Sum of interior angles

Every polygon 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, suppose a triangle has two interior angles of $50$ and $70$. You can find the third angle because the interior angles of any triangle add up to $180$ degrees. So the third angle must be $60$, since $50+70+60=180$. Here’s the algebra:

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

But what if the shape has four, five, or six sides? The interior angles won’t add up to $180$ anymore - they add up to a different total. You don’t need to memorize the list above, though, because there’s a simple formula for the sum of the interior angles of any polygon.

In this formula, $n$ is the number of sides. So if a polygon has $5$ sides, the sum of its interior angles is $540$:

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

This works for any 5-sided pentagon, no matter what it looks like. The individual angles may vary, but the total will always be $540$.

Interior angles of irregular polygons

Now let’s try a more challenging example, where you need to solve for an angle in an irregular polygon.

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

Try it on your own first, then read on to check your work.

Start by finding the sum of the interior angles. Count the sides: the shape has $6$ sides, so it’s a hexagon. Because the angles aren’t all the same, it’s an irregular hexagon.

No matter the shape, you can use the interior-angle-sum formula with $n=6$:

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

So the interior angles must add up to $720$ degrees. Next, add the angles shown in the diagram and set their sum equal to $720$. Remember that each right angle is $90$ degrees.

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