What is a tangent to a circle and what is its fundamental property?

A tangent to a circle is a straight line that touches the circle at exactly one point (the point of tangency). The fundamental geometric property of a tangent is that it is always strictly perpendicular (90°) to the circle's radius at the point of contact.

What is the formula for the length of a tangent from an external point to a circle?

For an external point P at distance d from the circle's center with radius r, the tangent segment length L is given by the Pythagorean theorem: L = sqrt(d^2 - r^2).

What is the equation of a tangent line to a circle at a point (x1, y1)?

For a circle with center (h, k) and radius r defined by (x - h)^2 + (y - k)^2 = r^2, the tangent line equation at point (x1, y1) on the circle is (x1 - h)(x - h) + (y1 - k)(y - k) = r^2. For a circle centered at the origin (0, 0), it simplifies to x1*x + y1*y = r^2.

What is the Tangent-Secant Theorem (Power of a Point) formula?

If a tangent segment PT touches the circle at T and a secant line from P passes through the circle at points A and B, then PT^2 = PA * PB => PT = sqrt(PA * PB).

How do you calculate the angle between two tangents drawn from an external point?

The angle theta between two tangents from point P at distance d to a circle of radius r is theta = 2 * arcsin(r / d) = 2 * arctan(r / L). The supplementary central angle formed by the radii is 180° - theta.

What is the chord of contact formula between two points of tangency?

When two tangents are drawn from external point P (distance d, tangent length L) to a circle of radius r, the chord connecting the two points of tangency T1 and T2 has length c = (2 * r * L) / d = (2 * r * sqrt(d^2 - r^2)) / d.

What is the difference between direct (external) and transverse (internal) common tangents of two circles?

Direct (external) common tangents touch both circles on the same side without crossing the center line (length L_ext = sqrt(D^2 - (r1 - r2)^2)). Transverse (internal) common tangents cross between the two circles (length L_int = sqrt(D^2 - (r1 + r2)^2)).

How many common tangents can two circles have?

(1) 4 tangents if separate (D > r1 + r2); (2) 3 tangents if externally tangent (D = r1 + r2); (3) 2 tangents if intersecting (|r1 - r2| < D < r1 + r2); (4) 1 tangent if internally tangent (D = |r1 - r2|); and (5) 0 tangents if one circle is inside the other (D < |r1 - r2|).

What is the Alternate Segment Theorem?

The Alternate Segment Theorem states that the angle between a tangent to a circle and a chord drawn from the point of tangency is equal to the inscribed angle subtended by the chord in the opposite (alternate) segment of the circle.

How does this Tangent of a Circle Calculator work and what problem types does it solve?

The calculator features 4 dedicated calculation engines: (1) External Point Tangent Lengths, Angles & Contact Chords (L = sqrt(d^2 - r^2)); (2) Tangent Line Equation at (x1, y1); (3) Tangent-Secant Power of a Point (PT^2 = PA * PB); and (4) Two-Circle Common Tangents & Pulley Belt Lengths.

100% Free • Tangent Length, Line Equation & Angle Solver

Tangent of a Circle Calculator

Calculate tangent segment lengths ($L = \sqrt{d^2 - r^2}$), tangent line equations ($(x_1-h)(x-h) + (y_1-k)(y-k) = r^2$), external vertex angles ($2\arcsin(r/d)$), chord of contact lengths, tangent-secant theorem lengths ($PT = \sqrt{PA \cdot PB}$), and common tangents of two circles.

Tangent Presets: Tap to load

Circle Tangent Solver

Select calculation mode to compute tangent properties.

L = √(d² - r²)

Calculation Mode

Circle Radius ($r$)

Distance from Center to Point ($d$)

Measurement Unit

Key Tangent Telemetry Glance

Tangent Length ($L$) 8.660 m

Vertex Angle ($\theta_P$) 60.00°

Central Angle ($\theta_C$) 120.00°

Contact Chord ($c$) 8.660 m

Tangent Segment Length ($L$)

8.660 m

Vertex Angle: 60.00° • Contact Chord: 8.660 m • Distance: 10.000 m

Tangent Geometry Telemetry Matrix

📏 Tangent Length ($L$)	8.660 m (√(d² - r²))
📐 Vertex Angle ($\theta_P$)	60.000° (2 arcsin(r/d))
🔄 Central Angle ($\theta_C$)	120.000° (180° - θ_P)
📏 Chord of Contact ($c$)	8.660 m (2rL/d)
🪁 Enclosed Kite Area	43.301 m² (r·L)
📐 Line Equation	3x + 4y = 25
🔄 Direct Common Tangent	38.730 m
🔀 Transverse Tangent	34.641 m

Why Use Our Tangent of a Circle Calculator? Real-World Problems It Solves

Tangents to circles are fundamental across mechanical engineering, civil transportation, satellite line-of-sight tracking, and coordinate geometry. Our Tangent of a Circle Calculator solves critical mathematical problems:

1. Tangent Segment & Distance Solving:

Calculates exact tangent segment lengths from any external point ($L = \sqrt{d^2 - r^2}$) using the right-angle radius-tangent property ($\text{Radius} \perp \text{Tangent}$).

2. Cartesian Tangent Line Equations:

Generates explicit linear equations in standard form ($Ax + By + C = 0$) and slope-intercept form ($y = mx + c$) for any contact point on circles centered at $(h, k)$.

3. Pulley Belt & Gear Common Tangents:

Computes direct common tangents ($L_{\text{ext}} = \sqrt{D^2 - (r_1 - r_2)^2}$) and transverse common tangents ($L_{\text{int}} = \sqrt{D^2 - (r_1 + r_2)^2}$) for mechanical belt drive sizing.

4. Horizon & Satellite Line-of-Sight:

Models distance to the geometric horizon for an observer at altitude $h$ above a spherical planet of radius $R$ using $d_{\text{horizon}} = \sqrt{2Rh + h^2}$.

How to Calculate Tangent of a Circle Properties (Step-by-Step Guide)

Step 1: Choose Tangent Mode

Select External Point, Line Equation, Tangent-Secant Theorem, or Two-Circle Common Tangents.

Step 2: Input Values

Enter radius ($r$), center distance ($d$), coordinates $(x_1, y_1)$, or secant lengths.

Step 3: Analyze Geometry

Review calculated segment lengths, vertex angles ($\theta_P$), contact chord ($c$), and kite area.

Step 4: Copy Full Telemetry

Click Copy Tangent Telemetry Card to copy a formatted report for engineering CAD or homework.

All Tangent of a Circle Formulas & Equations

1. Tangent Segment Length & Angles

$$L = \sqrt{d^2 - r^2}, \quad \theta_P = 2\arcsin\left(\frac{r}{d}\right) = 2\arctan\left(\frac{r}{L}\right)$$ $$\theta_{\text{central}} = 180^\circ - \theta_P, \quad c = \frac{2rL}{d}$$

2. Tangent Line Equation at $(x_1, y_1)$

$$(x_1 - h)(x - h) + (y_1 - k)(y - k) = r^2$$ $$\text{At origin } (0, 0): \quad x_1 x + y_1 y = r^2$$

3. Tangent-Secant Theorem (Power of a Point)

$$PT^2 = PA \cdot PB \implies PT = \sqrt{PA \cdot PB}$$

4. Two Circles Common Tangents

$$L_{\text{direct (ext)}} = \sqrt{D^2 - (r_1 - r_2)^2}$$ $$L_{\text{transverse (int)}} = \sqrt{D^2 - (r_1 + r_2)^2}$$

Classroom Practice & Study Guide: Circle Tangent Problems

Worked Tangent Geometry Practice Problems with Solutions:

Problem 1: Tangent Length from External Point

A circle has radius $r = 5\,\text{cm}$. A point $P$ is located $d = 13\,\text{cm}$ from the center. Find the tangent segment length and chord of contact.

Solution: $L = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = \mathbf{12.000\,\text{cm}}$. Chord of contact: $c = \frac{2 \times 5 \times 12}{13} = \frac{120}{13} \approx \mathbf{9.231\,\text{cm}}$.

Problem 2: Tangent Line Equation at Contact Point

Find the equation of the tangent line to the circle $x^2 + y^2 = 25$ at point $(3, 4)$.

Solution: Using $x_1 x + y_1 y = r^2$: $3x + 4y = 25 \implies y = -\frac{3}{4}x + \frac{25}{4}$. Slope of radius is $m_r = \frac{4}{3}$; tangent slope is $m_t = -\frac{3}{4}$ ($m_r \cdot m_t = -1$, confirming perpendicularity).

Problem 3: Direct Common Tangent for Pulley System

Two pulleys with radii $r_1 = 15\,\text{cm}$ and $r_2 = 5\,\text{cm}$ have their centers separated by $D = 26\,\text{cm}$. Find the direct common tangent length.

Solution: $L_{\text{direct}} = \sqrt{26^2 - (15 - 5)^2} = \sqrt{676 - 100} = \sqrt{576} = \mathbf{24.000\,\text{cm}}$.

Master Benchmark Circle Tangent Matrix Table

Standard geometric configurations from right-angle tangents to mechanical pulley spans.

Tangent Benchmarks

Configuration	Radius ($r$)	Distance ($d$)	Tangent Length ($L$)	Vertex Angle ($\theta_P$)	Contact Chord ($c$)
Point at $2\times$ Radius	$5.000$	$10.000$	$8.660$ ($5\sqrt{3}$)	$60.00^\circ$	$8.660$
$\sqrt{2}\times$ Radius (Square)	$5.000$	$7.071$	$5.000$	$90.00^\circ$	$7.071$
Point at $3\times$ Radius	$10.000$	$30.000$	$28.284$ ($20\sqrt{2}$)	$38.94^\circ$	$18.856$
5-12-13 Right Triangle	$5.000$	$13.000$	$12.000$	$45.24^\circ$	$9.231$
8-15-17 Right Triangle	$8.000$	$17.000$	$15.000$	$56.15^\circ$	$14.118$

Real-World Mechanical, Civil & Astronomical Applications

Everyday Engineering Implementations:

Mechanical Belt & Chain Drives: Sizing open and crossed serpentine drive belts for engines and industrial conveyor pulleys.
Highway & Railway Transition Curves: Designing smooth tangent transitions from straight highway segments into circular curves to prevent abrupt lateral acceleration.
Planetary Horizon Distance: Calculating visual horizon line-of-sight for maritime navigation and satellite ground station communications.
Optical Mirror Reflection & Laser Alignment: Aligning laser beams tangent to curved cylindrical and parabolic optical elements.

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about circle tangent formulas, line equations, vertex angles, chord of contact, and common tangents.

Configuration	Radius (\(r\))	Distance (\(d\))	Tangent Length (\(L\))	Vertex Angle (\(\theta_P\))	Contact Chord (\(c\))
Point at \(2\times\) Radius	\(5.000\)	\(10.000\)	\(8.660\) (\(5\sqrt{3}\))	\(60.00^\circ\)	\(8.660\)
\(\sqrt{2}\times\) Radius (Square)	\(5.000\)	\(7.071\)	\(5.000\)	\(90.00^\circ\)	\(7.071\)
Point at \(3\times\) Radius	\(10.000\)	\(30.000\)	\(28.284\) (\(20\sqrt{2}\))	\(38.94^\circ\)	\(18.856\)
5-12-13 Right Triangle	\(5.000\)	\(13.000\)	\(12.000\)	\(45.24^\circ\)	\(9.231\)
8-15-17 Right Triangle	\(8.000\)	\(17.000\)	\(15.000\)	\(56.15^\circ\)	\(14.118\)

📏 Tangent Length (\(L\))	8.660 m (√(d² - r²))
📐 Vertex Angle (\(\theta_P\))	60.000° (2 arcsin(r/d))
🔄 Central Angle (\(\theta_C\))	120.000° (180° - θ_P)
📏 Chord of Contact (\(c\))	8.660 m (2rL/d)
🪁 Enclosed Kite Area	43.301 m² (r·L)
📐 Line Equation	3x + 4y = 25
🔄 Direct Common Tangent	38.730 m
🔀 Transverse Tangent	34.641 m