How this instrument works
A tangent line touches a circle at exactly one point and, at that single point of contact, meets the radius drawn to it at a perfect right angle — this is the tangent-radius perpendicularity theorem, and it is the whole reason this calculator works with nothing more than the Pythagorean theorem. Draw the radius to the point of tangency, draw the line from the external point to the center, and the tangent segment itself: those three segments close into a right triangle every single time, no matter where the external point sits or how large the circle is.
Because that triangle is always right-angled, its hypotenuse is the distance from the external point to the center (d), one leg is the radius (r), and the other leg is exactly the tangent length you want. The Pythagorean theorem d² = r² + length² rearranges to length = √(d² − r²) — no trigonometry needed, just the same relationship that ties together the sides of any right triangle. Two tangent lines can always be drawn from a single external point, one to each side of the circle, and by symmetry both have this identical length.
The formula also marks its own edge cases cleanly. When d equals r exactly, the point sits on the circle itself, the 'tangent' has shrunk to a single point, and length comes out to zero. If d were ever smaller than r, the point would be trapped inside the circle where no tangent line exists at all — the square root would need a negative number, which is the calculator's way of saying the geometry has broken down.
- Enter how far the outside point sits from the circle's center into Distance from external point to center.
- Enter the circle's radius into Circle radius.
- Read the result in Tangent length — the length of the line from your point to where it just grazes the circle.
- Keep Distance from external point to center at least as large as Circle radius; a smaller distance means the point is inside the circle and no tangent exists.
Worked example — a 13-unit sightline to a 5-unit circle
A surveyor stands 13 units from the center of a circular plot whose radius is 5 units and needs to know how far along the ground a tangent sightline runs before it grazes the boundary. The right triangle has hypotenuse d = 13 and one leg r = 5, so length = √(13² − 5²) = √(169 − 25) = √144 = 12 exactly — the familiar 5-12-13 triple showing up inside a circle problem rather than a plain triangle.
A second point 10 units from a circle of radius 6 gives length = √(10² − 6²) = √(100 − 36) = √64 = 8, the 6-8-10 triple, which is just the 3-4-5 triple doubled. Whenever d and r happen to be the two shorter legs of a known Pythagorean triple, the tangent length lands on a whole number, which makes these two cases handy for checking that a calculator — or a hand sketch — is set up correctly before trusting it on messier figures.
Questions
Why is the radius always perpendicular to a tangent line?
Because the radius to the point of tangency is the shortest possible segment from the center to the tangent line — any other segment from the center to that line would have to be longer, since the point of tangency is the line's single closest point to the center. A shortest-distance segment to a line always meets it at 90°, which is exactly the tangent-radius theorem.
What if the distance to the center is smaller than the radius?
Then the point lies inside the circle, and no tangent line exists — every line through an interior point crosses the boundary twice rather than touching it once. The formula reflects this: d² − r² goes negative and the square root has no real result, which is the calculator's signal that the inputs describe an impossible geometry.
How many tangent lines can be drawn from one external point?
Exactly two, one touching each side of the circle, and by the symmetry of the figure both have the identical length given by √(d² − r²). The two points of tangency, together with the external point and the center, form a kite whose diagonal — the line from the external point to the center — bisects the angle between the two tangents.
How is this different from the arc length or chord length between two points on a circle?
Arc length and chord length both measure distances between two points that sit on the circle itself. Tangent length measures a straight segment from a point completely outside the circle to where a grazing line just touches the boundary — a different geometric setup that uses the Pythagorean theorem rather than an angle-based arc or chord formula.
Does the formula work if the circle radius is entered as zero?
Yes — with r = 0 the 'circle' collapses to a single point at the center, and length = √(d² − 0) = d, meaning the tangent length is simply the straight-line distance to that point, since a line from any external point to a single point is trivially tangent to it.