Tangent of a Circle Calculator
Solve tangent acircle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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.