An inscribed circle, also known as an incircle, is a circle that fits perfectly inside a triangle, touching all three sides. The radius of this circle is an important geometric property that can be calculated using a straightforward formula.
Key Concepts
Semi-Perimeter
First, let’s understand the concept of the semi-perimeter of a triangle. The semi-perimeter, denoted as $s$, is half the perimeter of the triangle. If the sides of the triangle are $a$, $b$, and $c$, then the semi-perimeter is given by:
$s = frac{a + b + c}{2}$
Area of the Triangle
The area of the triangle, denoted as $A$, can be calculated using various methods, like Heron’s formula if the sides are known:
$A = sqrt{s(s-a)(s-b)(s-c)}$
Formula for the Radius
The radius $r$ of the inscribed circle can be found using the formula:
$r = frac{A}{s}$
where $A$ is the area of the triangle, and $s$ is the semi-perimeter.
Step-by-Step Example
Let’s go through an example to make this clearer.
Example Problem
Suppose we have a triangle with sides $a = 7$ units, $b = 8$ units, and $c = 9$ units. We want to find the radius of the inscribed circle.
Calculate the Semi-Perimeter
$s = frac{7 + 8 + 9}{2} = 12$ unitsCalculate the Area
Using Heron’s formula:
$A = sqrt{12(12-7)(12-8)(12-9)}$
$A = sqrt{12 times 5 times 4 times 3}$
$A = sqrt{720}$
$A = 12 sqrt{5}$ square unitsCalculate the Radius
$r = frac{12 sqrt{5}}{12}$
$r = sqrt{5}$ units
Thus, the radius of the inscribed circle is $sqrt{5}$ units.
Conclusion
Understanding how to find the radius of an inscribed circle is a useful geometric skill. By using the formula $r = frac{A}{s}$, where $A$ is the area of the triangle and $s$ is the semi-perimeter, you can easily determine the radius. This knowledge is not only useful in geometry but also in various applications in science and engineering.
3. Wikipedia – Incircle and Excircle