How long is the tangent?
The length of a tangent is measured from the outside point P to the point of contact.
Tangent length
From an outside point P, a tangent touches the circle at a point T. The piece PT is the part of the tangent between P and the circle.
Now move P away from the circle.
As P moves farther from the circle, what happens to the length PT?
What this lesson covers
The idea
The length of the tangent from an external point P to a circle is the length of the segment of the tangent between P and the point of contact.
Tangent length
From an outside point P, a tangent touches the circle at a point T. The piece PT is the part of the tangent between P and the circle.
Now move P away from the circle.
As P moves farther from the circle, what happens to the length PT?
- It gets longer
- It gets shorter
- It stays the same
Set the length
The orange segment PT₁ is the length of the tangent. Drag P so that PT₁ is the length asked for. The circle has the radius shown.
From P to the point of contact
The length of the tangent from an external point P to a circle is the length of the segment of the tangent between P and the point of contact.
The radius OT is perpendicular to the tangent at T, so triangle OTP is a right triangle. That gives PT² = OP² − r². A radius of 4 cm and OP = 5 cm gives PT = 3 cm. A radius of 12 cm and OP = 13 cm gives PT = 5 cm.
Notes
The length of the tangent from an external point P is the length of the segment of the tangent between P and the point of contact: PT² = OP² − r².
Check yourself
What is the length of the tangent from an external point P to a circle?
A circle has radius 6 cm, and P is a point 10 cm from the centre. What is the length of the tangent from P (in cm)?
Answer: 8
PT² = OP² − OT² = 10² − 6² = 100 − 36 = 64, so PT = 8 cm.
A circle has radius 7 cm, and P is a point 25 cm from the centre. What is the length of the tangent from P (in cm)?
Answer: 24
PT² = 25² − 7² = 625 − 49 = 576, so PT = 24 cm.
A circle has a fixed radius. P moves farther from the centre. What happens to the length of the tangent from P?
- The distance from P to the centre. That is OP. The tangent length stops at the circle, at the point of contact.
- The length of the segment from P to the point of contact — correct. Yes! It is the part of the tangent between P and the point of contact.
- The radius of the circle. The radius is the length OT. The tangent length is PT.
- It stays the same. PT² = OP² − r². If OP grows and r is fixed, PT grows.
- It gets shorter. PT² = OP² − r². If OP grows and r is fixed, PT does not shrink.
- It gets longer — correct. Yes! PT² = OP² − r², so a larger OP gives a longer tangent.