Tangent of a Circle Calculator
Solve tangent acircle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Tangent of a Circle Calculator
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Formula: Tangent Length = sqrt(d^2 - r^2)
Worked example — Tangent Length = 12 | Power of Point = 144 | Angle = 45.24 deg
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 Tangent length = sqrt(d^2 - r^2) = sqrt(169 - 25) = sqrt(144) = 12 Power of point = 13^2 - 5^2 = 169 - 25 = 144 Angle 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) Radius slope = sin(30)/cos(30) = tan(30) = 0.5774 Tangent slope = -1/tan(30) = -sqrt(3) = -1.7321 Tangent equation: y - 2.5 = -1.7321*(x - 4.3301) y = -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.
How are tangent lines used in calculus?
In calculus, the tangent line concept extends from circles to any smooth curve. The derivative of a function at a point gives the slope of the tangent line at that point, making tangent lines foundational to differential calculus. The tangent line provides the best linear approximation of a curve near the point of tangency. For a circle defined implicitly by x^2 + y^2 = r^2, implicit differentiation gives dy/dx = -x/y, which is the tangent slope at any point. This connection between geometry and calculus was crucial in the development of both fields. Newton and Leibniz both used tangent line problems as motivation for developing the calculus.
What is the angle between two tangent lines from an external point?
When two tangent lines are drawn from an external point P to a circle of radius r, the angle between them depends on the distance d from P to the center. The half-angle between the tangents equals arcsin(r/d), so the full angle between the two tangent lines is 2*arcsin(r/d). As the point moves farther from the circle, the angle decreases toward zero (the tangent lines become nearly parallel). As the point approaches the circle, the angle approaches 180 degrees. At distance d = r*sqrt(2), the angle between tangents is exactly 90 degrees. This relationship is used in surveying, optics, and mechanical engineering for designing cam profiles and gear tooth shapes.
How do tangent circles work and what is tangency between circles?
Two circles are tangent to each other when they touch at exactly one point. External tangency occurs when the circles touch on the outside (distance between centers equals the sum of radii: d = r1 + r2). Internal tangency occurs when one circle touches the other from inside (distance equals the absolute difference: d = |r1 - r2|). At the point of tangency, both circles share the same tangent line. Tangent circles are fundamental in the study of Apollonius circles and Descartes circle theorem. In engineering, tangent circles appear in gear design, ball bearing arrangements, and Wankel engine geometry. The concept extends to spheres in three dimensions.
What are the properties of tangent segments from an external point?
Tangent segments drawn from the same external point to a circle have several important properties. First, both tangent segments have equal length, which can be proven using congruent triangles (the two right triangles formed share the hypotenuse and have equal legs equal to the radius). Second, the line from the external point to the center bisects the angle between the two tangent lines. Third, the line from the external point to the center also passes through the midpoint of the chord connecting the two tangent points. Fourth, the tangent segments and the chord form an isosceles triangle. These properties are used extensively in geometric constructions, circle packing problems, and the design of smooth curves in CAD software.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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