About

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained within it, touching (tangent to) all three sides. The center of this circle is called the incenter, and it is the point where the triangle's angle bisectors intersect.

This calculator is a versatile tool for students and engineers. It solves for the Inradius using Heron's Formula (Side Mode) or calculates the precise (X, Y) coordinates of the Incenter (Coordinate Mode). This is particularly useful in surveying, computer graphics, and mechanical linkage design where clearance constraints are critical.

Formulas

1. Radius Formula (Given Sides a, b, c):

First, calculate the semi-perimeter s:

s = a + b + c2

Then, the Area (A) using Heron's Formula:

A = √s(s−a)(s−b)(s−c)

Finally, the Inradius r:

r = As

2. Incenter Coordinates I(x,y):

I_x = ax_A + bx_B + cx_CP

Reference Data

Triangle Type	Example Sides (a, b, c)	Area	Semi-perimeter (s)	Inradius (r)
Equilateral	10, 10, 10	43.30	15.0	2.887
Right (3-4-5)	3, 4, 5	6.00	6.0	1.000
Isosceles	5, 5, 8	12.00	9.0	1.333
Scalene	7, 8, 9	26.83	12.0	2.236
Obtuse	4, 5, 8	8.18	8.5	0.962
Degenerate	1, 2, 3	0.00	3.0	0.000

Frequently Asked Questions

Yes, every non-degenerate triangle (where the sum of any two sides is greater than the third) has exactly one unique inscribed circle.

The calculator will return an error or a radius of 0. For three sides to form a triangle, they must satisfy the Triangle Inequality Theorem: a + b > c.

No. The Centroid is the center of mass (average of coordinates), while the Incenter is the center of the inscribed circle. They are only the same point in an Equilateral Triangle.