Inscribed Circle (Incircle) Calculator
The incircle touches all three sides of a triangle. MiniWebtool’s Inscribed Circle (Incircle) Calculator finds its radius, center, and contact geometry from three side lengths or vertex coordinates, with a diagram and formula steps.
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Enter three side lengths or three vertex coordinates below to get the inradius, incenter, tangent points and area, with the steps.
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.
At a Glance
Inscribed Circle (Incircle) Calculator finds the inradius and incenter of a triangle from three side lengths or three vertex coordinates. It also gives tangent points, excircle radii and a diagram with formula steps.
| Feature | What you get | Good for |
|---|---|---|
| Two input modes | Enter side lengths a, b and c, or the x and y coordinates of three vertices. Quick examples fill in equilateral, right 3-4-5, isosceles 5-5-8 and obtuse 3-4-6 triangles. | Triangles given by side lengths or by coordinates |
| Inradius and incenter | The inradius is r = K / s, with area K from Heron's formula and semi-perimeter s. The incenter is where the three angle bisectors meet. | Geometry homework on triangle circles |
| Incircle measures | Shows the incircle area (pi r squared) and circumference (2 pi r) alongside the inradius. | Sizing the largest circle that fits inside a triangle |
| Tangent points | Gives the three points where the incircle touches the sides, the tangent lengths s - a, s - b and s - c, and the contact triangle. | Constructions that need the touch points |
| Excircles | Reports the three excircle radii and the R/r ratio, which Euler's inequality bounds below by 2. | Comparing the incircle with the circumcircle |
| Diagram overlays | Toggle the incircle, angle bisectors, tangent points, contact triangle and labels on the triangle diagram. | Visualizing how the pieces fit together |
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:
| 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
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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