Introduction to Inradius of Inscribed Circle in a Triangle

The inradius (r) of a circle inscribed within a triangle is the radius of the largest circle that fits within the triangle and touches all three sides. This article aims to explore the methods and formulas for calculating the inradius of a circle inscribed in a triangle, focusing specifically on the scenario where the triangle is inscribed within a square or an equilateral triangle. Additionally, we will discuss the relationship between the inradius and the sides of the triangle, providing detailed mathematical derivations and examples.

Calculating the Inradius for a Triangle with Given Perimeter

Let's consider a triangle with a perimeter of (P 23x). The semi-perimeter (s) is given by:

[s frac{P}{2} frac{23x}{2} 11.5x]

The area (A) of the triangle can be calculated using the formula:

[A sqrt{s(s-a)(s-b)(s-c)}]

Given the specific side lengths and simplifying the expression, we can find the area and subsequently the inradius (r). Let's assume the sides are (a x), (b 2.50.5x), and (c 3.25x). Plugging these into the area formula:

[A sqrt{11.5x(11.5x - x)(11.5x - 2.50.5x)(11.5x - 3.25x)}]

Further simplification yields:

[A sqrt{11.5x cdot 10.5x cdot 9x cdot 8.25x}]

Then the inradius (r) is given by:

[r frac{A}{s}] [r frac{sqrt{11.5x cdot 10.5x cdot 9x cdot 8.25x}}{11.5x}]

For a more simplified example, let's consider a right-angled isosceles triangle with equal legs of length (a). The perimeter is given by:

[P 2x sqrt{2}a]

The inradius of a right-angled isosceles triangle with legs of length (a) and hypotenuse (c asqrt{2}) is:

[r frac{a}{2 sqrt{2}}]

Inradius of an Equilateral Triangle

In an equilateral triangle, the incenter (where the inradius meets the sides) is the same as the centroid and orthocenter. Given the side length of the equilateral triangle (s), the inradius (r) is:

[r frac{ssqrt{3}}{6}]

For a square inscribed within the same equilateral triangle, the inradius can also be found using the side length of the square (s). The diameter of the inscribed circle is equal to the side length of the square. Therefore, the inradius is:

[r frac{s}{2}]

Geometrical Relationship for Any Triangle

A key relationship in triangle geometry is that the inradius (r) of a circle inscribed in a triangle can be derived using trigonometric functions. Specifically, for a triangle with sides (a), (b), and (c), and area (A), the inradius (r) is given by:

[r frac{A}{s}]

Where (s) is the semi-perimeter of the triangle. This relationship is particularly useful for calculating the inradius in various types of triangles, including right-angled triangles.

Conclusion

Understanding the inradius of a circle inscribed in a triangle is crucial for various geometric problems. By leveraging trigonometric relationships and the semi-perimeter area formula, one can easily find the inradius regardless of the type of triangle. Whether dealing with right-angled triangles, equilateral triangles, or squares, the inradius can be accurately calculated with the right formulas and steps.