# Interior & Exterior Angles - Beyond the Hexagon

## INSTRUCTIONS

## Heptagon - 7 sides

## Heptagon

Before you move the slider, predict how many triangles can be created inside the heptagon? Write your answer as a number.

## Interior Angles in a Heptagon - Making connections

Using our previous answer and the fact that the sum of the interior angles in a triangle = 180^{o
}we can determine the sum of the interior angles of a heptagon.
Which is the correct calculation?

## Exterior angles in a Heptagon

Now move the slider "Show Exterior Angles" slowly to the right and notice what happens when it gets to the end. What is the sum of the exterior angles in degrees? Enter your answer as a number.

## Octagon - 8 sides!

## Sum of interior angles

What calculation will give the sum of the interior angles for this shape? You can use the slider to help you see the number of triangles inside.

## Exterior Angles of an Octagon

Select "Show Exterior Angles" and move the slider to the right. What is the shape of the exterior angles when they are all added, and what is the angle measurement?

## BEYOND THE OCTAGON - GENERALIZING

We have explored interior and exterior angles for six different polygons, let's see if we can come up with some rules that we can use for all polygons. Interior angles. If a polygon has 12 sides, how many triangles can we make inside? Write your answer as a number

Based on your previous answer, determine the sum of the interior angles for a polygon with 12 sides. You can do your calculations separately, just enter your answer as a number.

What if a polygon has * n* sides?
What would be the formula to calculate the sum (

**S**) of the

*interior*angles for a polygon with

**n**sides?

Explain why you chose your previous answer. You can use an example in your answer.

## Exterior Angles

What did you learn about the sum of the *exterior* angles of any polygon?
Connect your answer to the explorations we have done so far.

## CONNECTING EXTERIOR AND INTERIOR ANGLES

Click on "Show Exterior Angles" on the above triangle applet, but DO NOT move the slider. Look at the three exterior angles and the interior angle next to them. Focus on the red exterior angle and its corresponding interior angle to answer the following: If we add the exterior and interior angle at any vertex, the answer is...

Now click on "Show Exterior Angles" for the Heptagon or Octagon, but DO NOT move the slider. Look at each of the exterior angles and the interior angle next to them. You can move the vertices of the polygon if you want. TRUE or FALSE: Each exterior and interior angle pair make a straight line (=180^{o})

If we know that the exterior angle + its corresponding interior = 180^{o}, this means that if we know one of them we can figure out the other.
If a polygon has an exterior angle = 105^{o}, what is the measurement of its corresponding interior angle?

All interior angles in a regular hexagon measure 120^{o
}What is the measurement of one exterior angle?

## EXTENDING

Since the sum of the * exterior* angles for all polygons = 360

^{o}, we can figure out how many sides are there in a regular polygon if we know the measure of one exterior angle. For example, if we know that the exterior angle of a polygon = 40

^{o}, then it has sides. How many sides are there in a regular polygon with an exterior angle measuring 18

^{o}? Do your calculations separately and enter your answer as a number.

What is the name for the polygon you just found out?

## Finding interior angles if we know how many sides.

We can also determine the measurement of an exterior angle of a regular polygon easily if we know how many sides it has, because we know that all * exterior* angles add up to 360

^{o}. For example, for a polygon with 24 (icositetragon) sides , each exterior angle measures degrees. This means that its corresponding interior angle = 180

^{o}- 15

^{o }= 165

^{o}. What is the measurement of one interior angle of a regular polygon with 15 sides? Do your calculations separately and enter you answer as a number (we know that the units are degrees)

What is the measurement of one * interior* angle for a regular decagon (10-sided polygon)?
Enter your answer as a number (we know the measurement is degrees)