Calculate the bisector length, opposite side split, or missing segment with the angle bisector theorem calculator.

Angle Bisector Calculator

Calculate bisector length, split the opposite side, or solve a missing opposite-side segment using the Angle Bisector Theorem.

Bisector Length
Opposite Side Split
Missing Segment
Use this when you know the two sides around the angle and the included angle. The calculator returns the angle bisector length.
Use the Angle Bisector Theorem to divide the opposite side into two segments. The split follows the ratio a : b.
Use this when you know side a, side b, and one of the two opposite-side segments. Choose whether the known segment is x or y, where x : y = a : b.

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Angle Bisector Formula

The following formula is used to calculate the length of the internal angle bisector from the angle θ between sides a and b to the opposite side of the triangle.

l=\frac{2ab\cos(\theta/2)}{a+b}

Variables:

  • l is the length of the angle bisector
  • a is the length of one side of the triangle
  • b is the length of the other side of the triangle
  • θ is the angle between sides a and b in degrees

To calculate the length of the angle bisector, multiply 2ab by cos(θ/2), then divide by (a + b).

What is an Angle Bisector?

An angle bisector is a line or segment that divides an angle into two equal parts. In the context of a triangle, the angle bisector of one of the triangle's angles will intersect the opposite side, dividing it into two segments that are proportional to the other two sides of the triangle. The angle bisector has important properties and applications in geometry, including its use in constructing the incenter of a triangle, which is the point where the angle bisectors of all three angles of the triangle intersect.

How to Calculate Angle Bisector?

The following steps outline how to calculate the length of the angle bisector in a triangle.

  1. First, determine the lengths of the two sides of the triangle (a and b).
  2. Next, measure the angle (θ) between these two sides in degrees.
  3. Use the formula l = (2ab cos(θ/2)) / (a + b) to calculate the length of the angle bisector.
  4. Insert the values of a, b, and θ into the formula and solve for l.
  5. After calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Side a = 5 units

Side b = 7 units

Angle θ = 45 degrees