Tangent and Angle Bisector Calculator Guide

The Tangent and Angle Bisector Calculator is a focused circle-geometry tool for finding a tangent length, locating the meeting point of two tangents, or measuring the bisector of an isosceles triangle formed by two radii. This guide explains this specific calculator page, including its five choices in Choose a calculation, its changing input labels, its unit selector, its result panel, and the controls that appear after a calculation.

The calculator page itself is intentionally short so that the working area remains easy to use on a phone or computer. This longer guide supplies the definitions, restrictions, formulas, examples, and checking habits that the compact tool leaves out. Use the calculator for the numerical result, then use the explanation here to decide whether the diagram and the selected relationship match your geometry problem.

What This Calculator Calculates

A tangent touches a circle at one point. If OT is the radius to the contact point T, and PT is a tangent from an outside point P, then OT is perpendicular to PT. That right triangle is the basis of the two tangent-length choices.

The two-tangent choice uses a circle with centre O, contact points A and B, and an outside intersection point Q. The calculator finds OQ, the distance from the centre to the point where the tangents meet, and also reports the equal tangent lengths QA and QB.

The two angle-bisector choices use triangle AOB, where OA and OB are equal radii and AB is a chord. The perpendicular bisector from the centre to the chord is also the angle bisector of the central angle. The calculator calls that segment OM.

Tangent Length PT

Tangent length PT is the straight-line distance from an outside point P to the point T where its tangent touches the circle. It is not the length of an arc and it does not run through the circle. In the right triangle OPT, the radius OT is one leg, the tangent PT is the other leg, and OP is the hypotenuse.

Centre-to-Point Distance OP

Centre-to-point distance OP is the distance from the circle centre to the external point that produces the tangent. For the distance-based tangent choice, OP is an input. For the angle-based tangent choice, the calculator derives OP and displays it as an output.

Distance from Centre to Tangent Intersection OQ

Distance from centre to tangent intersection is the length OQ from the circle centre to the point where two tangents meet. It is different from either tangent length QA or QB. The two tangent segments are equal because they are drawn from the same outside point.

Angle-Bisector Length OM

Angle-bisector length OM is the segment from the centre O to the midpoint M of chord AB. In the isosceles triangle AOB, OM divides the central angle AOB into two equal angles and meets AB at a right angle.

Related Values Shown in the Result

Depending on the selected calculation, the result panel can also show the angle between OP and PT, the angle between two tangents, the central angle between contact radii, the chord length, the half-angle, and the triangle area. A length is reported in the selected unit; the triangle area is reported in the corresponding square unit, such as cm².

How the Calculator Page Is Organized

Open the Tangent and Angle Bisector Calculator.
Use Choose a calculation to select one of the five relationships. The two fields below it change immediately to match the choice.
Enter two positive numbers in the displayed fields. The inputs are number fields with the placeholders Enter radius, Enter centre-to-point distance, Enter angle between OP and tangent PT (°), Enter angle between the two tangents (°), Enter central angle between radii (°), or Enter chord length, as applicable.
Choose Unit. The available choices are mm, cm, m, in, and ft. The default selection is cm.
Press Click to calculate. The result panel reports the requested answer and the derived quantities for that mode.
After a successful calculation, Click to visualize it, Click to copy result, and Click to show steps become useful. The visualization is a diagrammatic check, while the numeric result remains the precise value.

Press Clear to empty the current inputs and remove the result, steps, visualization, and enabled result controls. Changing Choose a calculation also redraws the relevant fields and resets the current answer.

All Five Choices and Their Exact Inputs

Choice in “Choose a calculation”	First field	Second field	Main result	Required relationship
Tangent length from radius and centre distance	Radius	Centre-to-point distance	Tangent length PT and angle between OP and PT	Centre-to-point distance must be greater than radius
Tangent length from radius and tangent angle	Radius	Angle between OP and tangent PT (°)	Tangent length PT and centre-to-point distance OP	Angle must be greater than 0° and less than 90°
Two tangent intersection from radius and tangent angle	Radius	Angle between the two tangents (°)	Distance from centre to tangent intersection OQ	Angle must be greater than 0° and less than 180°
Angle bisector from radius and central angle	Radius	Central angle between radii (°)	Angle-bisector length OM	Central angle must be greater than 0° and less than 180°
Angle bisector from radius and chord	Radius	Chord length	Angle-bisector length OM and the corresponding central angle	Chord must be positive and no longer than the diameter

Every mode begins by checking that both entered values are finite and greater than zero. If a value is missing, zero, negative, or not a usable number, the calculator displays the error message Enter two values greater than zero. The restrictions in the final column prevent an impossible tangent or chord geometry.

Meaning of the Units and the Mathematical Constant π

The Unit selector changes the label attached to the calculator's length results. Select mm for millimetres, cm for centimetres, m for metres, in for inches, or ft for feet. Keep every length input in the same unit: for example, do not enter a radius in cm and a chord in inches unless you have converted one first. The tool does not perform a hidden conversion between different inputs.

The calculator accepts angles in degrees, shown with the symbol °. It converts degrees to radians internally before using sine, cosine, tangent, or inverse trigonometric functions. The conversion is angle in radians = angle in degrees × π ÷ 180, where π (pi) is the circle constant, approximately 3.141592653589793. You do not type π into any field on this page; enter an ordinary degree value such as 30 or 60.

Pi is relevant because the browser's trigonometric functions use radians, even though the visible input labels use degrees. The displayed values are rounded to a practical number of decimal places, so a hand calculation may show 8.660254 while the calculator displays 8.6603.

Formulas Used by the Tangent Choices

Tangent Length from Radius and Centre Distance

When the inputs are radius r and centre-to-point distance p, the right triangle gives:

PT = √(OP² − OT²2

− r²)

The angle between OP and PT is:

β = sin⁻¹

The page requires p > r. If p equals r, the outside point is on the circle and the tangent length collapses to zero; if p is smaller, a real tangent from that point does not exist.

Tangent Length from Radius and Tangent Angle

When the radius r and the angle β between OP and PT are known, the right-triangle ratios are:

PT = r ÷ tan(β)

OP = r ÷ sin(β)

The page accepts an angle strictly between 0° and 90°. At an angle close to 0°, the tangent length becomes very large; at 90°, the configuration no longer represents the intended finite external point.

Two Tangent Intersection from Radius and Tangent Angle

Let φ be the angle between the two tangents at their intersection Q. The central angle between the radii to the contact points is:

θ = 180° − φ

The result panel uses:

OQ = r ÷ sin(φ ÷ 2)

QA = QB = r ÷ tan(φ ÷ 2)

The calculator accepts 0° < φ < 180°. Notice that OQ is not the same output as QA or QB; the first is centre-to-intersection distance and the latter two are tangent lengths.

Formulas Used by the Angle-Bisector Choices

Angle Bisector from Radius and Central Angle

For radius r and central angle θ, the isosceles triangle AOB is split into two right triangles. The calculator finds:

OM = r cos(θ ÷ 2)

It also reports the chord and the area of triangle AOB:

AB = 2r sin(θ ÷ 2)

Area of triangle AOB = ½r²sin(θ)

The entered central angle must be strictly between 0° and 180°. The result also includes the half-angle θ ÷ 2.

Angle Bisector from Radius and Chord

For radius r and chord length c, the calculator first recovers the central angle:

θ = 2 sin⁻¹

It then uses that angle to calculate:

OM = r cos(θ ÷ 2)

The chord cannot be longer than the diameter, so the page rejects c > 2r. It also rejects a zero or negative chord. The output includes the recalculated chord, central angle, half-angle, and triangle area, which lets you check that the chord input is geometrically consistent.

Practical Examples Using Every Calculator Option

The following examples use centimetres throughout. Each example names the exact choice, uses the exact field labels shown by the page, and identifies the control to press. The values are deliberately simple enough to verify with the formulas above.

Example 1: Find a Tangent from Radius and Centre Distance

Set Choose a calculation to Tangent length from radius and centre distance.
Enter Radius = 5 and Centre-to-point distance = 13.
Set Unit to cm, then press Click to calculate.

The calculator reports Tangent length PT = 12 cm, because √(13² − 5²) = √144 = 12. It also reports Angle between OP and PT = 22.6199°. The result identifies the radius OT as 5 cm and OP as 13 cm.

Example 2: Find a Tangent from an Angle

Choose Tangent length from radius and tangent angle.
Enter Radius = 5 and Angle between OP and tangent PT (°) = 30.
Choose cm in Unit, then press Click to calculate.

The result is Tangent length PT = 8.6603 cm and Centre-to-point distance OP = 10 cm. The calculation is PT = 5 ÷ tan(30°) and OP = 5 ÷ sin(30°). The result panel also repeats the radius and the tangent angle so that the selected triangle is clear.

Example 3: Locate the Meeting Point of Two Tangents

Choose Two tangent intersection from radius and tangent angle.
Enter Radius = 5 and Angle between the two tangents (°) = 60.
Leave Unit as cm and press Click to calculate.

The calculator gives Distance from centre to tangent intersection = 10 cm. It reports a Central angle between contact radii = 120° and Each tangent length from the intersection = 8.6603 cm. The 120° central angle is the supplement of the 60° angle between the tangents.

Example 4: Find a Bisector from Radius and Central Angle

Choose Angle bisector from radius and central angle.
Enter Radius = 5 and Central angle between radii (°) = 60.
Choose cm and press Click to calculate.

The calculator returns Angle-bisector length OM = 4.3301 cm, Chord AB = 5 cm, Half-angle = 30°, and Triangle area AOB = 10.8253 cm². These values follow from OM = 5 cos(30°), AB = 2(5) sin(30°), and ½(5²)sin(60°).

Example 5: Find a Bisector from Radius and Chord

Choose Angle bisector from radius and chord.
Enter Radius = 5 and Chord length = 5.
Set Unit to cm, then press Click to calculate.

The tool recovers Central angle AOB = 60°, then reports Angle-bisector length OM = 4.3301 cm, Half-angle = 30°, and Triangle area AOB = 10.8253 cm². This is the same geometry as Example 4 reached from a different known value, so the two modes provide a useful cross-check.

Compare the Outputs Before Choosing a Mode

Output	What it measures	Appears in	Do not confuse it with
Tangent length PT	Outside point to circle contact point	Both tangent-length modes	Radius or centre-to-point distance
Centre-to-point distance OP	Centre to the point from which one tangent is drawn	Both tangent-length modes	Tangent length PT
Distance from centre to tangent intersection OQ	Centre to the meeting point of two tangents	Two tangent intersection mode	Equal tangent lengths QA and QB
Central angle	Angle at the centre between two radii	Intersection and bisector modes	Angle between the tangent lines
Angle-bisector length OM	Centre to the midpoint of the chord	Both angle-bisector modes	Chord AB
Chord AB	Straight segment joining two points on the circle	Both angle-bisector modes	Arc length along the circle
Triangle area AOB	Area of the two-radii-and-chord triangle	Both angle-bisector modes	Area of the whole circle or sector

Visualization, Steps, Copy, Clear, and History

View the Diagram

After pressing Click to calculate, press Click to visualize it. For a tangent result, the diagram shows the circle, the radius to the contact point, the tangent, the centre-to-point line, a right-angle marker, and labelled values. For two tangents, it shows the two contact radii, the two tangent segments, their intersection, and the central and tangent angles. For a bisector result, it shows the two radii, chord AB, and the dashed bisector OM.

The page labels the drawing as a visual aid and advises you to use the numeric result for exact measurements. The drawing is therefore best used to confirm which points and segments you intended, not to measure pixels from the screen.

Show the Working

Press Click to show steps after a successful calculation. The button changes to Hide steps while the working is open. The displayed steps substitute your values into the relevant Pythagorean, sine, cosine, or tangent relationship. If you have not calculated a valid result, the steps control has no completed calculation to reveal.

Copy the Result

Press Click to copy result to copy the principal result to the clipboard. Depending on the mode, the copied text is the tangent length, the centre-to-intersection distance, or the angle-bisector length. The button may briefly change to Copied. If the browser does not grant clipboard access, it reports Select and copy result; in that case, copy the visible result manually.

Clear the Current Calculation

Press Clear when you want a fresh calculation. It removes both entered numbers, the error message, the result, the steps, and the drawing, and disables the visualization and copy controls until another valid calculation is made.

Review or Clear History

Press Click to view history to open the Recent calculations panel. The page stores up to ten recent calculations on that device and browser, and displays the date and time beside each item. Selecting a history item restores its mode, values, and unit, then runs the calculation again. Inside the open panel, press Clear history to remove those saved entries. The history is device-local; it is not a substitute for a saved project file.

Where Do We Use Tangent Lengths, Intersection Distances, and Angle-Bisector Lengths in Real Life?

Calculator output	Distinct practical use
Tangent length PT	Use it when a straight cable, sight line, belt, ramp edge, or road boundary must touch a circular tank, wheel, round wall, or safety zone without crossing it. The result gives the straight clearance segment from the outside point to the contact point.
Centre-to-point distance OP	Use it to place an external anchor, camera, sensor, or inspection point at a known distance from a circular object while preserving a tangent line of sight.
Distance from centre to tangent intersection OQ	Use it when two straight boundaries meet outside a round plot, sign, reservoir, or turning feature. OQ locates the meeting point relative to the centre, while QA and QB give the equal boundary lengths.
Angle-bisector length OM	Use it to locate the midpoint of a chord and the shortest centre-to-chord clearance. This is useful for setting out a circular opening, positioning a brace symmetrically, or finding the depth of a circular segment.
Central angle and chord	Use them to convert between an angular layout and a straight span across a circular part, such as a window, curved panel, flange, or surveyed arc boundary.
Triangle area AOB	Use it as the triangular component in a sector or segment calculation, for example when separating a pie-shaped region from the two-radii triangle beneath it.

When This Is Not the Right Tool

This calculator is appropriate when the geometry is specifically a circle tangent, a pair of tangents, or the isosceles triangle formed by two radii and a chord. Choose another tool when the unknown is a different circle measurement or when the shape is not circular.

For a complete circle's area, use the Circle Area Calculator. The triangle area reported here is only the area of AOB, not the entire disk.
For the distance around a circle, use the Circle Circumference Calculator. A tangent is a straight segment, not circumference.
For a chord from radius, angle, or distance information, use the Circle Chord Calculator. It is better when the chord itself is the main unknown rather than a supporting value of an angle bisector.
For arc length or an arc's central-angle relationship, use the Circle Arc Calculator. This tangent tool does not calculate distance along the curved arc.
For a sector, circular segment, or annular ring, use the Sector, Segment and Ring Calculator. Those regions require area and arc formulas beyond the tangent triangle.
For a general triangle's circle relationships, use the Triangle Circle Calculator. The present tool assumes the special equal-radius geometry of a circle.
For a coordinate equation or a plotted circle, use the Circle Equation Calculator. This page does not solve for a centre and radius from x-y coordinates.

Final Verification Workflow

Identify the actual geometry: one tangent, two tangents meeting outside the circle, or the bisector in triangle AOB.
Write down the known quantities and convert all lengths to one unit before entering them.
Open Choose a calculation and select the option whose wording matches those known quantities, not merely the output you hope to see.
Read the changing labels carefully. Confirm that Centre-to-point distance, Angle between OP and tangent PT (°), Angle between the two tangents (°), Central angle between radii (°), or Chord length is the value you intend to enter.
Check the domain before calculating: OP must exceed r for the distance tangent mode; the single tangent angle must lie between 0° and 90°; the two-tangent angle must lie between 0° and 180°; the central angle must lie between 0° and 180°; and a chord must be no longer than 2r.
Select Unit, press Click to calculate, and read the bold principal result together with the supporting values.
Press Click to show steps and compare the substitution with the appropriate formula in this guide. For a geometric check, press Click to visualize it.
Check units one last time. Lengths retain the selected unit, while triangle area uses its square, such as m² or ft².
If the result is needed elsewhere, press Click to copy result; otherwise record the inputs, mode, unit, and date rather than saving only an unexplained number.
For an independent check, calculate the same geometry through the alternative mode where possible. For example, radius 5 cm and a 60° central angle should agree with radius 5 cm and chord 5 cm in the two angle-bisector choices.

With that workflow, the short calculator page remains quick to operate while every number can still be traced to a labelled segment, a valid domain, and a specific circle-geometry formula.