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Angle Bisector Length

Geometry

Enter the three side lengths of the triangle. Side a is opposite vertex A (segment BC), b is opposite B (AC), and c is opposite C (AB).

Side a (opposite vertex A / segment BC)

Side b (opposite vertex B / segment AC)

Side c (opposite vertex C / segment AB)

Bisector from vertex

Length Unit

Decimal Places (0–10)

AcuteScalene

triangle

Bisector t_a (from vertex A)

5.9822 cm

Division ratio (BD : DC) = 7 : 6

Bisector t_b (from vertex B)

4.2251 cm

Division ratio (AE : EC) = 5 : 6

Bisector t_c (from vertex C)

5.1235 cm

Division ratio (AF : FB) = 5 : 7

Triangle Properties

Semi-perimeter (s)9.0000 cm
BC division (b : c)7 : 6
AC division (a : c)5 : 6
AB division (a : b)5 : 7

Triangle Diagram with Bisector(s)

t_a=5.9822t_b=4.2251t_c=5.1235ABCabcDashed lines = angle bisector(s) to division point

About This Tool

Angle Bisector Length Calculator – Compute Triangle Bisector Lengths Instantly

An angle bisector is a line segment from a vertex of a triangle that divides the vertex angle into two equal halves, ending on the opposite side. This calculator uses Stewart's Theorem to compute the exact bisector length from any vertex — or all three simultaneously — given just the three side lengths. It also applies the Angle Bisector Theorem to display where the bisector divides the opposite side.

What Is an Angle Bisector?

In a triangle with vertices A, B, and C and opposite sides a (BC), b (AC), and c (AB):

  • t_a — the bisector from vertex A bisects angle A and meets side BC at a point D.
  • t_b — the bisector from vertex B bisects angle B and meets side AC at a point E.
  • t_c — the bisector from vertex C bisects angle C and meets side AB at a point F.

The three angle bisectors of any triangle always meet at a single interior point called the incenter (I), which is equidistant from all three sides and serves as the center of the inscribed circle (incircle). Unlike the centroid (intersection of medians) or orthocenter (intersection of altitudes), the incenter is always strictly inside the triangle.

The Formula: Stewart's Theorem Derivation

The length of the angle bisector from vertex A is given by:

t_a = √(b × c × ((b+c)² − a²)) / (b+c)

An equivalent form using the semi-perimeter s = (a+b+c) / 2 is:

t_a = (2 / (b+c)) × √(b × c × s × (s−a))

The bisectors from vertices B and C follow the same pattern with sides substituted accordingly:

t_b = √(a × c × ((a+c)² − b²)) / (a+c)
t_c = √(a × b × ((a+b)² − c²)) / (a+b)

All formulas use only the three side lengths — no angles or coordinates are required. The calculator performs all arithmetic in JavaScript's 64-bit floating-point (IEEE 754 double precision), giving approximately 15 significant digits of accuracy.

The Angle Bisector Theorem and Division Ratios

Beyond the bisector length, the Angle Bisector Theorem tells us exactly where the bisector meets the opposite side:

  • Bisector from A meets BC at D, dividing it in the ratio BD : DC = c : b.
  • Bisector from B meets AC at E, dividing it in the ratio AE : EC = c : a.
  • Bisector from C meets AB at F, dividing it in the ratio AF : FB = b : a.

These ratios are automatically shown alongside each bisector length in the results. For example, with sides a = 5, b = 7, c = 6, the bisector from A divides BC in the ratio 6 : 7 — so if BC = 5, point D is at 5 × 6/13 ≈ 2.31 from B and 5 × 7/13 ≈ 2.69 from C.

Special Triangle Cases

Certain triangle types produce notable bisector properties:

  • Equilateral triangle (a = b = c): All three angle bisectors are equal in length and coincide with the medians, altitudes, and perpendicular bisectors.
  • Isosceles triangle (two equal sides): The bisectors from the two base vertices are equal; the bisector from the apex vertex also serves as the median and altitude to the base.
  • Right triangle: The bisector from the right-angle vertex has a particularly clean formula relating to the two legs and the hypotenuse.

How to Use This Calculator

  1. Enter side lengths a, b, and c. Remember: a is opposite vertex A (segment BC), b is opposite vertex B (AC), and c is opposite vertex C (AB).
  2. Select the vertex to compute the bisector from: A, B, C, or All three simultaneously.
  3. Choose a length unit (mm, cm, m, in, or ft).
  4. Adjust decimal precision between 0 and 10 places.
  5. Results appear instantly: bisector length(s), division ratio(s), and the triangle's semi-perimeter.
  6. Use the Copy button to copy any result value, or expand the Step-by-Step panel to see the full formula substitution for each bisector.
  7. The SVG diagram shows the triangle with the selected bisector(s) drawn as dashed lines from vertex to the division point on the opposite side.

Validation and Error Handling

The calculator validates all inputs before computing:

  • All three sides must be positive numbers greater than zero.
  • The triangle inequalitymust hold: a+b > c, a+c > b, and b+c > a. If not, a clear error message is shown.
  • Non-numeric input is rejected with an explanatory message.

Practical Applications

Angle bisector calculations appear in several real-world contexts:

  • Geometry education: Angle bisectors are central to high school and university triangle geometry, appearing in Ceva's theorem, the incircle, and olympiad problems. This tool lets students verify hand-computed results instantly.
  • CAD and structural design: Triangular structural members and trusses often require bisector lengths to place joints, gusset plates, or reinforcements at the correct angle-bisecting position.
  • Computer graphics: The incenter and its associated bisectors are used in mesh smoothing, triangle subdivision algorithms, and level-of-detail (LOD) transitions in 3D rendering pipelines.
  • Competitive mathematics: Triangle center problems in math olympiads regularly require angle bisector lengths and division ratios. This calculator confirms numeric results during practice.

Angle Bisector vs. Median vs. Altitude

It is worth distinguishing the three main cevians of a triangle: the median (vertex to midpoint of opposite side), the altitude (perpendicular from vertex to opposite side), and the angle bisector (from vertex, bisecting the vertex angle). Each set of three concurrent lines meets at a different triangle center: the centroid (G), the orthocenter (H), and the incenter (I) respectively. In a scalene triangle all three cevians from a given vertex are distinct; in an equilateral triangle they all coincide.

Frequently Asked Questions

Is the Angle Bisector Length free?

Yes, Angle Bisector Length is totally free :)

Can I use the Angle Bisector Length offline?

Yes, you can install the webapp as PWA.

Is it safe to use Angle Bisector Length?

Yes, any data related to Angle Bisector Length only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is an angle bisector in a triangle?

An angle bisector is a line segment drawn from a vertex of a triangle that divides the angle at that vertex into two equal halves. It ends on the opposite side of the triangle. Every triangle has exactly three angle bisectors, one from each vertex. Unlike the median (which goes to the midpoint of the opposite side) or the altitude (which is perpendicular to the opposite side), the angle bisector bisects the vertex angle.

What is the formula for the angle bisector length?

The length of the angle bisector from vertex A (with adjacent sides b and c, opposite side a) is given by Stewart's Theorem as: t_a = √(b × c × ((b+c)² − a²)) / (b+c). Equivalently, using the semi-perimeter s = (a+b+c)/2: t_a = (2/(b+c)) × √(b×c×s×(s−a)). The same structure applies for bisectors from B and C by substituting the appropriate sides.

What does the Angle Bisector Theorem state?

The Angle Bisector Theorem states that when a bisector from vertex A meets the opposite side BC, it divides BC in the ratio of the two adjacent sides: BD:DC = AB:AC = c:b. For example, if AB = 7 and AC = 6, the bisector from A divides BC in the ratio 7:6. This ratio is displayed alongside each bisector length in the calculator results.

How is the angle bisector different from the median and altitude?

All three are line segments from a vertex of a triangle, but they serve different purposes. The median connects a vertex to the midpoint of the opposite side. The altitude is perpendicular to the opposite side. The angle bisector divides the vertex angle into two equal halves. In a general scalene triangle, these three line segments are distinct. In an equilateral triangle, all three coincide for each vertex.

Can a triangle have all three angle bisectors of the same length?

Yes — in an equilateral triangle (where all three sides are equal), all three angle bisectors have the same length and also coincide with the medians and altitudes. In an isosceles triangle, the two bisectors from the base vertices are equal. In a scalene triangle, all three bisectors have different lengths. This calculator automatically detects and flags equilateral and isosceles cases.

What validation does the calculator perform?

The calculator checks that all three side lengths are positive numbers, and that they satisfy the triangle inequality: the sum of any two sides must be greater than the third side (a+b > c, a+c > b, b+c > a). If these conditions are not met, a clear error message is displayed. Numeric input is required for all fields.