GEOMETRYCircleMathematics Calculator
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Tangent to a Circle

A tangent touches the circle at exactly one point and is perpendicular to the radius at that point. From an external point, two tangents of equal length can be drawn—essential for gear design, optics, and coordinate geometry.

Concept Fundamentals
At point of tangency
Tangent ⊥ radius
m = -(px-h)/(py-k)
Slope at point
L = √(d² - r²)
Length (external)
Equal length from external
Two tangents

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Tangent is perpendicular to radius at the point of tangency—proven by contradiction. Two tangents from an external point have equal length. Used in gear teeth design—contact force acts along the tangent.

Key quantities
At point of tangency
Tangent ⊥ radius
Key relation
m = -(px-h)/(py-k)
Slope at point
Key relation
L = √(d² - r²)
Length (external)
Key relation
Equal length from external
Two tangents
Key relation

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Why: Tangents appear in gear design (contact perpendicular to radius), optics (light rays), and navigation (sight lines). The tangent-radius theorem is foundational in circle geometry.

How: At point on circle: tangent slope m = -(px-h)/(py-k). From external point: tangent length L = √(d²-r²) where d = distance to center. Tangent is always perpendicular to radius.

Tangent is perpendicular to radius at the point of tangency—proven by contradiction.Two tangents from an external point have equal length.
Sources:Khan Academy - Tangent LinesWolfram MathWorld - Tangent

Run the calculator when you are ready.

Start CalculatingEnter circle center, radius, and point to find tangent properties
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TANGENT GEOMETRY

Tangent Line to Circle Calculator

Enter circle center (h,k), radius r, and point (px,py). Get tangent slope, equation, and length (from external point).

Circle →Equation →

◔ Real-World Examples — Click to Load

Input Dimensions

tangent_calc.sh
CALCULATED
$ tangent_circle --center=(0,0) --radius=5 --point=(3,4)
Distance
5
units
Position
on
Tangent Slope
-0.75
Tangent Length
—
units
Tangent Equation
y = -0.75x + 6.25
Share:
Tangent of Circle
C(0,0), r = 5 | P(3,4)
y = -0.75x + 6.25
Position: on
numbervibe.com/calculators/mathematics/circle/tangent-of-circle

Tangent Properties Radar

Property Comparison

Distance Breakdown

Step-by-Step Breakdown

INPUT
Circle center
(0, 0)
INPUT
Radius
5 units
INPUT
Point
(3, 4)
INPUT
Distance from center
d = 5 units
d = √((px-h)² + (py-k)²)
RESULT
Point position
on
RESULT
Tangent slope
-0.75
m = -(px-h)/(py-k)
RESULT
Tangent equation
y = -0.75x + 6.25

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

⊥

Tangent is perpendicular to the radius at the point of tangency.

— Tangent-radius theorem

—

From external point: tangent length L = √(d² - r²).

— Pythagorean

📋 Key Takeaways

  • • A tangent touches the circle at exactly one point
  • • Tangent is perpendicular to the radius at the point of tangency
  • • Tangent slope at point (px,py): m=−fracpx−hpy−km = -\\frac{px-h}{py-k} (when py ≠ k)
  • • Tangent length from external point: L=sqrtd2−r2L = \\sqrt{d^2 - r^2}
  • • Point inside: no tangents; on circle: one tangent; outside: two tangents

💡 Did You Know?

🛣️Road design uses tangent lines where straight roads meet circular curves — the transition point is the point of tangency.
🔬In optics, the tangent line defines the angle of incidence for light rays grazing a curved surface.
🧭Navigation uses tangent lines to find the shortest distance from a point to a circular region (e.g., city boundary).
📐The word "tangent" comes from Latin tangere meaning "to touch" — it touches the circle at exactly one point.
⚡From any external point, the two tangent lines to a circle are equal in length.
🎯The power of a point theorem: PT² = d² - r² where PT is tangent length, d is distance to center, r is radius.

📖 How It Works

A tangent to a circle is a line that touches the circle at exactly one point. The tangent is always perpendicular to the radius at the point of tangency.

Tangent at Point on Circle

For circle center (h,k), radius r, and point (px,py) on the circle: slope of radius = (py-k)/(px-h). Tangent slope = negative reciprocal = -(px-h)/(py-k). If py = k, tangent is vertical: x = px.

Tangent from External Point

Distance from point to center: d = √((px-h)² + (py-k)²). Tangent length from external point: L = √(d² - r²). Two tangents from an external point are equal in length.

Point Position

If d < r: point is inside; no tangent lines. If d = r: point is on circle; one tangent. If d > r: point is outside; two tangent lines.

Tangent Equation

Point-slope form: y - py = m(x - px) where m = -(px-h)/(py-k). Or: (x-h)(px-h) + (y-k)(py-k) = r².

🎯 Expert Tips

Vertical Tangent

When the point is directly to the right or left of the center (py = k), the tangent is vertical: x = px.

Unit Circle

For unit circle (0,0) radius 1, tangent at (1,0) is vertical x=1; tangent at (cos θ, sin θ) has slope -cot θ.

External Point

When the point is outside, use L = √(d² - r²) for tangent length. Points of tangency are found using angle geometry.

Numerical Precision

Use a small tolerance when checking if a point is on the circle (d ≈ r) due to floating-point precision.

Point Position Comparison

PositionConditionTangent LinesTangent Length
Insided < r0N/A
Ond = r1N/A
Outsided > r2L = √(d² - r²)

Frequently Asked Questions

How many tangent lines can be drawn from a point to a circle?

External point: 2; point on circle: 1; point inside: 0.

Why is the tangent perpendicular to the radius?

If the tangent were not perpendicular, it would intersect the circle at a second point, contradicting the definition of a tangent.

What is the tangent equation formula?

For point (px,py) on circle (h,k): slope m = -(px-h)/(py-k). Equation: y - py = m(x - px). If py = k: x = px.

What is the tangent length from an external point?

L = √(d² - r²) where d is distance from point to center and r is radius.

Are the two tangents from an external point equal?

Yes. Both tangents from an external point to a circle are equal in length.

How do I find the points of tangency?

Use angle geometry: base angle = atan2(py-k, px-h), offset = asin(r/d). Tangent points are at center + r·(cos(θ±α), sin(θ±α)).

What is the power of a point?

Power of point P = PT² = d² - r² where PT is tangent length, d is distance to center, r is radius.

Are tangents used in real-world applications?

Yes: road design, optics, navigation, computer graphics, collision detection, and architectural design.

Tangent by the Numbers

1
Point of Contact
90°
Angle with Radius
√(d²-r²)
Tangent Length
2
Tangents from Outside

Disclaimer: This calculator provides mathematically precise results based on standard geometric formulas. Results are limited by floating-point precision (~15 significant digits). For critical engineering or architectural applications, verify with domain-specific tools. Not a substitute for professional analysis.

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