Triangle Incenter Calculator
Solve triangle incenter problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
I = (a*A + b*B + c*C) / (a + b + c)
The incenter I is the weighted average of the three vertices A, B, C, where the weights a, b, c are the lengths of the sides opposite to each vertex. The inradius r = Area / s, where s is the semi-perimeter.
Worked Examples
Example 1: Incenter of a 3-4-5 Right Triangle
Problem:Find the incenter and inradius of a right triangle with vertices A(0,0), B(4,0), C(0,3).
Solution:Side a (opposite A) = sqrt(16+9) = 5\nSide b (opposite B) = sqrt(0+9) = 3\nSide c (opposite C) = sqrt(16+0) = 4\nPerimeter = 12, Semi-perimeter s = 6\nArea = (1/2)(4)(3) = 6\nInradius r = 6/6 = 1\nIx = (5*0 + 3*4 + 4*0)/12 = 12/12 = 1\nIy = (5*0 + 3*0 + 4*3)/12 = 12/12 = 1
Result:Incenter: (1, 1) | Inradius: 1
Example 2: Incenter of an Equilateral Triangle
Problem:Find the incenter of an equilateral triangle with vertices A(0,0), B(6,0), C(3, 5.196).
Solution:All sides equal: a = b = c = 6\nPerimeter = 18, Semi-perimeter s = 9\nArea = (sqrt(3)/4)(36) = 15.588\nInradius r = 15.588/9 = 1.732\nIx = (6*0 + 6*6 + 6*3)/18 = 54/18 = 3\nIy = (6*0 + 6*0 + 6*5.196)/18 = 31.176/18 = 1.732
Result:Incenter: (3, 1.732) | Inradius: 1.732
Frequently Asked Questions
What is the incenter of a triangle and how is it defined?
The incenter is the point where all three interior angle bisectors of a triangle meet. It is the center of the inscribed circle (incircle), which is the largest circle that fits entirely inside the triangle and is tangent to all three sides. Unlike the circumcenter, the incenter always lies inside the triangle regardless of whether it is acute, right, or obtuse. The incenter is equidistant from all three sides of the triangle, and that distance is called the inradius. It is one of the four classical triangle centers and plays an essential role in geometric constructions and proofs.
How do you calculate the incenter from vertex coordinates?
The incenter is calculated as the weighted average of the three vertex coordinates, where the weight for each vertex equals the length of the opposite side. If the vertices are A, B, C with opposite side lengths a, b, c respectively, then the incenter I = (a*Ax + b*Bx + c*Cx) / (a+b+c) for the x-coordinate and similarly for the y-coordinate. This weighting ensures the point lies on all three angle bisectors. The side lengths are computed using the distance formula between pairs of vertices. This approach is computationally efficient and numerically stable, making it the preferred method in most software implementations.
What is the inradius and how is it related to the triangle area?
The inradius (r) is the radius of the inscribed circle and equals the perpendicular distance from the incenter to any side of the triangle. It is calculated using the elegant formula r = Area / s, where s is the semi-perimeter (half the perimeter). This relationship can be rearranged to show that the triangle area equals r times s, which provides an alternative way to compute triangle area. For an equilateral triangle with side length a, the inradius simplifies to r = a / (2 * sqrt(3)). The inradius is always positive and is maximized (relative to the area) for equilateral triangles, making it a useful measure of how close a triangle is to being equilateral.
What are exradii and how do they relate to the incenter?
Exradii are the radii of the three excircles (escribed circles) of a triangle. Each excircle is tangent to one side of the triangle and to the extensions of the other two sides. The exradius opposite to vertex A is calculated as ra = Area / (s - a), where s is the semi-perimeter. Similarly, rb = Area / (s - b) and rc = Area / (s - c). There is a beautiful relationship: 1/r = 1/ra + 1/rb + 1/rc, where r is the inradius. The exradii are always larger than the inradius, and their product relates to the triangle area through ra * rb * rc = Area * s. These relationships connect the incircle and excircles in fundamental ways.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy