## Interior Angles of Polygons

A short chart of additional polygons with their interior angles provides the following data:Pentagon 5 interior angles of 108º (540º or 3π radians total) Hexagon 6 interior angles of 120º (720º or 4π radians total) Heptagon 7 interior angles of about 128.57º (900º or 5π radians total) Octagon 8 interior angles of 135º (1080º or 6π radians total)

Clearly, the sum of the interior angles of a regular polygon obeys the relationship: An n-sided polygon has a total interior angle measure, ф,ф = (n-2)π radians or ф = (n-2)180º

But how do we know, without resorting to a formula, that a triangle possesses an interior angle total of 180º or π radians? And how can we use this example to establish at least one other polygon’s angular total?## Determining Interior Angles Total of a Triangle

We begin with drawing a Cartesian Coordinate system with two variables, x and y. We will choose the x-axis (y = 0) as the simplest line for our discussion. The y-axis is perpendicular to it. So the angles between the x-axis and the y-axis for the four quadrants are (by definition) each 90° or π/2 radians. Above y = 0, the angles add up to 180° or π radians. Below that line, the same is true. Now draw a line that cuts across the x-axis at any angle. The figure demonstrates this. The two angles at the intersection above that axis add up to 180° or π radians. So if there are three angles total, say α and β and γ , then α + β + γ = 180°. That being said, note our second image.## But If All Is Not Clear

But if all of this still leaves you a bit in the dark, see the video below. And even if it doesn’t leave you in the dark, but you wish put an in-depth impress into mind… see the Khan Academy video below.## Consider the Hexagon

Now let us consider a regular hexagon. The hexagon possesses six identical sides, six identical interior angles. According to the formula for the interior angles of a regular polygon, we should arrive at the result of 4π radians. But we don’t have to draw them that way. We can, in fact, take a short cut and arrive at the same answer. Consider our figure featuring the regular hexagon…

**Note:**You might also enjoy Is a Circle a Polygon or Not? Implications for Calculus**← Back to Math-Logic-Design**

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