Incircle Radius Calculator
Free Incircle radius Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
r = Area / s = sqrt(s(s-a)(s-b)(s-c)) / s
Where r is the inradius, s is the semi-perimeter (a+b+c)/2, and a, b, c are the three side lengths. The area is computed using Heron's formula. The inradius equals the area divided by the semi-perimeter.
Worked Examples
Example 1: Incircle of a 5-6-7 Triangle
Problem:Find the inradius and related properties of a triangle with sides 5, 6, and 7.
Solution:Semi-perimeter s = (5 + 6 + 7) / 2 = 9\nArea = sqrt(9 * (9-5) * (9-6) * (9-7)) = sqrt(9 * 4 * 3 * 2) = sqrt(216) = 14.6969\nInradius r = Area / s = 14.6969 / 9 = 1.6330\nCircumradius R = (5 * 6 * 7) / (4 * 14.6969) = 210 / 58.7878 = 3.5707\nR/r ratio = 3.5707 / 1.6330 = 2.186
Result:Inradius = 1.6330, Area = 14.6969, Circumradius = 3.5707, R/r = 2.186
Example 2: Incircle of a 3-4-5 Right Triangle
Problem:Calculate the inradius of the classic 3-4-5 right triangle using the simplified formula.
Solution:Using the right triangle formula: r = (a + b - c) / 2 = (3 + 4 - 5) / 2 = 1\nVerification: s = (3+4+5)/2 = 6, Area = 0.5*3*4 = 6\nr = Area/s = 6/6 = 1 (confirmed)\nCircumradius R = c/2 = 5/2 = 2.5\nR/r = 2.5
Result:Inradius = 1, Area = 6, Circumradius = 2.5, R/r = 2.5
Frequently Asked Questions
What is the incircle of a triangle?
The incircle (or inscribed circle) of a triangle is the largest circle that fits entirely inside the triangle, touching all three sides. The center of the incircle is called the incenter, which is the point where all three angle bisectors of the triangle intersect. The incircle is tangent to each side of the triangle at exactly one point, meaning it just barely touches each side without crossing it. Every triangle, regardless of its shape, has exactly one incircle, making it a fundamental geometric property. The incircle is particularly important in computational geometry, triangle centers research, and various engineering applications where fitting the largest possible circle inside a triangular region is required.
What special properties does the incircle have for right triangles?
For right triangles, the inradius has an especially simple formula: r = (a + b - c) / 2, where a and b are the legs and c is the hypotenuse. This can also be written as r = a + b - c all divided by 2. For the classic 3-4-5 right triangle, the inradius is (3 + 4 - 5) / 2 = 1. The incircle of a right triangle touches the hypotenuse at a point that divides it into two segments equal to s - a and s - b, where s is the semi-perimeter. Another neat property is that the incenter of a right triangle lies at coordinates (r, r) from the right-angle vertex, meaning it is equidistant from both legs by exactly the inradius value.
Can the incircle be used in practical engineering applications?
The incircle has numerous practical applications in engineering, manufacturing, and design. In machining, the inradius determines the largest circular hole or shaft that can be cut from a triangular piece of material, minimizing waste. In structural engineering, the incircle helps analyze stress distribution in triangular cross-sections of beams and trusses. In mesh generation for finite element analysis, the incircle quality metric (ratio of inradius to circumradius) determines element quality, with values closer to 0.5 indicating better-shaped elements. In packaging design, the incircle determines the largest cylindrical object that fits inside a triangular container. The concept also extends to polygon incircles used in computational geometry for collision detection and spatial partitioning algorithms.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy