Sum of interior angles | Geometry | Quantitative reasoning

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:

$50 + 70 + x 120 + x x = 180 = 180 = 60$

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.

Definitions

Sum of interior angles for an $n$ -sided shape: $(n - 2) \times 180$ degrees

For instance, a triangle (3 sides) has interior angles with a sum of $(n - 2) \times 180 = (3 - 2) \times 180 = 1 \times 180 = 180$ degrees.

A square (4 sides) has interior angles with a sum of $(n - 2) \times 180 = (4 - 2) \times 180 = 2 \times 180 = 360$ degrees.

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

$sum of interior angles = (n - 2) \times 180 = (5 - 2) \times 180 = 3 \times 180 = 540 degrees$

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.

Irregular hexagon sum of interior angles 720 degrees

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:

$sum of interior angles = (n - 2) \times 180 = (6 - 2) \times 180 = 4 \times 180 = 720 degrees$

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 $x$ .

$90 + 90 + 4 x + x + 2 x + 2 x 180 + 9 x 9 x x = 720 = 720 = 540 = 60$