Inscribed Circle Calculator
Find the inradius of a triangle from its three sides.
The inscribed circle is the largest circle that fits inside a triangle, touching all three sides.
How the Math Works
The inradius (r) of a triangle is calculated using the formula r = Area ÷ s, where s is the semi-perimeter. First, compute the semi-perimeter (s) by adding all three sides (a, b, c) and dividing by 2: s = (a + b + c)/2. The area is derived using Heron's formula: Area = √[s(s - a)(s - b)(s - c)]. Divide this area by the semi-perimeter to find the inradius, which represents the radius of the largest circle that fits perfectly inside the triangle, touching all three sides.
Practical Applications
This calculation is vital in fields like architecture, engineering, and design. For instance, when creating triangular structures or layouts, knowing the inradius helps determine the size of circular components that fit within the triangle, such as columns, decorative elements, or utility placements. It’s also used in optimizing space, like designing a circular fountain in a triangular courtyard or fitting a circular base for a triangular table. In computer graphics, it aids in modeling objects with precise geometric constraints.
Day-to-Day Use
While not always obvious, this concept appears in everyday scenarios like DIY projects, landscaping, or art. Imagine designing a triangular garden bed and wanting to place a circular water feature centered perfectly within it. Similarly, carpenters might use it to ensure a circular tabletop fits snugly inside a triangular base. Even in cooking, shaping dough into a triangle and calculating the inscribed circle could help create evenly sized circular snacks or pastries. It simplifies problems requiring efficient, symmetrical space usage in practical, tangible ways.
Worked example
Sides 3, 4, 5 → area 6, semi-perimeter 6, inradius 1.
FAQ
Where is the centre?
At the incentre, where the triangle's angle bisectors meet.