How to Find the Length of a Tangent from an External Point

What formula gives the tangent length?

If P is outside a circle with centre O and radius r, and PT touches the circle at T, then PT = √(OP² − r²). This follows because the radius OT is perpendicular to the tangent at the point of contact. The external-point condition is OP > r.

Worked example

Suppose OP = 13 cm and the radius is 5 cm. Right triangle OTP has hypotenuse OP, so PT² = 13² − 5² = 144. The tangent segment is therefore 12 cm long.

Check by squaring: 5² + 12² = 13². Taking 13 − 5 = 8 cm would calculate the shortest distance from P to the circle along the line through the centre, not the tangent length.

Two tangents from the same point

If PA and PB are tangents from the same external point to the same circle, PA = PB. Both right triangles share OP and have equal radii OA and OB. The same formula consequently gives the same positive tangent length.

This equality does not mean every tangent segment in a diagram is equal. The segments must start at the same external point and end at their points of contact with that circle.

Working backwards and checking the location

If the tangent is 15 cm and the radius is 8 cm, the centre-to-point distance is √(225 + 64) = 17 cm. Here you add the squares because you are finding the hypotenuse.

For a point inside the circle, OP < r and no real tangent can be drawn from it. A point on the circle has a tangent line at that point, but there is no positive segment from the point to itself.

Practice with answers

• OP = 10 cm and r = 6 cm: tangent length 8 cm.
• OP = 25 cm and r = 7 cm: tangent length 24 cm.
• Tangent length 12 cm and radius 9 cm: OP = 15 cm.

Sketch and label the right angle before choosing whether to add or subtract squares.

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