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

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

Related MiniWebtools:

Circumscribed Circle (Circumcircle) CalculatorNew
General Triangle Solver
Inscribed Circle (Incircle) CalculatorNew
Law of Cosines Calculator

Geometry Calculators:

Top & Updated:

Random PickerRandom Name PickerBatting Average CalculatorRelative Standard Deviation CalculatorLine CounterFPS ConverterSort NumbersERA CalculatorRemove SpacesMAC Address GeneratorInstagram User ID LookupWord to Phone Number ConverterMAC Address LookupFeet and Inches to Cm ConverterFacebook User ID LookupRandom Truth or Dare GeneratorSum CalculatorPercent Off CalculatorBitwise CalculatorSHA256 Hash GeneratorOPS CalculatorMP3 LooperSlope and Grade CalculatorRandom Quote GeneratorSlugging Percentage CalculatorSalary Conversion CalculatorNumber of Digits CalculatorLog Base 10 CalculatorSquare Root (√) CalculatorSun, Moon & Rising Sign Calculator 🌞🌙✨Octal CalculatorSaturn Return CalculatorAudio SplitterPhone Number ExtractorRoman Numerals ConverterRandom IMEI GeneratorVertical Jump CalculatorOn Base Percentage CalculatorCm to Feet and Inches ConverterMerge VideosWAR CalculatorVideo to Image ExtractorDecimal to BCD ConverterCompound Growth CalculatorRandom Activity GeneratorBCD to Decimal ConverterFirst n Digits of PiUpgrade to Pro or PremiumRandom Writing Prompt GeneratorBinary to Gray Code ConverterRandom Poker Hand GeneratorCompare Two StringsOutlier CalculatorRandom Fake Address GeneratorWHIP CalculatorCaffeine Overdose CalculatorFile Size ConverterCM to Inches ConverterRandom Movie PickerText FormatterAI ParaphraserTime Duration CalculatorPER CalculatorRandom Superpower GeneratorQuotient and Remainder CalculatorRemove AccentAdd Prefix and Suffix to TextRandom Number PickerRemove Leading Trailing SpacesBinary to BCD ConverterVideo CropperNumber to Word ConverterImage SplitterWord Ladder GeneratorLove Compatibility CalculatorDay of Year CalendarImage CompressorYouTube Channel StatisticsWhat is my Lucky Number?Gray Code to Binary ConverterConnect the Dots GeneratorReverse VideoRandom Birthday GeneratorAI Punctuation AdderRandom Loadout GeneratorProportion CalculatorSocial Media Username CheckerPercent Growth Rate CalculatorIP Address to Hex ConverterInvisible Text GeneratorStair CalculatorSRT Time ShiftLeap Years ListSHA512 Hash GeneratorRandom Object GeneratorVideo SplitterArc Length CalculatorName Number CalculatorShort Selling Profit CalculatorEmail ExtractorURL ExtractorList of Prime NumbersDay of the Year Calculator - What Day of the Year Is It Today?Video CompressorSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterMedian CalculatorStandard Error CalculatorList RandomizerBreak Line by CharactersAverage CalculatorModulo CalculatorPVIFA CalculatorHypotenuse CalculatorRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterNumber ExtractorAngel Number CalculatorLog Base 2 CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence Expander📅 Date CalculatorLbs to Kg ConverterHex to Decimal ConverterRandom Group GeneratorConvolution 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 CalculatorBoiling Point CalculatorTitration CalculatorMole/Gram/Particle ConverterLED Resistor CalculatorVoltage Divider CalculatorParallel Resistor CalculatorCapacitor Calculator555 Timer CalculatorWire Gauge CalculatorTransformer CalculatorRC Time Constant CalculatorPower Factor CalculatorDecibel (dB) CalculatorImpedance CalculatorResonant Frequency CalculatorGrade CalculatorFinal Grade CalculatorWeighted Grade CalculatorTest Score CalculatorSignificant Figures CalculatorStudy Timer (Pomodoro)Long Division CalculatorRounding CalculatorCompleting the Square CalculatorRatio Calculatorp-Value CalculatorNormal Distribution CalculatorPercentile CalculatorFive Number Summary CalculatorCross Multiplication CalculatorLumber CalculatorRebar CalculatorPaver CalculatorInsulation CalculatorHVAC Sizing CalculatorRetaining Wall CalculatorCarpet CalculatorSquare Footage Calculator⏱️ Countdown Timer⏱️ Online Stopwatch⏱️ Hours Calculator🕐 Military Time Converter📅 Date Difference Calculator⏰ Time Card Calculator⏰ Online Alarm Clock🌐 Time Zone Converter🌬️ Wind Chill Calculator🌡️ Heat Index Calculator💧 Dew Point CalculatorFuel Cost CalculatorTire Size Calculator👙 Bra Size Calculator🌍 Carbon Footprint Calculator⬛ Aspect Ratio CalculatorOnline Notepad🖱️ Click Counter🔊 Tone Generator📊 Bar Graph Maker🥧 Pie Chart Maker📈 Line Graph Maker📷 OCR / Image to Text🔍 Plagiarism Checker🚚 Moving Cost Estimator❄️ Snow Day Calculator🎮 Game Sensitivity Converter⚔️ DPS Calculator🎰 Gacha Pity Calculator🎲 Loot Drop Probability Calculator🎮 In-Game Currency ConverterMultiplication Table GeneratorLong Multiplication CalculatorLong Addition and Subtraction CalculatorOrder of Operations Calculator (PEMDAS)Place Value Chart GeneratorNumber Pattern FinderEven or Odd Number CheckerAbsolute Value CalculatorCeiling and Floor Function CalculatorUnit Rate CalculatorSkip Counting GeneratorNumber to Fraction ConverterEstimation CalculatorCubic Equation SolverQuartic Equation SolverLogarithmic Equation SolverExponential Equation SolverTrigonometric Equation SolverLiteral Equation SolverRational Equation SolverSystem of Nonlinear Equations SolverPoint-Slope Form CalculatorStandard Form to Slope-Intercept ConverterEquation of a Line CalculatorParallel and Perpendicular Line CalculatorDescartes' Rule of Signs CalculatorRational Root Theorem CalculatorSigma Notation Calculator (Summation)Product Notation Calculator (Pi Notation)Pascal's Triangle GeneratorBinomial Theorem Expansion CalculatorParabola CalculatorHyperbola CalculatorConic Section IdentifierRegular Polygon CalculatorIrregular 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 CalculatorRelative Risk CalculatorEffect Size CalculatorPermutations with Repetition CalculatorModular Exponentiation CalculatorPrimitive Root CalculatorPerfect Number CheckerAmicable Number CheckerTwin Prime FinderMersenne Prime CheckerGoldbach Conjecture VerifierMöbius Function CalculatorEgyptian Fraction CalculatorFibonacci Number CheckerDigital Root CalculatorPartition Function CalculatorBoolean Algebra SimplifierKarnaugh Map (K-Map) SolverLogic Gate SimulatorGraph Coloring CalculatorTopological Sort CalculatorAdjacency Matrix CalculatorRecurrence Relation SolverInclusion-Exclusion CalculatorLinear Programming SolverTraveling Salesman Solver (TSP)Hamiltonian Path CheckerPlanar Graph CheckerNetwork Flow Calculator (Max Flow)Stable Marriage Problem SolverFirst-Order ODE SolverSecond-Order ODE SolverDirection Field / Slope Field PlotterEuler's Method CalculatorBernoulli ODE SolverSystem of ODEs SolverGroup Theory Order CalculatorRing and Field CalculatorJordan Normal Form Calculator