Angle Bisector Length Calculator
Free Angle bisector length Calculator for linear algebra. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Angle Bisector Length Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: t_a = sqrt(b*c*[(b+c)^2 - a^2]) / (b+c)
Worked example — t_a = 11.155 | t_b = 9.435 | t_c = 5.625
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) = sqrt(140 * [576 - 64]) / 24 = sqrt(140 * 512) / 24 = sqrt(71680) / 24 = 267.73 / 24 = 11.155 Bisector from B: t_b = sqrt(8 * 14 * [(8+14)^2 - 10^2]) / (8+14) = sqrt(112 * [484 - 100]) / 22 = sqrt(112 * 384) / 22 = 9.435 Bisector from C: t_c = sqrt(8 * 10 * [(8+10)^2 - 14^2]) / (8+10) = 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): t = sqrt(10 * 10 * [(10+10)^2 - 10^2]) / (10+10) = sqrt(100 * [400 - 100]) / 20 = sqrt(100 * 300) / 20 = sqrt(30000) / 20 = 173.21 / 20 = 8.660 All three bisectors equal sqrt(3)/2 * 10 = 8.660 Each 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.
How are angle bisectors used in geometric constructions?
Angle bisectors are fundamental tools in classical geometric constructions using compass and straightedge. Bisecting an angle is one of the basic construction problems solvable with these tools. Practical applications include: constructing the incircle of a triangle (by finding the intersection of two angle bisectors), dividing angles into equal parts, creating regular polygons, and solving locus problems. In engineering and architecture, angle bisectors determine the optimal placement of features equidistant from two edges. In computer graphics, angle bisectors are used for polygon offsetting (shrinking or expanding shapes) and computing Voronoi diagrams.
What happens to the bisector length as the triangle becomes degenerate?
As a triangle approaches degeneracy (when the three vertices become collinear), the angle bisector lengths exhibit specific limiting behavior. If one angle approaches 180 degrees, the bisector from that vertex approaches zero length because the two adjacent sides become nearly antiparallel. If one angle approaches zero, the bisector from that vertex approaches the length of the adjacent sides. If the triangle becomes very thin (one side nearly equals the sum of the other two), the bisectors from the endpoints of the longest side become very short. These limiting cases are important for numerical stability in computational geometry algorithms.
How does the angle bisector formula relate to Stewarts Theorem?
Stewarts Theorem provides a general formula relating the length of a cevian (any line segment from a vertex to the opposite side) to the sides of the triangle. The angle bisector length formula is a special case of Stewarts Theorem. Stewarts Theorem states: b^2 * m + c^2 * n - a * m * n = a * d^2, where d is the cevian length and m, n are the segments it creates on the opposite side (m + n = a). For the angle bisector, the Angle Bisector Theorem gives m/n = c/b, so m = ac/(b+c) and n = ab/(b+c). Substituting these specific values into Stewarts Theorem and simplifying yields the standard angle bisector length formula.
Are there applications of angle bisectors in real-world problems?
Angle bisectors have numerous practical applications beyond pure mathematics. In urban planning, the angle bisector determines the optimal path equidistant from two roads meeting at an intersection. In optics, the angle bisector of incoming and reflected light rays defines the normal to a reflective surface (law of reflection). In robotics, angle bisectors help plan paths that maintain equal clearance from two obstacles. In surveying, bisecting angles is essential for triangulation and boundary determination. In art and design, angle bisectors create aesthetically pleasing symmetry. Even in nature, crystal growth patterns often follow angle bisector directions at grain boundaries.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Pyramid Angle Calculator
Calculate pyramid angle with inputs, formulas, and instant results.
🧮Angle Between Two Vectors Calculator
Calculate angle between two vectors with inputs, formulas, and instant results.
🧮Central Angle Calculator
Calculate central angle with inputs, formulas, and instant results.
🧮Clock Angle Calculator
Calculate clock angle with inputs, formulas, and instant results.
🧮Coterminal Angle Calculator
Calculate coterminal angle with inputs, formulas, and instant results.
🧮Reference Angle Calculator
Calculate reference angle with inputs, formulas, and instant results.
🧮Angle Converter
Calculate angle converter with inputs, formulas, and instant results.