Tangents from a Point to a Circle
Find the tangent contact points and tangent length from an entered point to a circle. Distinguish interior, boundary and exterior points.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Compare squared center-to-point distance with radius squared. An exterior point has two tangents, a boundary point one and an interior point none. At each contact the radius is perpendicular to the tangent.
Tangent length=√(d²−r²); exterior contact vectors=(r²/d²)p±[r√(d²−r²)/d²]perp(p)
Worked example
For a circle centered at(0,0), radius5 and external point(13,0), both tangent segments have length12. Contact points are(25/13,±60/13).
Assumptions and input details
Coordinate inputs accept fractions or decimals up to six places, with magnitude≤1e6 and reduced denominator≤1e6. All coordinates share one abstract coordinate unit. Classification can be exact while displayed coordinates and diagrams are approximate. Radius must be positive. For a point on the circle the tangent line exists with zero segment length. Very distant points can make the circle visually small on the common-scale diagram.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Interior, boundary and exterior classification uses exact fractions. Tangent contacts, lengths and the diagram are approximate.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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