Angle Bisector Calculator for Triangles
This angle bisector calculator takes the three sides of a triangle and returns the length of all three angle bisectors, the half angles and the two segments into which each bisector divides the opposite side. Example: for sides 5, 6 and 7 the bisector of the angle opposite side a is 5.9822 long and divides that side into 2.6923 and 2.3077.
Example: Side a 5 · Side b 6 · Side c 7 → Result: Angle bisectors: w_α = 5.9822 · w_β = 5.1235 · w_γ = 4.2251. Source: Weisstein, Eric W.: Angle Bisector. MathWorld – A Wolfram Web Resource (length of the angle bisector in a triangle). Updated: .
How it is calculated
An angle bisector divides an angle into two equal halves. In a triangle, the bisector of each angle runs from the vertex to the opposite side – and where it lands follows a simple rule. This calculator works from the three side lengths alone; no angles need to be known.
Length of an angle bisector
For the bisector of the angle α (opposite side a, enclosed by sides b and c): w_α = √(b · c · [(b + c)² − a²]) ÷ (b + c). The same value comes from w_α = 2 · b · c · cos(α ÷ 2) ÷ (b + c). For the other two bisectors, swap the letters accordingly.
Angle bisector theorem
The bisector divides the opposite side in the ratio of the two adjacent sides. The bisector of α splits side a into a · c ÷ (b + c), the part next to side c, and a · b ÷ (b + c), the part next to side b. The longer adjacent side always gets the longer part.
Worked example
Triangle with a = 5, b = 6, c = 7. Angle: cos α = (36 + 49 − 25) ÷ 84 = 0.7143, so α = 44.4153° and half of it is 22.2077°. Length: w_α = √(6 × 7 × (169 − 25)) ÷ 13 = √6048 ÷ 13 = 5.9822. Division of side a: 5 × 7 ÷ 13 = 2.6923 and 5 × 6 ÷ 13 = 2.3077; the two parts add up to 5 and are in the ratio 7 : 6. The other two bisectors are 5.1235 and 4.2251 long.
The incentre
The three angle bisectors meet in one point, the incentre. It has the same distance from all three sides and is the centre of the inscribed circle. Its radius is the area divided by half the perimeter: here 14.6969 ÷ 9 = 1.6330.
- In an equilateral triangle, bisector, median and altitude coincide; each is (√3 ÷ 2) times the side.
- In an isosceles triangle this is true only for the bisector of the apex angle.
Frequently asked questions
How do I calculate the length of an angle bisector in a triangle?
With the three sides: w_α = √(b · c · [(b + c)² − a²]) ÷ (b + c), where a is the side opposite the bisected angle. For a = 5, b = 6, c = 7 the result is 5.9822.
What does the angle bisector theorem say?
The bisector of an angle divides the opposite side in the ratio of the two sides that enclose the angle. In the 5-6-7 triangle, side a = 5 is split into 2.6923 and 2.3077, a ratio of 7 : 6.
Is an angle bisector the same as a median or an altitude?
Usually not. The median goes to the midpoint of the opposite side, the altitude meets it at a right angle, and the bisector halves the angle. All three coincide only in an equilateral triangle, or for the apex of an isosceles triangle.
Where do the three angle bisectors meet?
At the incentre, the centre of the inscribed circle. Its radius equals the area divided by half the perimeter – 1.6330 for the 5-6-7 triangle.
How do I bisect an angle with compass and straightedge?
Draw an arc around the vertex that cuts both arms. From the two intersection points draw two arcs with the same radius so that they cross. The line from the vertex through that crossing point is the bisector.
Sources and legal basis
- Weisstein, Eric W.: Angle Bisector. MathWorld – A Wolfram Web Resource (length of the angle bisector in a triangle)
- Weisstein, Eric W.: Angle Bisector Theorem. MathWorld – A Wolfram Web Resource (division of the opposite side in the ratio of the adjacent sides)
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