Sum of interior angles

Every shape has a unique sum of its interior (inside) angles:

• Triangle (3 sides): $180$ degrees
• Square (4 sides): $360$ degrees
• Pentagon (5 sides): $540$ degrees
• Hexagon (6 sides): $720$ degrees
• Heptagon (7 sides): $900$ degrees
• Octagon (8 sides): $1080$ degrees

For example, imagine a triangle with two interior angles of $50$ and $70$. That’s enough information to know that the third angle will be $60$, since $50+70+60=180$. Here’s the math:

$$\begin{array}{rl}50+70+x& =180\\ 120+x& =180\\ x& =60\end{array}$$

But what if the shape had four, five, or even six sides? Would those interior angles add up to $180$ as well? No… they would add up to a different number, but you don’t need to memorize the list above, since there’s a simple equation to find the sum of the interior angles of any shape.

Since $n$ in this equation represents the number of sides, if we had $5$ sides, the sum of interior angles would be $540$:

$$\begin{array}{rl}\text{sum\hspace{0.33em}of interior angles}& =(n-2)\times 180\\ & =(5-2)\times 180\\ & =3\times 180\\ & =540\text{degrees}\end{array}$$

And this works for any 5-sided pentagon, regardless of the shape! The individual angles might be different, but the total sum will always be $540$.

Interior angles of irregular polygons

Let’s try an example that’s a bit more difficult, where we need to find the angles for an irregular polygon.

Solve for the variable $x$ in the figure below.

Take a moment to try it yourself, then continue reading to check your work.

The first step to solving for $x$ is to determine the sum of the interior angles of this shape. Counting them up, we can see that the shape has $6$ sides and a few different angles, making this an irregular hexagon.

Regardless of what it’s called, we now know we can plug in $6$ for the $6$ sides into the interior angles formula:

$$\begin{array}{rl}\text{sum\hspace{0.33em}of interior angles}& =(n-2)\times 180\\ & =(6-2)\times 180\\ & =4\times 180\\ & =720\text{degrees}\end{array}$$

The sum of the interior angles in this hexagon must be $720$. Now, we can write an equation that equates all the interior angles to $720$, remembering that right angles are 90 degrees, and solve for $x$.

$$\begin{array}{rl}90+90+4x+x+2x+2x& =720\\ 180+9x& =720\\ 9x& =540\\ x& =60\end{array}$$