Angle Bisector Length Calculator
Free Angle bisector length Calculator for linear algebra. Enter values to get step-by-step solutions with formulas and graphs.
Calculator
Adjust values & calculateFormula
Where t_a is the bisector length from vertex A, a is the opposite side, and b, c are the adjacent sides. The formula is derived from Stewarts Theorem combined with the Angle Bisector Theorem which states BD/DC = c/b.
Last reviewed: December 2025
Worked Examples
Example 1: Angle Bisector in a Scalene Triangle
Example 2: Equilateral Triangle Bisectors
Background & Theory
The Angle Bisector Length Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Angle Bisector Length Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
t_a = sqrt(b*c*[(b+c)^2 - a^2]) / (b+c)
Where t_a is the bisector length from vertex A, a is the opposite side, and b, c are the adjacent sides. The formula is derived from Stewarts Theorem combined with the Angle Bisector Theorem which states BD/DC = c/b.
Worked Examples
Example 1: Angle Bisector in a Scalene Triangle
Problem: Find the angle bisector lengths for a triangle with sides a = 8, b = 10, c = 14.
Solution: Bisector from A: t_a = sqrt(10 * 14 * [(10+14)^2 - 8^2]) / (10+14)\n= sqrt(140 * [576 - 64]) / 24 = sqrt(140 * 512) / 24\n= sqrt(71680) / 24 = 267.73 / 24 = 11.155\n\nBisector from B: t_b = sqrt(8 * 14 * [(8+14)^2 - 10^2]) / (8+14)\n= sqrt(112 * [484 - 100]) / 22 = sqrt(112 * 384) / 22 = 9.435\n\nBisector from C: t_c = sqrt(8 * 10 * [(8+10)^2 - 14^2]) / (8+10)\n= sqrt(80 * [324 - 196]) / 18 = sqrt(80 * 128) / 18 = 5.625
Result: t_a = 11.155 | t_b = 9.435 | t_c = 5.625
Example 2: Equilateral Triangle Bisectors
Problem: Find the angle bisector lengths for an equilateral triangle with side length 10.
Solution: For equilateral triangle (a = b = c = 10):\nt = sqrt(10 * 10 * [(10+10)^2 - 10^2]) / (10+10)\n= sqrt(100 * [400 - 100]) / 20\n= sqrt(100 * 300) / 20\n= sqrt(30000) / 20 = 173.21 / 20 = 8.660\nAll three bisectors equal sqrt(3)/2 * 10 = 8.660\nEach angle = 60 degrees, bisected into 30 degrees each
Result: t_a = t_b = t_c = 8.6603 | All angles = 60 degrees
Frequently Asked Questions
What is an angle bisector of a triangle?
An angle bisector of a triangle is a line segment that divides an interior angle of the triangle into two equal parts. Every triangle has three angle bisectors, one from each vertex. The angle bisector extends from the vertex to the opposite side, creating two smaller angles that are each exactly half the original angle. The three angle bisectors of any triangle always intersect at a single point called the incenter, which is equidistant from all three sides. The incenter is the center of the inscribed circle (incircle) that fits perfectly inside the triangle, tangent to all three sides.
What is the angle bisector length formula?
The length of the angle bisector from vertex A to side a (the side opposite to A) can be calculated using the formula: t_a = sqrt(b * c * [(b + c)^2 - a^2]) / (b + c), where a, b, and c are the side lengths of the triangle. An equivalent form uses the cosine of the half-angle: t_a = (2bc * cos(A/2)) / (b + c). There is also a form using the semi-perimeter s: t_a = (2/(b+c)) * sqrt(bcs(s-a)). All three formulas produce identical results and are chosen based on which information is most readily available for the computation.
What is the Angle Bisector Theorem?
The Angle Bisector Theorem states that the angle bisector from any vertex of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Specifically, if the bisector from vertex A meets side BC at point D, then BD/DC = AB/AC, or equivalently BD/DC = c/b. This theorem is one of the fundamental results in Euclidean geometry and has applications in triangle construction, similarity proofs, and computational geometry. It can be proved using the area method or by applying the law of sines to the two sub-triangles created by the bisector.
How does the angle bisector relate to the incircle?
The three angle bisectors of a triangle always meet at a single point called the incenter, which is the center of the triangle's inscribed circle (incircle). The incircle is the largest circle that fits entirely inside the triangle, touching all three sides. The radius of the incircle equals the area of the triangle divided by its semi-perimeter: r = Area / s. The incenter is equidistant from all three sides, and that distance equals the inradius. This relationship makes angle bisectors essential in circle-packing problems, geometric constructions, and the study of triangle centers in advanced Euclidean geometry.
Can the angle bisector length exceed any side of the triangle?
No, the angle bisector from any vertex is always shorter than each of the two adjacent sides of the triangle. This can be understood geometrically: the bisector cuts across the triangle interior and cannot be longer than either side enclosing the angle. More precisely, the bisector length t_a satisfies t_a < min(b, c), where b and c are the adjacent sides. The bisector approaches the length of the shorter adjacent side when the triangle is very flat (the bisected angle approaches zero). In an equilateral triangle, all three bisectors have equal length, and each equals (sqrt(3)/2) times the side length.
What is the difference between internal and external angle bisectors?
An internal angle bisector divides the interior angle of a triangle into two equal parts and always lies inside the triangle. An external angle bisector divides the exterior angle (the supplement of the interior angle) into two equal parts and lies outside the triangle. Every triangle has three internal and three external angle bisectors. The external bisector from vertex A divides side BC externally in the ratio AB:AC = c:b (negative ratio). An interesting property is that an internal bisector from one vertex is perpendicular to the external bisector from the same vertex. The three external bisectors, taken in pairs with the internal bisectors, define important geometric constructions called excircles.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy