Calculator inputs
The page stays in one stacked flow. Inside the calculator, fields switch to three columns on large screens, two on medium screens, and one on mobile.
Plotly graph
This graph shows how the bisector length changes as the included angle changes while the adjacent sides stay fixed. Your current result is marked.
Example data table
These worked examples show common input styles and expected outputs.
| Mode | Inputs | Bisector | Split segments | Selected angle |
|---|---|---|---|---|
| Three sides | a=8, b=7, c=6, vertex=A | 5.1083 | BD=3.6923, DC=4.3077 | 73.3985° |
| Three sides | a=10, b=9, c=7, vertex=B | 7.0979 | AD=3.7059, DC=5.2941 | 58.4119° |
| Adjacent sides + angle | side1=5, side2=9, angle=50°, vertex=C | 5.8263 | 2.4782 and 4.4608 | 50.0000° |
Formula used
For a bisector from vertex A to side a:
tₐ = √(bc[(b + c)² − a²]) / (b + c)
t = 2bc cos(A / 2) / (b + c)
If the bisector meets the opposite side, then the two segments are proportional to the two adjacent sides.
segment₁ / segment₂ = adjacent side 1 / adjacent side 2
a = √(b² + c² − 2bc cos A)
How to use this calculator
- Select whether you know three sides or two adjacent sides with the included angle.
- Choose the vertex whose angle bisector you want to evaluate.
- Enter your lengths, choose a unit, and set the display precision.
- Press Calculate Bisector to show the result above the form.
- Review the bisector length, opposite side segments, perimeter, area, and ratio check.
- Use the CSV or PDF buttons to save a neat copy of the result.
- Study the Plotly chart to see how angle size changes the bisector length.
FAQs
1) What does a triangle bisector calculator find?
It finds the length of a chosen internal angle bisector. It also reports how that bisector divides the opposite side, plus perimeter, area, and consistency checks when enough data is available.
2) Which inputs are enough for a bisector calculation?
You can use all three side lengths, or use the two sides adjacent to the chosen angle with the included angle. Both input paths lead to the bisector length and side split.
3) Why does the calculator validate the triangle first?
A triangle must satisfy the triangle inequality. Without a valid triangle, the geometry cannot exist, so any bisector length would be meaningless. Validation prevents misleading outputs and catches impossible entries early.
4) What is the angle bisector theorem?
The theorem says an internal angle bisector divides the opposite side into two segments proportional to the lengths of the two adjacent sides. This is why the result includes a ratio check.
5) Does this work for scalene, isosceles, and equilateral triangles?
Yes. The formulas apply to any valid triangle. In symmetric cases, like an equilateral triangle, the bisector also acts as a median and altitude, so several special lines overlap.
6) Why might the bisector be shorter than the adjacent sides?
That is normal. A bisector is an internal segment, and its length depends on both the adjacent sides and the opening angle. Larger angles often reduce the bisector length for fixed adjacent sides.
7) What do the CSV and PDF downloads contain?
They contain the current result summary displayed on the page, including the chosen mode, vertex, bisector length, angle, split segments, perimeter, area, and ratio statement for quick sharing or records.
8) Can I use different measurement units?
Yes. The calculator treats units consistently, so you may use centimeters, meters, feet, or any custom label. Just keep every length input in the same unit for correct results.