Tangent of a Circle Calculator
Solve tangent acircle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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The tangent length from an external point at distance d from the center of a circle with radius r equals the square root of (d squared minus r squared). The tangent line is always perpendicular to the radius at the point of tangency.
Last reviewed: December 2025
Worked Examples
Example 1: Tangent Length from External Point
Example 2: Tangent Line at a Point on the Circle
Background & Theory
The Tangent of a Circle Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Tangent of a Circle Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
Tangent Length = sqrt(d^2 - r^2)
The tangent length from an external point at distance d from the center of a circle with radius r equals the square root of (d squared minus r squared). The tangent line is always perpendicular to the radius at the point of tangency.
Worked Examples
Example 1: Tangent Length from External Point
Problem: Find the tangent length from point (13, 0) to a circle centered at the origin with radius 5.
Solution: Distance from point to center = sqrt(13^2 + 0^2) = 13\nTangent length = sqrt(d^2 - r^2) = sqrt(169 - 25) = sqrt(144) = 12\nPower of point = 13^2 - 5^2 = 169 - 25 = 144\nAngle between tangents = 2*arcsin(5/13) = 2*22.62 = 45.24 degrees
Result: Tangent Length = 12 | Power of Point = 144 | Angle = 45.24 deg
Example 2: Tangent Line at a Point on the Circle
Problem: Find the tangent line equation at the point where angle = 30 degrees on a circle of radius 5 centered at origin.
Solution: Point on circle: (5*cos(30), 5*sin(30)) = (4.3301, 2.5)\nRadius slope = sin(30)/cos(30) = tan(30) = 0.5774\nTangent slope = -1/tan(30) = -sqrt(3) = -1.7321\nTangent equation: y - 2.5 = -1.7321*(x - 4.3301)\ny = -1.7321x + 10
Result: Tangent at (4.33, 2.5): y = -1.732x + 10.000
Frequently Asked Questions
What is a tangent line to a circle?
A tangent line to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius drawn to that same point. Unlike a secant line which intersects the circle at two points, a tangent makes contact at precisely one location and does not cross into the interior of the circle. The word tangent comes from the Latin 'tangere' meaning 'to touch.' Every point on a circle has exactly one tangent line, and from any external point, exactly two tangent lines can be drawn to the circle. Tangent lines are fundamental in calculus, physics, and engineering.
How do you find the length of a tangent from an external point?
The tangent length from an external point to a circle is found using the Pythagorean theorem. If the external point is at distance d from the center of a circle with radius r, the tangent length is L = sqrt(d^2 - r^2). This works because the tangent line, the radius to the point of tangency, and the line from the center to the external point form a right triangle, with the tangent and radius meeting at 90 degrees. For example, from a point 13 units from the center of a circle with radius 5: L = sqrt(169 - 25) = sqrt(144) = 12 units. Both tangent lines from the same external point have equal length, which is a fundamental property of circle geometry.
Why is the tangent perpendicular to the radius at the point of tangency?
The perpendicularity of the tangent and radius is a fundamental theorem in Euclidean geometry that can be proved by contradiction. Assume the tangent is not perpendicular to the radius at point P. Then there would be a shorter distance from the center to some point Q on the tangent line (the perpendicular distance). But since Q is on the tangent line and outside the circle, the distance OQ must be greater than r. This creates a contradiction because if the tangent were not perpendicular, the perpendicular from the center would be shorter than the radius, placing a point of the tangent inside the circle. Since the tangent cannot enter the circle, it must be perpendicular to the radius.
What is the power of a point with respect to a circle?
The power of a point is a number that measures the relationship between a point and a circle. For a point P at distance d from the center of a circle with radius r, the power is d^2 - r^2. If P is outside the circle, the power is positive and equals the square of the tangent length. If P is on the circle, the power is zero. If P is inside the circle, the power is negative. The power of a point has a remarkable property: for any line through P that intersects the circle at points A and B, the product PA * PB equals the absolute value of the power. This invariance makes the power of a point a fundamental concept in projective geometry.
How do you find the equation of a tangent line to a circle?
To find the equation of a tangent line to a circle x^2 + y^2 = r^2 at a point (x1, y1) on the circle, use the formula x*x1 + y*y1 = r^2. For a general circle (x-h)^2 + (y-k)^2 = r^2, the tangent at point (x1, y1) is (x-h)*(x1-h) + (y-k)*(y1-k) = r^2. Alternatively, find the slope of the radius from center to the point (which is (y1-k)/(x1-h)), then the tangent slope is the negative reciprocal: -(x1-h)/(y1-k). Use point-slope form to write the line equation. From an external point, you may need to solve a system of equations to find the tangent points first, then compute each line equation.
What are common tangent lines between two circles?
Two circles can have common tangent lines that touch both circles. External common tangents do not pass between the circles, while internal common tangents cross between them. Two separate circles have 4 common tangents (2 external, 2 internal). Two externally tangent circles have 3 common tangents. Two overlapping circles have 2 common tangents (both external). Two internally tangent circles have 1 common tangent. Concentric circles have 0 common tangents. The number of common tangents depends on the relationship between the distance between centers and the sum and difference of radii. Common tangents are used in engineering for belt and pulley systems and in computational geometry for visibility calculations.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy