Library / Geometry, coordinates, surveying and trigonometry / GEO-10
Geometry, coordinates, surveying and trigonometry
Tangent length from an external point
Length of the tangent from a point outside a circle to the point where it touches, from the distance to the centre and the radius.
- Ref
- GEO-10
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- PT [cm]
A tangent touches a circle at one point, and the radius to that point is perpendicular to it. The centre, the touching point and the external point P form a right triangle with hypotenuse d and one leg r, so PT² = d² − r². Factorising gives (d − r)(d + r). When d = r the point is on the circle and the tangent has zero length.
Length of the tangent from a point outside a circle to the point where it touches, from the distance to the centre and the radius. The radius to the touching point is perpendicular to the tangent.
Calculator
- Angle at the centre between the radius and the line to P67.38 °
- Angle between the two tangents from P45.24 °
- Length of the pair of tangents24 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- From 13 cm away, a tangent to a 5 cm circle touches after 12 cm; the two tangents from the point meet at 45.24°.
- The classic 5–12–13 triangle appears here when d = 13 and r = 5 give a tangent of 12.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d Distance from the point to the centre | 5 to 40 cm | 0 | 39.69 | 39.69 | +10% | +11.6 % |
| r Radius | 1 to 12.9 cm | 1.609 | 12.96 | 11.35 | +10% | −1.84 % |
Sweep of Distance from the point to the centre
| Distance from the point to the centre (cm) | Tangent length (cm) |
|---|---|
| 5 | 0 |
| 9.375 | 7.93 |
| 13.75 | 12.81 |
| 18.13 | 17.42 |
| 22.5 | 21.94 |
| 26.88 | 26.41 |
| 31.25 | 30.85 |
| 35.63 | 35.27 |
| 40 | 39.69 |
Sweep one variable
Variables and how each one contributes
Tangent length
What the equation gives you. Everything else feeds it.
Distance from the point to the centre
The straight-line distance from the external point to the centre of the circle, in cm. It must be at least r. A larger distance gives a longer tangent, almost one for one when d is much bigger than r. Raising it by 10 % (to 14.3) moves the result by +11.6 %, from 12 to 13.4 cm.
Radius
The radius of the circle, in cm. A bigger circle shortens the tangent for the same distance. Raising it by 10 % (to 5.5) moves the result by −1.84 %, from 12 to 11.78 cm.
Worked example with the values above
- Factors: d − r = 13 − 5 = 8 and d + r = 18.
- Product: 8 × 18 = 144.
- PT = √144 = 12 cm.
Assumptions and limits
- A true circle, with angles in radians where the formula uses them.
- Lengths are measured from the same centre, in one consistent unit.
Common mistakes
- Entering a central angle in degrees where radians are needed.
- Using the full chord in place of the half chord when working with the right triangle.
- Using the major arc when the minor arc is meant.
Statistics and uncertainty
Circle results depend on the radius squared or on trigonometric functions of the angle, so errors in the radius are doubled in areas. The segment and rise formulas can lose digits for very small angles or very shallow arcs.
Where it is used
- Cutting, rolling and setting out curved plates, pipes and arches.
- Partly filled pipes and tanks, washers and flanges, and tangent lines past a circular object.
Maths you need
- Radians: angle measured as arc length over radius
- Right triangle: Pythagoras and sine over hypotenuse
- Circle area: π times radius squared