Length of the Angle Bisector Calculator
Enter the three side lengths of a triangle to find the length of the internal angle bisector — the segment that splits the angle between Side 1 and Side 2 into two equal parts.
Angle Bisector Length Calculator
Angle Bisector Length Formula
Where:
- \(l\) — length of the bisector of the angle between sides \(a\) and \(b\)
- \(a\) and \(b\) — the two sides of the triangle that form the bisected angle (Side 1 and Side 2 in the calculator)
- \(c\) — the side opposite the bisected angle (Side 3 in the calculator)
An equivalent form is \( l = \dfrac{2ab\cos(C/2)}{a+b} \), where \(C\) is the bisected angle. Both expressions give the length of the internal angle bisector measured from the vertex to the opposite side.
Angle Bisector Length – Calculation Example
Example: a triangle has sides \(a = 5\), \(b = 7\), and \(c = 6\). Find the length of the bisector of the angle between sides \(a\) and \(b\).
\[ l = \sqrt{5 \cdot 7 \left(1 - \frac{6^{2}}{(5+7)^{2}}\right)} = \sqrt{35\left(1 - \frac{36}{144}\right)} = \sqrt{35 \cdot 0.75} = \sqrt{26.25} \approx 5.123 \]
Result: the angle bisector has a length of approximately 5.123 units.
How to Use the Calculator
- Enter the length of the first side forming the bisected angle in the Side 1 field.
- Enter the length of the second side forming that angle in the Side 2 field.
- Enter the length of the side opposite the angle in the Side 3 field.
- The calculator instantly displays the Length of angle bisector — the segment from the vertex between Side 1 and Side 2 to the opposite side, rounded to three decimal places.
What Is an Angle Bisector?
An angle bisector of a triangle is a line segment that starts at a vertex, splits that vertex's angle into two equal angles, and ends on the opposite side. Every triangle has three internal angle bisectors, one from each vertex. They all meet at a single point called the incenter, which is the center of the triangle's inscribed circle.
The bisector has a useful property: it divides the opposite side into segments proportional to the two adjacent sides. If the bisector from the angle between sides \(a\) and \(b\) meets side \(c\), then the two segments of \(c\) are in the ratio \(a : b\). This property, together with the Law of Cosines or Stewart's theorem, leads directly to the length formula used above.
Knowing the bisector length matters in many practical situations: constructing triangles in geometry class, verifying symmetry in architectural and engineering drawings, computing areas by splitting a triangle into two smaller ones, or positioning elements in CAD models. Because the formula needs only the three side lengths, no angles have to be measured separately.
Note that the result is always shorter than both adjacent sides, and it can never exceed half the sum \(a + b\). If the computed value looks close to one of the sides, the triangle is likely very "flat" near that vertex.
Assumptions, Limitations and Tips
- Which angle is bisected: the calculator always bisects the angle between Side 1 and Side 2; Side 3 must be the side opposite that angle. To bisect a different angle, reorder the inputs accordingly.
- Valid triangle required: the three side lengths must satisfy the triangle inequality — each side must be smaller than the sum of the other two. Check your values first with the Triangle Inequality Checker.
- Positive lengths only: all inputs must be greater than zero. Zero or negative values do not form a triangle.
- Units: the result is expressed in the same units as the inputs. If you enter sides in centimeters, the bisector length is in centimeters.
- Internal bisector only: the formula gives the internal bisector. External angle bisectors follow a different formula and are not computed here.
If you need to work with the angles themselves rather than side lengths, the Angle Bisector calculator handles the related angle computations, and the 2D Vector Angle calculator is useful when your triangle is defined by coordinates instead of side lengths.
Frequently Asked Questions
How do I calculate the angle bisector length without the formula from the page?
You can use the trigonometric form instead: first find the bisected angle \(C\) with the Law of Cosines, \( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \), then compute \( l = \frac{2ab\cos(C/2)}{a+b} \). Another route is Stewart's theorem combined with the angle bisector proportionality. All methods give the same result, so you can use whichever fits the data you already have.
What is the maximum possible length of an angle bisector?
The bisector is always shorter than both adjacent sides \(a\) and \(b\), and it never exceeds their arithmetic mean \(\frac{a+b}{2}\). The largest values occur when the bisected angle is small, making the triangle long and narrow. The minimum approaches zero as the triangle degenerates toward a straight line. If your result falls outside these bounds, double-check the inputs — the sides may violate the triangle inequality.
Is there a simpler formula for isosceles or equilateral triangles?
Yes. In an isosceles triangle with \(a = b\), the bisector from the apex is also the median and the altitude, so \( l = \sqrt{a^2 - \frac{c^2}{4}} \). For an equilateral triangle of side \(s\), every bisector equals the altitude: \( l = \frac{s\sqrt{3}}{2} \approx 0.866s \). These are good quick sanity checks for the calculator's output.
What units does the result come in, and can I convert it?
The bisector length is returned in whatever unit you enter for the sides — if you type sides in inches, the answer is in inches. To convert, first convert the three input sides to the target unit, then calculate; that is more reliable than converting the result alone, because it keeps all values consistent. For example, enter 5, 7, and 6 cm and the answer 5.123 is in centimeters.
Why does my result look wrong? Common input mistakes
The most frequent error is putting the wrong side in the Side 3 field — it must be the side opposite the angle being bisected, not one of the sides forming it. Also verify the triangle inequality: each side must be smaller than the sum of the other two, or the square root becomes negative and the calculation fails. Finally, remember the tool computes the internal bisector only; external bisectors use a different formula.
How is the angle bisector related to the incenter and inradius?
All three internal angle bisectors of a triangle intersect at the incenter, the center of the inscribed circle. The distance from the incenter to any side equals the inradius \(r\). If you need \(r\), compute the area \(A\) (for example with Heron's formula) and use \( r = \frac{A}{s} \), where \( s = \frac{a+b+c}{2} \) is the semi-perimeter. The bisector length itself, however, is measured from the vertex, not from the incenter.
Can I find the bisector length if my triangle is given by coordinates?
Yes. First compute the three side lengths from the coordinates using the distance formula \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \), then enter those lengths here, making sure the side opposite the vertex of interest goes into Side 3. This two-step approach works for any triangle defined by points in 2D or 3D, since only the side lengths matter for the formula.