Inradius Calculator
Calculate the inradius (inscribed circle radius) of a triangle using multiple methods
Input Values
Results
Visual Diagram
Formula:
r = A / s
where s = (a + b + c) / 2
A = √[s(s-a)(s-b)(s-c)]
I = Incenter (intersection of angle bisectors)
Understanding the Inradius
The inradius is the radius of the inscribed circle (incircle) of a triangle. It represents the radius of the largest circle that can fit inside the triangle while touching all three sides.
Calculation Methods
- Using Three Sides: Calculate area with Heron's formula, then r = A/s
- Using Area & Perimeter: Directly apply r = A/s where s = perimeter/2
- Using Angles: For advanced cases, r = (s-a)tan(A/2) where s is semi-perimeter
- All methods yield the same result for the same triangle
Important Relationships
- Area = r × s (area equals inradius times semi-perimeter)
- The incenter divides angle bisectors in specific ratios
- For equilateral triangles: r = a/(2√3) where a is side length
- For right triangles: r = (a + b - c)/2 where c is the hypotenuse
- The inradius is always less than the circumradius
Applications
- Geometric design and optimization problems
- Collision detection in computer graphics
- Structural engineering and space utilization
- Mathematical competitions and problem solving
Frequently Asked Questions
What is the inradius of a triangle?
The inradius is the radius of the inscribed circle (incircle) that fits perfectly inside a triangle, touching all three sides. It's the largest circle that can be drawn inside the triangle.
How do you calculate inradius from side lengths?
First, calculate the semi-perimeter s = (a+b+c)/2. Then find the area using Heron's formula: A = √[s(s-a)(s-b)(s-c)]. Finally, calculate inradius: r = A/s.
What is the incenter and how is it related to inradius?
The incenter is the center of the inscribed circle, located at the intersection of the triangle's angle bisectors. The distance from the incenter to any side equals the inradius.
Can I calculate inradius if I only know the area?
Yes, but you also need the perimeter (or semi-perimeter). The formula r = Area/s requires both the area and the semi-perimeter to calculate the inradius.
What's the inradius of an equilateral triangle?
For an equilateral triangle with side length a, the inradius is r = a/(2√3) or approximately r = 0.289a. This can also be written as r = a√3/6.
How is inradius different from circumradius?
The inradius (r) is for the circle inside the triangle touching all sides, while the circumradius (R) is for the circle outside passing through all vertices. For any triangle, r ≤ R/2, with equality only for equilateral triangles.
What's the inradius formula for a right triangle?
For a right triangle with legs a and b and hypotenuse c, there's a simple formula: r = (a + b - c)/2. This is much simpler than using Heron's formula.
Is the incenter always inside the triangle?
Yes, unlike the circumcenter which can be outside for obtuse triangles, the incenter is always located inside the triangle regardless of the triangle's type (acute, right, or obtuse).