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 for accuracy by Manoj Kumar, Mathematics Educator
Incircle Radius Calculator
Calculator
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Formula: r = Area / s = sqrt(s(s-a)(s-b)(s-c)) / s
Worked example — Inradius = 1.6330, Area = 14.6969, Circumradius = 3.5707, R/r = 2.186
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 Area = sqrt(9 * (9-5) * (9-6) * (9-7)) = sqrt(9 * 4 * 3 * 2) = sqrt(216) = 14.6969 Inradius r = Area / s = 14.6969 / 9 = 1.6330 Circumradius R = (5 * 6 * 7) / (4 * 14.6969) = 210 / 58.7878 = 3.5707 R/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 Verification: s = (3+4+5)/2 = 6, Area = 0.5*3*4 = 6 r = Area/s = 6/6 = 1 (confirmed) Circumradius R = c/2 = 5/2 = 2.5 R/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.
How is the inradius calculated from side lengths?
The inradius formula uses two key components: the area of the triangle and the semi-perimeter. First, calculate the semi-perimeter s = (a + b + c) / 2, where a, b, and c are the three side lengths. Then find the area using Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)). The inradius r is simply the area divided by the semi-perimeter: r = Area / s. This elegant relationship arises because the triangle can be divided into three smaller triangles by connecting the incenter to each vertex, and each smaller triangle has a height equal to the inradius. The total area of these three triangles equals (1/2)(a)(r) + (1/2)(b)(r) + (1/2)(c)(r) = r times s, which gives us r = Area / s.
What is the relationship between the inradius and circumradius?
The inradius (r) and circumradius (R) of a triangle are related by Euler's formula: the distance d between the incenter and circumcenter satisfies d^2 = R(R - 2r). This means R is always at least 2r, with equality only for equilateral triangles. The ratio R/r provides insight into how close a triangle is to being equilateral. For an equilateral triangle, R/r = 2 exactly. As the triangle becomes more elongated or obtuse, this ratio increases. Another useful relationship is r = 4R times sin(A/2) times sin(B/2) times sin(C/2), where A, B, and C are the angles. These relationships are fundamental in triangle geometry and appear frequently in mathematical competitions and advanced geometric proofs.
What are the exradii and how do they relate to the inradius?
The exradii are the radii of the three excircles 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 side a is ra = Area / (s - a), opposite to side b is rb = Area / (s - b), and opposite to side c is rc = Area / (s - c). A beautiful relationship connects all four radii: 1/r = 1/ra + 1/rb + 1/rc. Additionally, the product of all four radii satisfies r times ra times rb times rc = Area^2. For an equilateral triangle with side length a, the inradius is a/(2 times sqrt(3)) and each exradius is a times sqrt(3)/2, making each exradius exactly three times the inradius.
How do you find the incenter coordinates?
The incenter coordinates are found using a weighted average of the triangle vertices, where the weights are the opposite side lengths. If the triangle has vertices A(x1,y1), B(x2,y2), C(x3,y3) with opposite side lengths a, b, c respectively, then the incenter I has coordinates Ix = (a*x1 + b*x2 + c*x3) / (a+b+c) and Iy = (a*y1 + b*y2 + c*y3) / (a+b+c). This is called a barycentric combination with barycentric coordinates (a:b:c). The incenter always lies inside the triangle regardless of triangle type, unlike the circumcenter which can lie outside for obtuse triangles. The incenter is equidistant from all three sides, and that distance equals the inradius.
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.
How does the inradius relate to triangle area?
The inradius provides one of the most elegant area formulas in geometry: Area = r times s, where r is the inradius and s is the semi-perimeter. This means if you know the inradius and the perimeter, you can instantly calculate the area. Conversely, knowing the area and perimeter immediately gives you the inradius. This relationship has a beautiful geometric interpretation: the triangle can be decomposed into three triangles formed by connecting the incenter to each vertex, each having height r and base equal to a side of the triangle. The sum of these three triangle areas equals (1/2)(a)(r) + (1/2)(b)(r) + (1/2)(c)(r) = r(a+b+c)/2 = rs. This formula is used extensively in computational geometry algorithms.
What is the inradius of an equilateral triangle?
For an equilateral triangle with side length a, the inradius is r = a / (2 times sqrt(3)), which simplifies to r = a times sqrt(3) / 6, approximately 0.2887 times a. The circumradius is R = a / sqrt(3) = a times sqrt(3) / 3, which is exactly twice the inradius. This 2:1 ratio between circumradius and inradius is the minimum possible for any triangle, making the equilateral triangle the most compact triangular shape. The area of an equilateral triangle is a^2 times sqrt(3) / 4, and using r = Area/s = (a^2 times sqrt(3)/4) / (3a/2) confirms the inradius formula. The equilateral triangle is the only triangle where the incenter, circumcenter, centroid, and orthocenter all coincide at the same point.
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.
How does the inradius change as the triangle shape changes?
The inradius is maximized for an equilateral triangle with a given perimeter, as the equilateral shape encloses the maximum area among all triangles with the same perimeter. As a triangle becomes more elongated (with one very long side and two shorter sides), the inradius decreases and approaches zero as the triangle degenerates into a line segment. For isosceles triangles, the inradius increases as the triangle becomes more equilateral and decreases as it becomes either very tall and narrow or very flat and wide. For a fixed area, decreasing the perimeter increases the inradius since r = Area/s. The inradius is also sensitive to obtuse angles: very obtuse triangles (angles approaching 180 degrees) have very small inradii relative to their longest side.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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