Library / Geometry, coordinates, surveying and trigonometry / GEO-11
Geometry, coordinates, surveying and trigonometry
Inradius of a triangle from area and perimeter
Radius of the circle that touches all three sides, from the triangle area and perimeter.
- Ref
- GEO-11
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- r [cm]
For a triangle with incircle radius r, joining the incentre to the three corners makes three triangles, each with height r above one side. Their areas add up to ½ r (a + b + c) = ½ r P, which equals A. So r = 2A/P. The incentre is where the three angle bisectors meet.
Radius of the circle that touches all three sides, from the triangle area and perimeter. Joining the incentre to the corners splits the triangle into three triangles of height r.
Calculator
- Diameter of the incircle, 4A/P2 cm
- Area of the incircle, π r²3.142 cm²
- Share of the triangle covered by its incircle0.5236
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A triangle of area 6 cm² and perimeter 12 cm has an incircle of radius 1 cm; the 3–4–5 triangle gives exactly 1 cm.
- The incircle covers 52.4 % of the triangle, so thin triangles waste more of it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Triangle area | 0.5 to 100 cm² | 0.08333 | 16.67 | 16.58 | +10% | +10 % |
| P Triangle perimeter | 2 to 80 cm | 0.15 | 6 | 5.85 | +10% | −9.09 % |
Sweep of Triangle area
| Triangle area (cm²) | Inradius (cm) |
|---|---|
| 0.5 | 0.08333 |
| 12.94 | 2.156 |
| 25.38 | 4.229 |
| 37.81 | 6.302 |
| 50.25 | 8.375 |
| 62.69 | 10.45 |
| 75.13 | 12.52 |
| 87.56 | 14.59 |
| 100 | 16.67 |
Sweep one variable
Variables and how each one contributes
Inradius
What the equation gives you. Everything else feeds it.
Triangle area
The area of the triangle, in cm². A larger area for the same perimeter gives a larger incircle. Raising it by 10 % (to 6.6) moves the result by +10 %, from 1 to 1.1 cm.
Triangle perimeter
The perimeter of the triangle, in cm. A longer perimeter for the same area gives a smaller incircle. Raising it by 10 % (to 13.2) moves the result by −9.09 %, from 1 to 0.9091 cm.
Worked example with the values above
- Double the area: 2A = 12 cm².
- Perimeter: P = 12 cm.
- r = 12 / 12 = 1 cm.
Assumptions and limits
- A non-degenerate triangle: the three sides satisfy the strict triangle inequality.
- All lengths and the area use one consistent unit.
Common mistakes
- Using an area that does not match the perimeter's triangle.
- Using the semi-perimeter in place of the perimeter, which doubles r.
- Mixing units, such as area in cm² and perimeter in mm.
Statistics and uncertainty
The inradius is a ratio of two measured quantities, so errors in the area and perimeter combine. A 1 % error in either gives about a 1 % error in r.
Where it is used
- Fitting the largest circle in a triangle and the circle through three points.
- Centre lines, balance points and bracing positions of triangular frames and plates.
Maths you need
- Triangle inequality: any two sides add to more than the third
- Heron’s formula: area from three sides
- Midpoint: the point halfway along a segment