Triangle Bisector Calculator

Calculate the length of angle bisectors and the segments they divide the opposite side into. Includes dynamic SVG visualization and Angle Bisector Theorem verification.

User Rating 0.0 ★★★★★
Total Usage 1 times
Category Geometry Planimetry
Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

In geometry and architectural design, precision is rarely optional. The Triangle Bisector Calculator is engineered for students, surveyors, and drafters who need to determine the exact length of an internal angle bisector and the proportional segments it creates on the opposing side. Whether you are solving for a Side-Side-Side (SSS) configuration or checking the structural integrity of a triangular truss, this tool eliminates the ambiguity of manual trigonometry.

Understanding the angle bisector is critical in construction for determining centers of in-circles (incenters) and optimizing material stress distribution. A small error in these calculations can lead to significant misalignments in complex geometric patterns or structural frames. This tool uses the Angle Bisector Theorem and the Law of Cosines to provide results with high decimal precision, accompanied by a scaled visual representation to verify your input data instantly.

triangle bisector svg

Formulas

The length of the angle bisector ta on side a can be calculated using the lengths of the adjacent sides b and c and the angle A:

ta = 2bcb + c × cos(A2)

Alternatively, using only side lengths (SSS) via Stewart's Theorem derived formula:

ta = √bc (1 − (ab + c)2)

Reference Data

VariableSymbolFormula / RelationshipTypical Application
Bisector Lengthta2bcb + c cos(A/2)Determining the distance from a vertex to the opposite side while splitting the angle.
Segment 1macb + cPartitioning the base side proportional to adjacent sides.
Segment 2nabb + cThe complementary partition of the base side.
Semi-perimeters(a + b + c) / 2Used in Heron's formula for area verification.
IncenterIIntersection of all bisectorsFinding the center of the inscribed circle.
Theorem—m / n = c / bFundamental ratio verification.
ConstraintSSSa + b > cTriangle Inequality Theorem (must be valid).
ConstraintSAS0° < A < 180°Angle validity range.

Frequently Asked Questions

The Angle Bisector Theorem states that the angle bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. Simply put, if you split angle A, the ratio of the segments on side 'a' equals the ratio of sides 'b' and 'c'.
For any three side lengths to form a valid closed triangle, the sum of any two sides must be greater than the third. If this condition isn't met (e.g., sides 1, 1, and 10), the lines never connect, and calculating a bisector is mathematically impossible.
Yes, indirectly. The incenter is the point where all three angle bisectors intersect. By calculating the bisector lengths and segments for all three angles (by rotating your inputs), you can determine the geometric coordinates of the incenter.
The SVG drawing is generated using precise coordinates derived from the input values. While scaled to fit the screen, the relative proportions and angles are mathematically exact, making it a reliable visual check for your data.