Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Geometry Calculators > Angle Bisector Calculator

Angle Bisector Calculator

Calculate the angle bisectors of a triangle. Enter three sides or three vertex coordinates to find bisector lengths, division points on opposite sides, incenter, inradius, and see an interactive diagram with step-by-step formulas.

Angle Bisector Calculator
Examples:
LIVE PREVIEW
Opposite vertex A
Opposite vertex B
Opposite vertex C
Examples:
LIVE PREVIEW
Vertex A
Vertex B
Vertex C

Embed Angle Bisector Calculator Widget

About Angle Bisector Calculator

The Angle Bisector Calculator computes the angle bisectors of any triangle. Enter three side lengths or three vertex coordinates, and the calculator finds all three bisector lengths, the points where each bisector meets the opposite side, the incenter, the inradius, and displays an interactive diagram. All computations include step-by-step MathJax formulas.

∠
Angle Bisector
A segment from a vertex to the opposite side that divides the angle into two equal halves.
◉
Incenter
The point where all three angle bisectors meet. It is the center of the inscribed circle.
⊙
Inradius
The radius of the inscribed circle, equal to the area divided by the semi-perimeter: r = K/s.
⚖
Bisector Theorem
The bisector from A divides side BC in the ratio AB:AC = c:b (proportional to adjacent sides).

Angle Bisector Formulas

PropertyFormulaDescription
Bisector Length (from A)\( t_a = \frac{2bc \cos(A/2)}{b+c} \)Length of the angle bisector from vertex A to side BC
Alternative Formula\( t_a = \frac{\sqrt{bc[(b+c)^2 - a^2]}}{b+c} \)Uses side lengths only, no trigonometry needed
Bisector Theorem\( \frac{BD}{DC} = \frac{c}{b} = \frac{AB}{AC} \)Division ratio of opposite side by the bisector
Division Segment\( BD = \frac{ac}{b+c} \)Length from B to division point D on BC
Incenter\( I = \frac{a \cdot A + b \cdot B + c \cdot C}{a+b+c} \)Weighted average of vertices using opposite side lengths
Inradius\( r = \frac{K}{s} \)Area K divided by semi-perimeter s

How to Use This Calculator

  1. Choose input mode: Select "Three Sides" if you know a, b, c, or "Three Vertices" if you have coordinates.
  2. Enter values: Type the three side lengths or the (x, y) coordinates for each vertex. Use the quick example buttons to try preset triangles.
  3. Click Calculate: Press the "Calculate Angle Bisectors" button to see results.
  4. Explore the diagram: Toggle layers (bisectors, division points, incircle, angle arcs, labels) to focus on specific properties.
  5. Review formulas: Scroll down to the step-by-step solution to see every formula with substituted values.

Understanding the Angle Bisector Theorem

The Angle Bisector Theorem is one of the fundamental results in triangle geometry. It states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides. Specifically, if the bisector from vertex A meets side BC at point D, then BD/DC = AB/AC = c/b.

This theorem has many practical applications: it is used in triangle construction, in proving properties of the incircle, and in coordinate geometry problems. The angle bisector length formula \( t_a = \frac{2bc \cos(A/2)}{b+c} \) can be derived by applying the cosine rule to the two sub-triangles created by the bisector.

Properties of Angle Bisectors

Frequently Asked Questions

What is an angle bisector of a triangle?
An angle bisector is a line segment from a vertex to the opposite side that divides the vertex angle into two equal halves. Every triangle has three angle bisectors, and they all meet at a single point called the incenter, which is the center of the inscribed circle (incircle).
How do you calculate the length of an angle bisector?
The length of the angle bisector from vertex A to side BC is given by the formula: ta = (2bc × cos(A/2)) / (b + c), where b and c are the sides adjacent to angle A. An alternative formula that uses only side lengths is: ta = √(bc[(b+c)² − a²]) / (b + c).
What is the angle bisector theorem?
The angle bisector theorem states that an angle bisector of a triangle divides the opposite side into two segments proportional to the adjacent sides. For the bisector from A to side BC at point D: BD/DC = AB/AC = c/b. This is a fundamental theorem used in many geometric proofs and constructions.
Where do the three angle bisectors of a triangle meet?
The three angle bisectors of a triangle always meet at a single point called the incenter, denoted I. The incenter is the center of the inscribed circle (incircle) and is always located inside the triangle, regardless of whether the triangle is acute, right, or obtuse.
What is the difference between angle bisector and perpendicular bisector?
An angle bisector divides an angle into two equal parts and goes from a vertex to the opposite side. A perpendicular bisector divides a side into two equal parts at a 90° angle. The three angle bisectors meet at the incenter (center of inscribed circle), while the three perpendicular bisectors meet at the circumcenter (center of circumscribed circle).

Reference this content, page, or tool as:

"Angle Bisector Calculator" at https://MiniWebtool.com/angle-bisector-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-03

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

3D Distance CalculatorNew
Absolute Value Equation Solver
Angle Between Vectors CalculatorNew
Angle Converter
Arc Length CalculatorFeatured
Area of A Sector Calculator
Area of An Ellipse Calculator
Circumscribed Circle (Circumcircle) CalculatorNew
Cone Flat Pattern (Template) GeneratorFeatured
Cross Product CalculatorNew

Geometry Calculators:

Top & Updated:

Random Name PickerRandom PickerInstagram User ID LookupImage ResizerLine CounterFPS ConverterSort NumbersRelative Standard Deviation CalculatorRandom Truth or Dare GeneratorRemove SpacesBatting Average CalculatorMAC Address GeneratorFacebook User ID LookupERA CalculatorWord to Phone Number ConverterFeet and Inches to Cm ConverterMAC Address LookupSun, Moon & Rising Sign Calculator 🌞🌙✨Sum CalculatorSlope and Grade CalculatorMP3 LooperRandom Quote GeneratorPercent Off CalculatorRandom Superpower GeneratorRandom IMEI GeneratorInvisible Text Generator📷 OCR / Image to TextAudio SplitterNumber of Digits CalculatorBitwise CalculatorVertical Jump CalculatorRoman Numerals ConverterWord Ladder GeneratorRandom Credit Card GeneratorSHA256 Hash GeneratorMerge VideosLog Base 10 CalculatorAI Text HumanizerRandom Birthday GeneratorMaster Number CalculatorSaturn Return CalculatorCm to Feet and Inches ConverterHalfway Date Calculator⬛ Aspect Ratio CalculatorRandom Meal GeneratorRandom Fake Address GeneratorPhone Number ExtractorOPS CalculatorSalary Conversion CalculatorWAR CalculatorRandom Activity GeneratorFile Size ConverterNumber to Word ConverterIP Subnet CalculatorSquare Root (√) CalculatorText FormatterMercury Retrograde CalendarRandom Writing Prompt GeneratorOn Base Percentage CalculatorYouTube Channel StatisticsCaffeine Overdose CalculatorRandom Movie PickerRandom Time GeneratorRandom Poker Hand GeneratorDecimal to BCD ConverterSlugging Percentage CalculatorCompound Growth CalculatorRandom Loadout GeneratorBinary to Gray Code ConverterBCD to Decimal ConverterVideo to Image ExtractorLove Compatibility CalculatorBattery Life CalculatorOctal Calculator🖱️ Click CounterStair CalculatorAdd Text to ImageLeap Years ListDNS LookupFirst n Digits of Pi🎰 Gacha Pity CalculatorCM to Inches ConverterCompare Two StringsConnect the Dots GeneratorLongest Day of the YearWeight Loss Calculator📅 Date CalculatorRemove AccentNumber ExtractorVideo CompressorProportion CalculatorImage SplitterImage CompressorOutlier CalculatorPercent Growth Rate CalculatorSHA512 Hash GeneratorSmall Text Generator ⁽ᶜᵒᵖʸ ⁿ ᵖᵃˢᵗᵉ⁾Gray Code to Binary ConverterLottery Number GeneratorAstrological Element Balance CalculatorWHIP CalculatorArc Length CalculatorBingo Card GeneratorWord Scramble GeneratorDay of the Year Calculator - What Day of the Year Is It Today?PER CalculatorTime Duration CalculatorDiff CheckerWhat is my Zodiac Sign?Bolt Torque CalculatorRandom Object GeneratorFirst Day of SummerQuotient and Remainder CalculatorRandom Emoji GeneratorFlip VideoMultiple Fraction CalculatorURL ExtractorList of Prime NumbersCoin FlipperAdd Prefix and Suffix to TextBreak Line by Characters🔍 Plagiarism CheckerIP Address to Hex ConverterAcreage CalculatorVideo SplitterYouTube Tag ExtractorPregnancy CalendarBroken Link CheckerName Number CalculatorModulo CalculatorAI Language DetectorCone Flat Pattern (Template) GeneratorTessellation GeneratorMandelbrot Set ExplorerDay of Year CalendarBcrypt Hash Generator / CheckerMD5 Hash GeneratorSocial Media Username CheckerRandom Chord GeneratorAngel Number CalculatorSun Position CalculatorEmail ExtractorIs it a Prime Number?Long Division CalculatorMiter Angle CalculatorRandom Number PickerSteel Weight CalculatorRandom Name GeneratorMolarity CalculatorRandom Chess Opening GeneratorBinary to BCD ConverterLunar Calendar ConverterRandom Group GeneratorMultiplication CalculatorRandom Tournament Bracket GeneratorShort Selling Profit CalculatorSummer Solstice DayHeight Percentile CalculatorMartingale Strategy CalculatorRandom Line PickerDMS to Decimal Degrees ConverterMorse Code GeneratorAntilog CalculatorPercentile CalculatorWhat is my Lucky Number?First n Digits of eArctan2 CalculatorRemove Leading Trailing SpacesYouTube Earnings EstimatorHTML Compressor🔊 Tone GeneratorRandomize NumbersCollage MakerRandom User-Agent GeneratorPizza Value CalculatorRandom RPG Character GeneratorAdjust Video SpeedRounding CalculatorFraction CalculatorTaco Bar CalculatorAI Punctuation AdderBirth Day of the Week CalculatorBoiling Point CalculatorHypotenuse CalculatorIncome Tax CalculatorBonus CalculatorReverse TextAPI TesterDue Date CalculatorFence Calculator🎲 Loot Drop Probability CalculatorColor InverterCrossword Puzzle MakerSquare Numbers ListHebrew Calendar ConverterPVIFA CalculatorInvisible Character RemoverJulia Set GeneratorAI ParaphraserSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterBCD to Hex ConverterMedian CalculatorStandard Error CalculatorList RandomizerAverage CalculatorReverse VideoRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterLog Base 2 CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterConvolution CalculatorMAC Address AnalyzerRandom String GeneratorAmortization CalculatorMarkup CalculatorPVIF CalculatorDecimal to Hex ConverterInstagram Font GeneratorSocial Media Image Size GuideTikTok Money CalculatorTwitter/X Character CounterTwitter/X Timestamp ConverterYouTube Watch Time CalculatorTwitch Earnings CalculatorYouTube Shorts Monetization CalculatorFacebook Ad Cost CalculatorSocial Media ROI CalculatorSocial Media Post Time OptimizerCTR CalculatorROAS CalculatorInfluencer ROI CalculatorForce CalculatorAcceleration CalculatorVelocity CalculatorMomentum CalculatorProjectile Motion CalculatorKinetic Energy CalculatorPotential Energy CalculatorWork and Power CalculatorDensity CalculatorPressure CalculatorIdeal Gas Law CalculatorFree Fall CalculatorTorque CalculatorHorsepower CalculatorDilution CalculatorChemical Equation BalancerStoichiometry CalculatorPercent Yield CalculatorEmpirical Formula CalculatorTitration CalculatorMole/Gram/Particle ConverterIrregular Polygon Area CalculatorFrustum CalculatorTorus Calculator3D Distance CalculatorGreat Circle Distance CalculatorCircumscribed Circle (Circumcircle) CalculatorInscribed Circle (Incircle) CalculatorAngle Bisector CalculatorTangent Line to Circle CalculatorHeron's Formula CalculatorCoordinate Geometry Distance CalculatorVolume of Revolution CalculatorSurface of Revolution CalculatorParametric Curve GrapherRiemann Sum CalculatorTrapezoidal Rule CalculatorSimpson's Rule CalculatorImproper Integral CalculatorL'Hôpital's Rule CalculatorMaclaurin Series CalculatorPower Series CalculatorSeries Convergence Test CalculatorInfinite Series Sum CalculatorAverage Rate of Change CalculatorInstantaneous Rate of Change CalculatorRelated Rates SolverOptimization Calculator (Calculus)Gradient Calculator (Multivariable)Divergence CalculatorCurl CalculatorLine Integral CalculatorSurface Integral CalculatorJacobian Matrix CalculatorNewton's Method CalculatorRREF Calculator (Row Echelon Form)Matrix Inverse CalculatorMatrix Multiplication CalculatorDot Product CalculatorCross Product CalculatorVector Magnitude CalculatorUnit Vector CalculatorAngle Between Vectors CalculatorNull Space CalculatorColumn Space CalculatorCramer's Rule CalculatorMatrix Diagonalization CalculatorQR Decomposition CalculatorCholesky Decomposition CalculatorMatrix Power CalculatorCharacteristic Polynomial CalculatorBayes' Theorem CalculatorF-Test / F-Distribution CalculatorHypergeometric Distribution CalculatorNegative Binomial Distribution CalculatorGeometric Distribution CalculatorExponential Distribution CalculatorWeibull Distribution CalculatorBeta Distribution CalculatorSpearman Rank Correlation CalculatorFisher's Exact Test CalculatorContingency Table CalculatorOdds Ratio CalculatorROT13 Encoder/DecoderAtbash Cipher ToolVigenère Cipher ToolPronunciation IPA ConverterHemingway-Style Readability EditorSentence Length Variance AnalyzerWord Frequency AnalyzerBusiness Days CalculatorAdd Business Days to DateDate Pattern GeneratorHow Long Until CalculatorHow Long Ago CalculatorBirthday Across Cultures CalculatorHijri Calendar ConverterInsulin Sensitivity Factor CalculatorCarb-to-Insulin Ratio CalculatorLean Body Mass to Strength CalculatorOne-Mile Walk Test (Rockport) CalculatorCooper 12-Minute Run CalculatorFFMI CalculatorAPGAR Score CalculatorGlasgow Coma Scale CalculatorWells Score Calculator (DVT/PE)Tennis Score TrackerSoccer xG (Expected Goals) CalculatorCricket Run Rate CalculatorRugby Points CalculatorBoxing Punch Power CalculatorRace Time PredictorSwimming SWOLF CalculatorYoga Pose Hold TimerFishing Knot Strength CalculatorBike Gear Ratio CalculatorClimbing Grade ConverterWine Pairing SuggesterStandard Drink CalculatorCaffeine Half-Life TrackerSpice Substitution FinderDietary Restriction Recipe FilterMarinade Time CalculatorFermentation Time CalculatorSmoking Wood Pairing GuideFreelance Project Pricing CalculatorSaaS Pricing CalculatorSubscription Cost TrackerSide Hustle ROI CalculatorRemote Work Savings CalculatorCoffee Habit Cost CalculatorGym vs Home Workout Cost CalculatorLunch Cost CalculatorWealth Growth Visualizer1031 Exchange CalculatorRental Yield CalculatorCash-on-Cash Return CalculatorBRRRR Method CalculatorSection 8 Rent CalculatorRoommate Rent SplitterAirbnb Pricing OptimizerStatute of Limitations CalculatorSentence Reduction CalculatorSales Tax Nexus CheckerPatent Filing Fee CalculatorTrademark Class FinderWill Asset Distribution CalculatorWorkers' Compensation CalculatorStopping Distance CalculatorTrip Cost SplitterVehicle Weight Distribution CalculatorTrailer Tongue Weight CalculatorTire Tread Wear CalculatorEngine Compression Ratio CalculatorHeadlight Beam Distance CalculatorCat Litter Box CalculatorAquarium Heater Wattage CalculatorBird Cage Size CalculatorReptile Habitat UVB CalculatorPet Travel Crate Size FinderHorse Hay CalculatorCrochet Hook Size ConverterKnitting Needle Size ConverterKnitting Pattern CalculatorCross-Stitch Floss CalculatorQuilt Binding CalculatorOrigami Paper Size CalculatorPottery Clay Shrinkage CalculatorBeading Pattern CalculatorResin Casting Volume CalculatorEmbroidery Thread Length CalculatorHiking Pace Calculator (Naismith's Rule)Backpacking Food Weight CalculatorTent Footprint Size CalculatorSleeping Bag Temperature Rating GuideKnot Tying Reference ToolStar Visibility CalculatorTide Time CalculatorReynolds Number CalculatorBernoulli Equation CalculatorHeat Transfer CalculatorThermal Expansion CalculatorSpecific Heat Capacity CalculatorGear Ratio Calculator (Mechanical)Pulley System CalculatorHydraulic Cylinder Force CalculatorBelt Length CalculatorCloset Capsule CalculatorStorage Unit Size CalculatorMoving Box Quantity CalculatorGift Card Tip CalculatorGas vs Electric Cost ComparisonPrint Cost CalculatorHair Dye Mixing CalculatorLaundry Detergent Dosage CalculatorDishwasher Load OptimizerTile Grout CalculatorPaint Color Mixing CalculatorFlashcard Spaced Repetition SchedulerLearning Curve CalculatorCornell Notes GeneratorVocabulary Quiz GeneratorLanguage Learning Hours to Fluency CalculatorCollege Cost CalculatorScholarship ROI CalculatorAI Recipe Generator (From Ingredients)AI Gift Idea GeneratorAI Meal Plan GeneratorAI Workout Plan GeneratorAI Reading List GeneratorAI Travel Itinerary GeneratorAI Excuse Generator (Polite)AI Apology Letter WriterAI Unit Converter (Natural Language)AI Resume / CV AnalyzerAI Text Tone AnalyzerAI Data Visualizer (Paste CSV)AI Regex GeneratorAI SQL Query Generator