Inscribed Circle Calculator
Calculate the inscribed circle (incircle) radius and area for any triangle using side lengths
Triangle Sides
Results
Visual Diagram
Formula:
r = Area / s
where s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]
About Inscribed Circles
An inscribed circle (incircle) is the largest circle that can be drawn inside a triangle, touching all three sides. The center of the incircle is called the incenter, which is the point where the three angle bisectors of the triangle meet.
Key Properties
- The incircle is tangent to all three sides of the triangle
- The incenter is equidistant from all three sides
- This distance equals the inradius (r)
- The inradius formula: r = Area / s, where s is the semi-perimeter
- Every triangle has exactly one inscribed circle
Applications
- Geometric constructions and proofs
- Architecture and design optimization
- Engineering and CAD applications
- Computer graphics and game development
- Mathematical problem solving
Frequently Asked Questions
What is an inscribed circle?
An inscribed circle (incircle) is the largest circle that fits inside a triangle, touching all three sides at exactly one point each. These points of contact are called the points of tangency.
How do you find the radius of an inscribed circle?
The inradius (r) is calculated using the formula r = Area / s, where Area is the triangle's area (found using Heron's formula) and s is the semi-perimeter (half the sum of all three sides).
What is the incenter of a triangle?
The incenter is the center of the inscribed circle. It's the point where the three angle bisectors of the triangle intersect. The incenter is always inside the triangle, regardless of the triangle's type.
Does every triangle have an inscribed circle?
Yes, every triangle has exactly one inscribed circle. This is true for all types of triangles: acute, right, and obtuse triangles all have a unique incircle.
What is the relationship between the inradius and triangle area?
The triangle's area equals the inradius multiplied by the semi-perimeter: Area = r × s. This relationship is fundamental and can be rearranged to solve for the inradius: r = Area / s.
How is an inscribed circle different from a circumscribed circle?
An inscribed circle (incircle) is inside the triangle and touches all three sides. A circumscribed circle (circumcircle) is outside the triangle and passes through all three vertices. The incircle is always smaller than the circumcircle.
Can you construct an inscribed circle with compass and straightedge?
Yes, you can construct an inscribed circle using classical geometric construction. Draw two angle bisectors to find the incenter, then draw a perpendicular from the incenter to any side to find the radius. Use this radius to draw the circle.
What happens to the inradius if you scale the triangle?
If you scale a triangle by a factor k (multiply all sides by k), the inradius also scales by the same factor k. This is because both the area and semi-perimeter scale proportionally.