Tangent of a Circle Calculator
Find the tangent length from an external point or the tangent line equation for any circle.
⭕ What is a Tangent of a Circle?
A tangent of a circle is a straight line that touches the circle at exactly one point, called the point of tangency. At that contact point, the tangent line is perpendicular to the radius drawn to that point. This 90-degree relationship between the radius and the tangent is one of the most fundamental properties in circle geometry and underpins every formula in this calculator.
The most common problem is finding the length of a tangent segment from an external point. If you stand at a point P outside a circle, you can draw two tangent lines to the circle. Each tangent touches the circle at one point, and the segment from P to that contact point is called a tangent length. Both tangent segments from the same external point are always equal in length. This equal-tangent theorem is used constantly in geometric constructions, engineering design, and coordinate geometry proofs.
A second important concept is the tangent line equation. In coordinate geometry, you often need to find the equation of a line that is tangent to a circle and has a specific slope. For any slope m and a circle with center (h, k) and radius r, exactly two parallel tangent lines with that slope exist, one on each side of the circle. Their equations follow from the condition that the perpendicular distance from the center to the line must equal the radius.
Tangent lines appear throughout applied mathematics and engineering. In road design, straight road sections meet circular curves as tangents to ensure smooth transitions. In mechanical engineering, belts running between two circular pulleys follow tangent segments. In optics, rays from a point source that graze a circular mirror travel along tangent lines. This calculator covers both the geometric (length and angle) and algebraic (line equation) aspects of circle tangents in one place.
📐 Formula
Tangent Length: L = √(d² − r²)
L = tangent length from external point to point of tangency
d = distance from external point to circle center
r = circle radius (must satisfy r < d)
Example: r = 5, d = 13 → L = √(169 − 25) = √144 = 12
Angle at External Point: θ = 2 × arcsin(r ÷ d)
θ = full angle between the two tangent lines at external point P
Central Angle = 180° − θ = 2 × arccos(r ÷ d)
Example: r = 5, d = 13 → θ = 2 × arcsin(5/13) ≈ 45.24°
Tangent Line Equation: y = mx + c where c = (k − mh) ± r√(1 + m²)
m = slope of the tangent line
h, k = center of the circle
r = radius of the circle
Two lines exist (one for + and one for −), on opposite sides of the circle
Example: center (0,0), r = 5, m = 1 → c = ±5√2 ≈ ±7.071
📖 How to Use This Calculator
Steps
1
Choose your calculation mode - Select Tangent Length to find the tangent segment from an external point, or Tangent Line Equation to get the algebraic form of a tangent with a chosen slope.
2
Enter the radius and distance - In Tangent Length mode, type or drag the sliders for the circle radius and the distance from the external point to the center. The distance must exceed the radius for a tangent to exist.
3
Read the results - The calculator instantly shows the tangent length, the angle between the two tangents at the external point, the central angle subtended at the circle, and the area of the quadrilateral formed by both tangents and both radii.
4
Switch to equation mode for algebraic tangents - In Tangent Line Equation mode, enter center coordinates (h, k), the radius, and the desired slope m. The calculator returns both parallel tangent line equations and their y-intercepts.
💡 Example Calculations
Example 1 - Classic 5-12-13 Right Triangle
Circle radius 5, external point 13 units from center
1
Tangent length: L = √(13² − 5²) = √(169 − 25) = √144 = 12 units
2
Angle at external point: 2 × arcsin(5/13) = 2 × 22.62° = 45.24°
3
Central angle: 180° − 45.24° = 134.76°
Tangent length = 12 units, angle between tangents = 45.24°
Try this example →Example 2 - 3-4-5 Pythagorean Triple
Circle radius 3, external point 5 units from center
1
Tangent length: L = √(5² − 3²) = √(25 − 9) = √16 = 4 units
2
Angle at external point: 2 × arcsin(3/5) = 2 × 36.87° = 73.74°
3
Quadrilateral area: r × L = 3 × 4 = 12 square units
Tangent length = 4 units, quadrilateral area = 12 sq units
Try this example →Example 3 - 8-15-17 Pythagorean Triple
Circle radius 8, external point 17 units from center
1
Tangent length: L = √(17² − 8²) = √(289 − 64) = √225 = 15 units
2
Angle at external point: 2 × arcsin(8/17) = 2 × 28.07° = 56.15°
3
Central angle: 180° − 56.15° = 123.85°
Tangent length = 15 units, central angle = 123.85°
Try this example →Example 4 - Tangent Line with Slope 2
Circle centered at (1, 2) with radius 3, tangent slope m = 2
1
Offset: r√(1 + m²) = 3 × √(1 + 4) = 3√5 ≈ 6.708
2
Base intercept: k − mh = 2 − 2(1) = 0
3
Line 1: c = 0 + 6.708, so y = 2x + 6.708. Line 2: c = 0 − 6.708, so y = 2x − 6.708.
Tangent lines: y = 2x + 6.7082 and y = 2x − 6.7082
Try this example →❓ Frequently Asked Questions
What is the formula for the tangent length from an external point?
The tangent length L from an external point to a circle equals the square root of (d squared minus r squared), where d is the distance from the external point to the center and r is the radius. This comes from the Pythagorean theorem applied to the right triangle formed by the radius, the tangent segment, and the line from the external point to the center.
Why is the angle between the radius and tangent always 90 degrees?
A tangent line touches a circle at exactly one point. At that point of tangency, the radius drawn to that point is perpendicular to the tangent line by definition. If the angle were anything other than 90 degrees, the line would either miss the circle entirely or cross through it as a secant, intersecting at two points instead of one.
Are the two tangent segments from an external point always equal?
Yes. From any external point, both tangent segments drawn to the same circle are equal in length. This is the equal tangent theorem. The proof uses congruent right triangles: both triangles share the hypotenuse (distance to center) and one leg (radius), so the third sides (tangent lengths) must be equal by the Hypotenuse-Leg theorem.
What is the angle between the two tangents drawn from an external point?
The full angle between two tangents from an external point equals 2 times arcsin(r/d), where r is the radius and d is the distance to the center. For r equal to 5 and d equal to 13, the angle is 2 times arcsin(5/13), approximately 45.24 degrees. The angle decreases as the external point moves farther away and approaches 180 degrees as the point moves toward the circle.
How do I find the equation of a tangent line with a given slope?
For a circle with center (h, k) and radius r, a tangent line with slope m has equation y = mx + c where c equals (k minus mh) plus or minus r times the square root of (1 plus m squared). The two signs give two parallel tangent lines on opposite sides of the circle. Use the Tangent Line Equation mode in this calculator to get both equations instantly.
What is the difference between a tangent and a secant of a circle?
A tangent line touches a circle at exactly one point and lies entirely outside except at that contact point. A secant line passes through the interior and intersects the circle at two distinct points. The discriminant of the system of equations (circle plus line) is zero for a tangent and positive for a secant. A secant can be thought of as a tangent that has been moved toward the center until it cuts through.
Can a tangent be drawn from a point inside the circle?
No. If the external point is inside the circle (d is less than r), the expression under the square root in L = sqrt(d squared minus r squared) becomes negative. This means no real tangent length exists. Every line through an interior point will intersect the circle at two points, making every such line a secant rather than a tangent.
What is the central angle between the two tangent points?
The central angle between the two points of tangency equals 180 degrees minus the angle at the external point. It can also be computed as 2 times arccos(r/d). For r equal to 5 and d equal to 13, this is 2 times arccos(5/13), approximately 134.76 degrees. The central angle and the angle at the external point always sum to 180 degrees.
What is the area enclosed by the two tangent segments and the two radii?
The area of the quadrilateral formed by the two tangent segments and the two radii (the quadrilateral has vertices at the external point and the center, with right angles at the two tangent points) equals the radius times the tangent length (r times L). For the 5-12-13 example this is 5 times 12 equal to 60 square units.
What are the real-world applications of tangent length calculations?
Circle tangents appear in road and rail design (tangent sections meeting circular curves), belt-and-pulley calculations in mechanical engineering (the belt length between two pulleys requires tangent segments), satellite orbit geometry, optics (rays grazing a circular lens), and nautical navigation (computing tangent distances to circular obstacles or safe passage distances around a buoy).
Does the tangent length depend on the direction of the external point?
No. The tangent length depends only on the distance d from the external point to the center and the radius r, not on the direction. Any external point located at distance d from the center has the same tangent length regardless of which direction from the center it lies. Moving the external point along a circle of radius d centered on the circle center keeps the tangent length constant.
How do vertical tangent lines fit into the formula?
Vertical tangent lines have undefined slope and cannot be expressed as y = mx + c. For a circle with center (h, k) and radius r, the two vertical tangents are the lines x = h + r and x = h minus r. These are a special case not covered by the slope formula, but they can be read directly from the center and radius values. This calculator handles all slopes except vertical lines.