Tangent Circle - Length, Radius, or Distance

A tangent circle is the configuration formed by a circle and a line that touches the circle at exactly one point, called the point of tangency. The segment from that contact point to any external point on the line is a tangent segment, and the radius drawn to the contact point is perpendicular to the tangent line. This calculator works on the right triangle built from the radius r, the tangent length l, and the segment d from the center to the external point, using r^2 + l^2 = d^2 to solve for whichever side you do not already know.

• • Find a tangent length from a radius and an external point: Enter the radius and the distance from the center to the external point, then read off the tangent length.
• • Recover the radius from a tangent and a known distance: When a sketch already has a tangent segment and a center-to-point distance marked, the radius mode solves for the circle that fits.
• • Verify a right triangle on a tangent construction: If a CAD tool or paper sketch claims a tangent is correct, plug the two known sides in and read the d^2 vs r^2 + l^2 check.
• • Plan a circle around a known contact point and offset: The distance mode confirms the center sits exactly r^2 + l^2 away from the external point.

The defining fact is the right angle: the radius drawn to the point of tangency is perpendicular to the tangent line, so r, l, and d always form a right triangle.

If you already know the radius and only need the area and circumference of the same circle, the Circle Calculator handles those values without retyping the geometry.

The calculator reads the two known sides of the right triangle from the form, then applies r^2 + l^2 = d^2 to recover the third side. It also reports d^2, r^2 + l^2, the residual between them, and a check column. Results update as you type or switch modes.

• r: Radius of the circle. Shown in the same length unit as the other sides.
• l: Length of the tangent segment from the tangent point to the external point.
• d: Distance from the center of the circle to the external point on the tangent line.

Tangent length solves l = sqrt(d^2 - r^2); Radius solves r = sqrt(d^2 - l^2); External distance solves d = sqrt(r^2 + l^2). The check column compares d^2 against r^2 + l^2 to 4 decimals. A clean 0 means an exact right triangle, a value under 0.001 means rounding is responsible, and a larger value means the configuration is geometrically inconsistent.

According to Wolfram MathWorld, a tangent line to a circle is perpendicular to the radius drawn to the point of tangency, and the radius, the tangent segment, and the segment from the center to the external point form a right triangle.

According to Math Open Reference, a tangent line to a circle touches the circle at exactly one point, and the radius drawn to that point is perpendicular to the tangent line.

When the line is a chord instead of a tangent and crosses the circle at two points, the Chord Length Calculator handles the same radius-to-distance Pythagorean relationship with a slightly different setup.

Four ideas show up every time a line is tangent to a circle. They are the minimum vocabulary needed to read the right triangle the calculator is solving.

The right angle is the load-bearing fact; everything else follows from the Pythagorean theorem applied to the triangle that the right angle produces.

Once the tangent point and the external point are known, the Arc Length Calculator extends the same circle to an arc or a sector when only a slice of the perimeter is needed.

Five short steps cover the three solve modes and the right-triangle check, from the default 5-12-13 example to an invalid configuration.

1. 1 Pick the solve mode: Use the dropdown to choose whether you want the tangent length l, the radius r, or the external distance d. The page solves the right triangle for whichever side you pick.
2. 2 Enter the two known values: Type the two inputs the mode leaves free. The defaults (5, 12, 13) start you on a clean 5-12-13 right triangle; replace them with the values from your sketch or paper problem.
3. 3 Read the solved value and the right-triangle sums: The primary row shows the value you solved for. The d^2, r^2 + l^2, and residual rows show the supporting arithmetic from the Pythagorean identity r^2 + l^2 = d^2.
4. 4 Check the status row for validity: A residual close to 0 and a 'Valid' status mean the configuration is a real right triangle. 'Invalid' means the external point sits inside the circle or the requested radius is too large; switch mode or change inputs.
5. 5 Reset or swap inputs to compare configurations: Use the Reset button to return to the 5-12-13 example, or change mode and edit a value to confirm that the same right triangle satisfies the identity in every direction.

Try a tangent-length solve with r = 8 and d = 17. The page returns l = sqrt(17^2 - 8^2) = sqrt(289 - 64) = sqrt(225) = 15 units, d^2 = 289, r^2 + l^2 = 289, residual 0, status 'Valid'. The (8, 15, 17) triple is the next Pythagorean triple after 5-12-13, so the residual stays exactly at 0.

When the sketch gives you the diameter and you need the radius for the right triangle, the Circle Diameter Calculator converts the diameter into the r value to plug into the form.

What the calculator returns and how each output helps in a geometry or construction workflow.

• • Solve for any side of the right triangle: A mode switch is enough to solve for the tangent length, the radius, or the external distance, so the same page covers the three common tangent problems.
• • See the residual between d^2 and r^2 + l^2: The check column shows the difference to 4 decimals, so a wrong configuration or rounding slip is obvious before it reaches a sketch.
• • Get an immediate validity flag: If the external point is inside the circle, the page flags the configuration as 'Invalid' instead of returning a complex length.
• • Use any consistent length unit: The calculator works in whatever unit you type, so centimeters, inches, and meters can all share the same form.
• • Rely on a single identity, not a memorised table: Every result comes from r^2 + l^2 = d^2, the same identity used in textbooks.
• • Pair with other circle calculators: Once the radius is in hand, the page links to circle area, circumference, diameter, chord, arc, and equation tools.

These benefits stack in the typical workflow. A student checking a homework answer gets the right triangle check in one pass. A designer laying out a tangent construction gets the third side plus the validity flag, so the geometry is verified before the line is drawn.

If the same circle is described by an equation instead of a tangent construction, the Circle Equation recovers the center and radius from the standard or general form of the same circle.

A few characteristics of the inputs shape the result. Knowing them tells you when the page is returning a real tangent and when it is forced into a degenerate or invalid case.

• • The calculator only handles the right triangle built from r, l, and d; it does not solve for the tangent line's equation in slope-intercept form.
• • Inputs are treated as plain numbers, not symbolic expressions. The page does not parse '5 sqrt(2)' or '12 pi'.

The math is exact, so rounding enters only in the decimal display. The residual column shows how much rounding contributed.

If the configuration is invalid, the page returns 0 for the unknown side and labels the result as 'Invalid' rather than a complex or negative length.

According to Wikipedia, the squared length of the tangent from an external point plus the squared radius equals the squared distance from the center to the external point, which is the form r^2 + l^2 = d^2.

Once the radius is recovered, the Circle Length Calculator uses the same r to find the circumference of the circle without retyping the geometry.