Free · Mathematical relationships

Tangent Length from an External Point

A radius to a point of tangency is perpendicular to the tangent. The external point’s center distance d must be strictly greater than the positive radius r.

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Formula and method

With radius = 3, length_d = 5, length_tan = 4.

l = √(d²−r²)

Worked example

Enter these known values and leave the other values blank.

Radius of circle (r): 3 cm
Length of side OT (d): 5 cm

Length of tangent (l): 4 cm

Assumptions and limitations

The radius and tangent length are strictly positive in this workflow. The external center distance must exceed the radius; the zero tangent boundary is excluded.
A radius to a point of tangency is perpendicular to the tangent. The external point’s center distance d must be at least the radius r.
Enter enough independent values and clear the quantity being solved. Locks preserve known values; contradictory inputs can make the system inconsistent. Unit controls convert physical quantities rather than merely changing their labels.
Displayed precision is numerical rounding, not an estimate of measurement uncertainty.

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Clear the missing value, enter the independent known values and retain compatible units. A nonunique inverse may require another known value.

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