Tangent Length from an External Point
Circles · Class X
From a point at distance d from the centre of a circle of radius r, find the tangent length.
- Tangent length t12
- Is the point outside?1
- Angle between two tangents (deg)45.2397
- Upper semicircle of radius r
- Lower semicircle
Formulas in this lab
- Tangent length t
- Is the point outside?
- Angle between two tangents (deg)
Frequently asked questions
▶What is the tangent length formula?
From an external point at distance $d$ from the centre of a circle of radius $r$, the tangent length is $t = \sqrt{d^2 - r^2}$. This works because the radius is perpendicular to the tangent at the point of contact, forming a right triangle.
▶How do I use this lab?
Slide the radius r and the distance d (with d > r). The lab draws both tangents and shows the tangent length. Try r = 5, d = 13 to see t = 12, a clean Pythagorean triple.
▶Common mistake on this topic
Students forget that the formula needs $d > r$; if $d \le r$, the point is on or inside the circle and no real tangent exists. Also, $\sqrt{d^2 - r^2}$ is not the same as $d - r$.
▶Where is this useful?
Architects designing curved walls calculate how far a straight beam can reach without piercing a circular pillar. In sports, a player throwing a ball tangent to a circular boundary uses the same right-triangle idea.