How this calculator works
A tangent touches a circle at exactly one point, and the radius drawn to that point is perpendicular to the tangent. That right angle creates a right triangle with three sides: the radius r from the centre to the touching point, the tangent length l from the external point to the touching point, and the distance d from the centre to the external point. The distance is the hypotenuse. Knowing any two sides gives the third through the Pythagorean theorem. The calculator also reports the two acute angles of the triangle. The angle between the tangent and the line from the external point to the centre is opposite the radius, and the angle at the centre is its complement.
Formula
The right triangle gives d² = r² + l². Therefore d = sqrt(r² + l²), l = sqrt(d² - r²), and r = sqrt(d² - l²). The angle between the tangent and the line to the centre is arcsin(r/d), and the angle at the centre is 90 degrees minus that angle.
Worked example
A circle of radius 4 m has a tangent of length 3 m from an external point. The distance from that point to the centre is sqrt(16 + 9) = 5 m. The angle between the tangent and the line to the centre is arcsin(4/5) ≈ 53.13 degrees, leaving about 36.87 degrees at the centre. If the distance is 5 m and the tangent is 4 m, the radius is 3 m.
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Frequently asked questions
Why is the radius perpendicular to the tangent?
That perpendicularity is the defining property of a tangent: the shortest distance from the centre to the tangent line is the radius, and the shortest path meets the line at a right angle. Without it the line would cut the circle at two points instead of touching once.
Can the tangent be longer than the distance to the centre?
No. The distance is the hypotenuse of the right triangle, so it is always the longest side. In the radius mode, a tangent longer than the distance is rejected.
What happens when the external point lies on the circle?
Then the distance equals the radius and the tangent length is zero. The tangent line still touches the circle at that point, but the segment from the point to the touching point has no length.
Is this the same as the tangent-secant theorem?
They are related but not identical. This page uses one tangent and the radius to the touching point. The tangent-secant theorem uses two lines from an external point, one tangent and one secant, and states tangent² = external part × whole secant.