Inscribed Circle (Incircle) Calculator

About Inscribed Circle (Incircle) Calculator

The Inscribed Circle (Incircle) Calculator finds the inscribed circle of any triangle. The inscribed circle — also known as the incircle — is the largest circle that fits entirely inside a triangle, tangent to all three sides. Enter three side lengths or three vertex coordinates to instantly compute the inradius, incenter location, tangent points, tangent lengths, contact triangle, excircle radii, and more, with an interactive SVG diagram and step-by-step formulas.

Key Concepts of the Inscribed Circle

Inscribed Circle Formulas

For a triangle with sides a, b, c and semi-perimeter s = (a + b + c) / 2:

Incircle vs. Circumcircle

The incircle and circumcircle are the two most fundamental circles associated with a triangle, but they have distinct properties:

• Incircle: Fits inside the triangle, tangent to all three sides. Found via angle bisectors. The incenter always lies inside the triangle.
• Circumcircle: Passes through all three vertices, usually larger. Found via perpendicular bisectors. The circumcenter can lie outside for obtuse triangles.
• Euler's inequality: For any triangle, \(R \geq 2r\), with equality only for equilateral triangles.

Tangent Lengths and the Contact Triangle

When the incircle touches side BC at point D, side CA at point E, and side AB at point F, the tangent lengths from each vertex are equal: from A, the distances AF = AE = s − a; from B, BF = BD = s − b; from C, CD = CE = s − c. The triangle DEF formed by connecting these tangent points is called the contact triangle (or intouch triangle). The contact triangle has special properties: its angles are related to the original triangle's angles by the formula ∠D = 90° − A/2.

Excircles: The Three Companion Circles

Every triangle has three excircles — circles that are tangent to one side of the triangle and to the extensions of the other two sides. The excircle opposite vertex A has radius r_A = K/(s−a), opposite B has r_B = K/(s−b), and opposite C has r_C = K/(s−c). An elegant identity connects all four: 1/r = 1/r_A + 1/r_B + 1/r_C. Excircles are essential in advanced triangle geometry and appear in the Nagel point construction.

How to Find the Inscribed Circle

1. Choose your input method: Select "Three Sides" if you know the side lengths a, b, c, or "Three Vertices" if you have the coordinates of each vertex.
2. Enter the values: Input the three side lengths or the (x, y) coordinates of vertices A, B, and C. Click a quick example to auto-fill sample values.
3. Click Calculate: Press the "Calculate Inscribed Circle" button.
4. Review the results: See the inradius r, incenter coordinates, incircle area and circumference, tangent points, tangent lengths, excircle radii, and the R/r ratio.
5. Explore the diagram: Toggle overlays for the incircle, angle bisectors, tangent points, contact triangle, and labels to visualize the geometry.

Practical Applications

The inscribed circle has many practical uses. In manufacturing, the inradius determines the largest circular component (bolt, drill bit, pipe) that fits within a triangular opening. In architecture, incircles help design maximal circular features within triangular floor plans. In computational geometry, the incircle and excircles are used in mesh refinement algorithms for finite element analysis. The incircle radius also serves as a measure of triangle "fatness" — thin triangles have small inradii relative to their circumradii, which is important for numerical stability in simulations.

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