Tangent of a Circle Calculator

The tangent length is t = √(d² - r²), where d is the distance from the external point to the center and r is the circle radius. This comes from the Pythagorean theorem: the tangent is perpendicular to the radius at the point of tangency, forming a right triangle.

This is a fundamental theorem of circle geometry. The tangent line touches the circle at exactly one point, and at that point, the radius drawn to the point of tangency is perpendicular to the tangent. This creates a right angle, which is why the Pythagorean theorem applies.

If the point is on the circle (d = r), the tangent length is 0. The tangent line still exists and is perpendicular to the radius at that point, but the "length from the point to the tangency" is zero since the point of tangency is the point itself.

If d < r, the formula gives t = √(d² - r²) = √(negative number), which is not real. There is no tangent line from a point inside the circle. Tangent lines can only be drawn from points on or outside the circle.

Exactly two tangent lines can be drawn from any external point to a circle. They are symmetric about the line connecting the external point to the center, and both have the same length.

The angle between the two tangent lines is 2θ, where θ is the angle each tangent makes with the line to the center. Since cos(θ) = r/d, the full angle is 2 × arccos(r/d).

The point of tangency lies on the circle at the foot of the perpendicular from the center to the tangent line. Using coordinates: if the center is at (0,0) and the external point is at (d, 0), the tangent touches at (r²/d, ±r√(d²-r²)/d).

The power of a point P with respect to a circle is PA × PB (where A and B are the intersection points of any line through P with the circle). For a tangent, the power equals t² = d² - r². The tangent length squared equals the power of the point.

The tangent at point (x₀, y₀) on circle x² + y² = r² is x₀x + y₀y = r². For a general circle (x-h)² + (y-k)² = r², the tangent at (x₀, y₀) is (x₀-h)(x-h) + (y₀-k)(y-k) = r².

The tangent concept extends to 3D: from an external point to a sphere, the tangent lines form a cone. The tangent length formula t = √(d² - r²) still applies, where d is the 3D distance from the point to the sphere center.

The tangent-secant theorem states that if a tangent and a secant are drawn from an external point, then t² = PA × PB, where t is the tangent length and A, B are the secant intersection points with the circle. This is equivalent to the power of a point.

Tangent lines are used in optics (light ray reflection), navigation (great circle routes), computer graphics (tangent vectors for surface rendering), and engineering (belt and pulley system design).

The formula t = √(d² - r²) still works as long as d is the distance from the external point to the actual center. The coordinate shifts do not affect the tangent length calculation.

For a point (x₁, y₁) outside circle x² + y² = r², the tangent lines have the form y - y₁ = m(x - x₁) where m satisfies the condition that the line touches the circle at exactly one point. Solve the quadratic discriminant = 0 for m.

The angle the tangent line makes with the positive x-axis depends on the geometry. For a tangent from (d, 0) to circle at origin: the tangent angle with the x-axis is ±arctan(r/√(d²-r²)). The two tangent lines are symmetric about the x-axis.

Yes. From the equation (x-h)² + (y-k)² = r², extract the center (h, k) and radius r. Compute d as the distance from the external point to (h, k). Then t = √(d² - r²).

Given two circles, find the lines tangent to both. These are called common tangents. There can be up to 4 common tangents (2 direct, 2 transverse) depending on the relative positions and sizes of the circles.

The angle between a tangent and a chord drawn from the point of tangency equals half the intercepted arc. This is the tangent-chord angle theorem, a key result in circle geometry that connects tangents to inscribed angles.

The calculator uses full floating-point precision (about 15 decimal places). For most practical purposes, 2-4 decimal places suffice. The formula t = √(d² - r²) is exact; any rounding is in the display, not the calculation.

Step 1: Find the midpoint M of the segment from the center O to the external point P. Step 2: Draw a circle with center M and radius OM = MP. Step 3: The intersections of this circle with the original circle are the points of tangency. Step 4: Draw lines from P to these points.

The Tangent of a Circle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Tangent of a Circle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.