Inscribed Circle Calculator
Inradius (r): -
Center Coordinates: -
Area of Triangle: -
Understanding Inscribed Circles
What is an Inscribed Circle?
An inscribed circle (incircle) is the largest circle that can be drawn inside a triangle touching all three sides. Key properties include:
- Tangent to all three sides of the triangle
- Center (incenter) is equidistant from all sides
- Center is the intersection of angle bisectors
- Radius is called the inradius
- Area relates directly to the triangle's semiperimeter
Key Formulas
Inradius Formula
r = A/s
where A is area and s is semiperimeter
Area Formula
A = √(s(s-a)(s-b)(s-c))
Semiperimeter
s = (a + b + c)/2
Advanced Properties
Area Relationship
Area = rs
Touch Points
Divide sides proportionally
Gergonne Point
Formed by lines from vertices to touch points
Euler's Theorem
Relates to circumradius: R ≥ 2r
Special Cases
Equilateral Triangle
r = a/(2√3) where a is side length
Right Triangle
r = (a + b - c)/2 where c is hypotenuse
Isosceles Triangle
Incenter lies on the height to base
Real-World Applications
Architecture
Used in designing circular features within triangular spaces
Engineering
Applied in optimal material usage and stress distribution
Design
Essential in logo design and geometric patterns
Construction
Used in planning circular structures within triangular plots