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Finding Area with the Inradius

Vocabulary

There are a few terms we need to cover before we can dive in.

Semi-Perimeter: One half of the perimeter, $s=\frac{a+b+c}{2}$

Incircle: Circle within a triangle that touches each side of the triangle in one place

Inradius: Radius of the incircle

These are all the terms we need to know to use this method of finding area.

Finding the Inradius

We can start with a triangle ABC.

We're going to draw a bisector from each angle. This is a line that splits the angle in half. The point where they meet will be the center point of our incircle.

Now, we need to find the radius of the circle. We're going to make a line from the center of the incircle perpendicular to each line in the triangle. The length of any of these lines is the inradius.

The length of the radius can be used to find the area.

Calculating the Area

Recall the equation for the semi-perimeter, $s=\frac{a+b+c}{2}$. We can multiply this by the length of the inradius to find the area. $$A=s\cdot r$$

Inradius, semi-perimeter, and area

The inradius of a triangle is formed by first dividing each of the three angles in half by a line (refer to dotted lines in the below image). The point at which these three lines meet is the center of the incircle, and the inradius is a line drawn from the center to perpendicularly intersect a side of the triangle.

The semi-perimeter is one half of the perimeter and is denoted s in the above image.

Once the inradius and semi-perimeter are known, the area of the triangle is easily solved.

Area = inradius x semi-perimeter

How we determine the inradius is rephrased in the below video.

References:

Agutie.homestead

Art of Problem Solving

(Video) Circumradius and Inradius of Triangle

by Lecture Videos for Entrance Exams

This video by Lecture Videos for Entrance Exams covers the circumradius and inradius of a circle.

$\large{\text{Summary}}$

The incircle is the largest circle that can fit inside of a triangle. It touches each side of the triangle at exactly one place. The radius of this circle is known as the inradius.

Inradius can be calculated with the following equation: $$r = \frac{A}{s}$$ Where $A$ is the area of the triangle, and $s$ is the semi-perimeter of the triangle, or one-half of the perimeter.

You can use this equation to find the radius of the incircle given the three side lengths of a triangle. Let's try it out.

What is the inradius of a right triangle with a height of $3$, base of $4$, and a hypotenuse of $5$

First, let's calculate the area using $A=\frac12 bh$. $$\begin{align} A &= \frac12 \cdot4 \cdot 3\\ A&= \frac{12}{2} \\ A&=6 \end{align}$$

Now, let's calculate the semi-perimeter, which is half of the perimeter. $$\begin{align}s &= \frac{a+b+c}{2}\\ s &= \frac{3+4+5}{2}\\ s &= \frac{12}{2}\\ s&=6 \end{align}$$

Finally, we can calculate the inradius with $r=\frac{A}{s}$. $$\begin{align}r&=\frac{A}{s}\\ r&=\frac{6}{6}\\ r&=1 \end{align}$$