Inscribed Circle (Incircle) Calculator

Calculate the inscribed circle (incircle) of a triangle. Enter three sides or three vertex coordinates to find the inradius, incenter, tangent points, tangent lengths, contact triangle, and see an interactive diagram with step-by-step formulas.

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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

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Incircle

The largest circle fitting inside a triangle, tangent to all 3 sides

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Incenter

Intersection of the three angle bisectors — always inside the triangle

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Tangent Points

Three points where the incircle touches each side of the triangle

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Contact Triangle

Triangle formed by connecting the three tangent points (intouch triangle)

Inscribed Circle Formulas

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

Property	Formula	Description
Triangle Area (Heron's)	\(K = \sqrt{s(s-a)(s-b)(s-c)}\)	Area from three sides using semi-perimeter
Inradius	\(r = \frac{K}{s}\)	Radius of the inscribed circle
Incircle Area	\(A = \pi r^2\)	Area enclosed by the incircle
Incircle Circumference	\(C = 2\pi r\)	Perimeter of the incircle
Incenter Coordinates	\(I = \frac{a \cdot A + b \cdot B + c \cdot C}{a+b+c}\)	Weighted average of vertices by opposite side lengths
Tangent Length from A	\(t_A = s - a\)	Distance from vertex A to nearest tangent points
Excircle Radius	\(r_A = \frac{K}{s-a}\)	Radius of the excircle opposite vertex A
Euler's Distance	\(d = \sqrt{R(R-2r)}\)	Distance between circumcenter and incenter

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

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.
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.
Click Calculate: Press the "Calculate Inscribed Circle" button.
Review the results: See the inradius r, incenter coordinates, incircle area and circumference, tangent points, tangent lengths, excircle radii, and the R/r ratio.
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.

FAQ

What is an inscribed circle (incircle)?

An inscribed circle, or incircle, is the largest circle that fits entirely inside a triangle and is tangent to all three sides. Its center is called the incenter (the intersection of the three angle bisectors), and its radius is called the inradius. Every non-degenerate triangle has exactly one inscribed circle.

How do you find the inradius of a triangle?

The inradius r is calculated as r = K / s, where K is the triangle area and s is the semi-perimeter s = (a+b+c)/2. First compute the area using Heron's formula: K = √(s(s−a)(s−b)(s−c)). Then divide the area by the semi-perimeter.

Where is the incenter of a triangle?

The incenter is the point where all three angle bisectors of a triangle meet. Unlike the circumcenter, the incenter always lies inside the triangle, regardless of whether the triangle is acute, right, or obtuse. Its coordinates are a weighted average of the vertices: I = (a·A + b·B + c·C) / (a+b+c).

What are the tangent lengths of an incircle?

The tangent lengths are the distances from each vertex to the two nearest tangent points where the incircle touches the sides. From vertex A, the tangent length is s − a; from B, it is s − b; from C, it is s − c. These tangent lengths satisfy the property that two tangent segments from the same external point to a circle are equal in length.

What is the contact triangle?

The contact triangle (also called the intouch triangle) is formed by connecting the three points where the incircle touches the sides of the triangle. Its vertices lie on the sides of the original triangle at the tangent points. The contact triangle has interesting properties — for example, its angles are each equal to 90° minus half the corresponding angle of the original triangle.

Reference this content, page, or tool as:

"Inscribed Circle (Incircle) Calculator" at https://MiniWebtool.com/inscribed-circle-incircle-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-03

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Inscribed Circle (Incircle) Calculator

About Inscribed Circle (Incircle) Calculator

Key Concepts of the Inscribed Circle

Inscribed Circle Formulas

Incircle vs. Circumcircle

Tangent Lengths and the Contact Triangle

Excircles: The Three Companion Circles

How to Find the Inscribed Circle

Practical Applications

FAQ

Geometry Calculators:

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