What this calculator does
The incentre is where the three angle bisectors of a triangle meet, and it is the centre of the largest circle that fits inside. Unlike the centroid, it is a weighted average of the vertices, with each weighted by the opposite side length.
The inradius follows from area and semiperimeter. A triangle of area 24.000 with a semiperimeter of 11.405 has an inradius of 2.104, and that relationship holds for every triangle.
The formula
Each vertex is weighted by the length of the side opposite it, and the weighted average gives the incentre. The inradius is the area divided by the semiperimeter.
| Term | Meaning |
|---|---|
| Incentre | Where the angle bisectors meet, and the centre of the inscribed circle. |
| Inradius | The radius of the largest circle fitting inside the triangle. |
| Semiperimeter | Half the perimeter, which appears throughout triangle geometry. |
The inputs explained
| Field | What to enter |
|---|---|
| Vertex A: x | Vertex A, x coordinate. |
| Vertex A: y | Vertex A, y coordinate. |
| Vertex B: x | Vertex B, x coordinate. |
| Vertex B: y | Vertex B, y coordinate. |
| Vertex C: x | Vertex C, x coordinate. |
| Vertex C: y | Vertex C, y coordinate. |
When to use it
Inscribing a circle
The incentre and inradius fully determine the inscribed circle.
Geometry problems
The incentre is one of the classical triangle centres.
Understanding weighted averages
The incentre weights each vertex by its opposite side.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does stretching the triangle change the inradius?
The third vertex raised progressively.
Questions
How does the incentre differ from the centroid?
The centroid is the plain average of the three vertices and is the centre of mass. The incentre weights each vertex by its opposite side length, and it is the point equidistant from all three sides rather than from the vertices.
Why is the inradius area over semiperimeter?
Because the triangle splits into three smaller triangles with apex at the incentre, each with height equal to the inradius. Their areas sum to the inradius times the semiperimeter, which must equal the total area.
Is the incentre always inside the triangle?
Always, without exception. It is the meeting point of internal angle bisectors, which cannot leave the triangle. The circumcentre and orthocentre can both fall outside for obtuse triangles, but the incentre never does.
What are the other triangle centres?
The centroid, circumcentre and orthocentre are the other three classical ones. Remarkably, those three always lie on a single straight line called the Euler line, though the incentre generally does not.
For polygon area and centroid, see the shoelace formula calculator. For triangle area from sides, see the triangle area calculator.