the tangent length. The constraint becomes d > diameter / 2.
Real-World Applications
The circle tangent length formula appears in a surprising variety of practical contexts:
- Engineering clearance — calculating the shortest distance a cable, pipe, or beam must reach around a circular obstacle from a fixed anchor point.
- Surveying and road design — computing tangent distances to circular curves on horizontal alignments, a standard calculation in highway geometry.
- Astronomy and orbital mechanics — a spacecraft or light ray skimming the surface of a planet follows a path that is tangent to the planetary sphere; the same formula applies.
- Optics — the apparent angular diameter of a spherical object is twice the tangent angle θ, which is derived from exactly this formula.
- Machine design — belt-drive geometry uses tangent lengths to determine the straight portion of a belt between two pulleys of different sizes.
Step-by-Step Example
Suppose an external point P is 13 cm from the center of a circle with radius 5 cm. To find the tangent length:
- Check validity:
d = 13 > r = 5✓ - Apply the formula:
t = √(13² − 5²) = √(169 − 25) = √144 = 12 cm - Tangent angle:
θ = arcsin(5 / 13) ≈ 22.62° - Full opening angle:
2θ ≈ 45.24°
This is a classic Pythagorean triple (5, 12, 13), which is why the answer is a whole number. In general, the result will be an irrational number, and the precision slider controls how many decimal places are shown.
Why Are Both Tangent Lengths Always Equal?
This follows directly from the symmetry of the right triangles formed. Both triangles OPT₁ and OPT₂ share the hypotenuse OP = d and one leg OT₁ = OT₂ = r (both radii). By the Pythagorean theorem, the remaining sides must be equal: PT₁ = PT₂ = t. This symmetry is the basis for the tangent construction in classical geometry and is used to prove many theorems about circles and inscribed polygons.
Multi-Unit Support
The calculator accepts inputs in millimetres, centimetres, metres, kilometres, inches, feet, yards, and miles. All values are converted to a common base unit for computation, so the unit you enter for d and r is the same unit you receive for t. Switch the unit selector at any time — the result updates immediately.