Tangent of a Circle Calculator
Enter the radius of a circle and the distance from the centre to an external point. This tool finds the tangent length using the Pythagorean theorem, shows the angle between the two tangent lines drawn from that point, and also solves two-circle tangent problems (external and internal common tangent lengths). Results update instantly as you type, and the show-your-work panel walks through every step.
What is a tangent to a circle?
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. The key geometric property is that the radius drawn to the point of tangency is always perpendicular to the tangent line, forming a 90-degree angle. This perpendicularity is the foundation for every formula this calculator uses. A secant line, by contrast, crosses the circle at two points; a chord connects two points on the circle but does not extend beyond it.
How to find the length of a tangent from an external point
When a point P lies outside a circle with centre O and radius r, two tangent lines can be drawn from P to the circle. By the two-tangent theorem, both tangent segments have equal length. Because the radius OT is perpendicular to the tangent PT at the contact point T, the three points O, T and P form a right triangle with the right angle at T. Applying the Pythagorean theorem gives PT squared equals OP squared minus OT squared, so the tangent length L equals the square root of (d squared minus r squared), where d is the distance from O to P. This quantity is also called the square root of the power of the point P with respect to the circle.
Angle between the two tangent lines
Because both tangent segments from P have the same length and the geometry is symmetric about the line OP, the half-angle alpha at P satisfies sin(alpha) equals r divided by d. The full angle between the two tangent lines is therefore 2 times arcsin(r / d). When P is very far from the circle d is much larger than r, alpha approaches zero and the two tangents are nearly parallel. As P approaches the circle surface d approaches r, alpha approaches 90 degrees and the angle between tangents approaches 180 degrees. At d exactly equal to r the point is on the circle, alpha is 90 degrees and there is only one tangent.
Common tangent lines between two circles
When two circles are given, common tangent lines touch both circles simultaneously. External common tangents lie outside both circles (both circles are on the same side), and internal common tangents cross between the circles. For two circles with radii r1 and r2 and centre-to-centre distance D, the external tangent length is the square root of (D squared minus (r1 minus r2) squared), and the internal tangent length is the square root of (D squared minus (r1 plus r2) squared). Internal tangents exist only when D is greater than r1 plus r2, meaning the circles do not overlap. This geometry is the basis of the engineering belt and pulley problem: the external tangent gives the straight-run length of an open belt, and the internal tangent gives the straight-run length of a crossed belt.
Tangent configuration summary
| Position of P | Condition | Number of tangent lines | Tangent length |
|---|---|---|---|
| Outside circle | d > r | 2 | L = √(d² − r²) |
| On circle | d = r | 1 | L = 0 |
| Inside circle | d < r | 0 | No real tangent |
| Two circles - external | Any D | 2 | L = √(D² − (r1 − r2)²) |
| Two circles - internal | D > r1 + r2 | 2 | L = √(D² − (r1 + r2)²) |
| Two circles - no internal | D ≤ r1 + r2 | 0 | No real internal tangent |
Key geometric cases for tangent lines relative to a circle.
Frequently asked questions
What is the tangent length formula?
The tangent length from an external point P to a circle with centre O and radius r is L = sqrt(d^2 - r^2), where d is the distance from P to O. This follows directly from the Pythagorean theorem applied to the right triangle formed by O, the contact point T and P, with the right angle at T.
Can I draw a tangent from a point inside the circle?
No. If the external point P is inside the circle, d is less than r, and d^2 - r^2 is negative, so there is no real square root and no real tangent. You need at least one real tangent line, and that requires the point to be on or outside the circle.
Why are both tangents from an external point equal in length?
Both tangent segments form congruent right triangles with the two radii to the contact points and the line from P to the centre. Because the hypotenuse (OP) and one leg (the radius r) are the same for both triangles, the other leg (the tangent length) must also be equal.
What is the power of a point?
The power of a point P with respect to a circle is defined as d^2 - r^2, where d is the distance from P to the centre and r is the radius. For an external point it equals the square of the tangent length. For any chord through P, the product of the two distances from P to the chord endpoints also equals the power of the point, regardless of the chord direction.
When do external and internal tangent lines exist for two circles?
External common tangents always exist when the two circles are not one inside the other (D > |r1 - r2|). Internal common tangents exist only when the circles are external to each other (D > r1 + r2). When D equals r1 + r2 the circles are externally tangent and share exactly one internal tangent. When D is less than r1 + r2 the circles overlap and internal tangents do not exist.
What is the belt problem and how does it use tangent lengths?
The belt problem asks for the total length of a belt or rope that fits tightly around two circular pulleys. The belt has two straight sections connecting the pulleys and two curved arc sections wrapping around each pulley. Each straight section is an external common tangent (open belt) or an internal common tangent (crossed belt). The tangent length formula gives the straight-run length; add the arc lengths computed from the wrapping angles to get the total belt length.