### Quick Definitions

Let"s go over a few key words therefore we"re every on the very same page. Remember that a **polygon** is a two-dimensional form with sides attracted by directly lines (no curves) i m sorry together type a closeup of the door area. Each allude on a polygon where 2 sides satisfy is dubbed a **vertex**. At each vertex, over there is an **interior angle** of the polygon. A square, for example, has 4 interior angles, each of 90 degrees. If the square stood for your classroom, the internal angles space the 4 corners the the room.

You are watching: A polygon has an angle sum of 360°, and each angle measures 90°. what is the polygon?

### Sum of the interior angles

To prolong that further, if the polygon has x sides, the sum, S, the the level measures of these x interior sides is given by the formula **S = (x - 2)(180)**.

For example, a triangle has 3 angle which add up come 180 degrees. A square has actually 4 angles which include up to 360 degrees. Because that every extr side you add, you have to add *another* 180 degrees to the complete sum.

Let"s talk around a diagonal for a minute. What is a **diagonal** anyway? A diagonal is a line segment connecting 2 *nonconsecutive* vertices of the polygon. It"s all the lines in between points in a polygon if girlfriend don"t count those the are likewise sides the the polygon. In the photo below, BD is a diagonal. As you can see, heat segment BD divides square ABCD into two triangles. The amount of the angle in those triangles (180+180=360) is the very same as the amount of every the angle actions of the rectangle (360).

## Example 1

Quadrilateral ABCD has, of course, 4 angles. Those 4 angles room in the ratio 2:3:3:4. Find the degree measure that the *biggest* angle of square ABCD.

### What perform we know?

We have 4 unknown angles, but information about their partnership to each other. Since we recognize the amount of all 4 angles *must* it is in 360 degrees, we simply need one expression which to add our four unknown angles and also sets them same to 360. Since they are in a ratio, lock must have actually some usual factor that we must find, called x.

### Steps:

include the state 2x + 3x + 3x + 4x Equate the sum of the state to 360 solve for x identify the angle steps in degrees.### Solve

Even despite we know x = 30 we aren"t excellent yet. Us multiply 30 times 4 to discover the best angle. Due to the fact that 30 times 4 = 120, the best angle is 120 degrees. Likewise, the other angles space **3***30=90, **3***30=90, and also **2***30 = 60.

### Regular Polygons

A constant polygon is equiangular. Every one of its angles have the exact same measure. The is likewise equilateral. All of its sides have actually the exact same length. A square is a continual polygon, and while a square is a kind of rectangle, rectangles which room *not* squares would not be continual polygons.

## Example 2

Find the amount of the level measures that the angle of a hexagon. Presume the hexagon is *regular*, find the degree measure that each inner angle.

### What carry out we know?

We deserve to use the formula S = (x - 2)(180) to sum the level measure of any type of polygon.

A hexagon has actually 6 sides, for this reason x=6.

### Solve

Let x = 6 in the formula and also simplify:

A **regular polygon** is *equiangular*, which means all angles space the exact same measure. In the instance of a consistent hexagon, the amount of 720 degrees would be dispersed evenly amongst the 6 sides.

So, 720/6 = 120. Over there are 6 angles in a regular hexagon, every measuring 120 degrees.

## Example 3

If the amount of the angle of a polygon is 3600 degrees, find the variety of sides of the polygon.

### Reversing the formula

Again, we have the right to use the formula S = (x - 2)(180), however this time we"re addressing for x instead of S. No large deal!

### Solve

In this problem, permit S = 3600 and solve for x.

A polygon through 22 sides has 22 angles whose amount is 3600 degrees.

### Exterior angles of a Polygon

At every vertex that a polygon, an exterior angle might be developed by prolonging one side of the polygon so that the interior and also exterior angles at that vertex room supplementary (add approximately 180). In the snapshot below, angles a, b c and d room exterior and the sum of their level measures is 360.

If a consistent polygon has x sides, climate the level measure of every exterior edge is 360 split by x.

Let"s look at 2 sample questions.

## Example 4

Find the level measure of every interior and exterior edge of a continuous hexagon.

Remember the formula for the amount of the interior angles is S=(x-2)*180. A hexagon has 6 sides. Since x = 6, the amount S deserve to be discovered by using S = (x - 2)(180)

There are six angles in a hexagon, and in a constant hexagon they room all equal. Every is 720/6, or 120 degrees. We now know that interior and also exterior angles space *supplementary* (add up to 180) at each vertex, for this reason the measure up of each exterior angle is 180 - 120 = 60.

## Example 5

If the measure up of each inner angle the a continual polygon is 150, uncover the number of sides that the polygon.

Previously we determined the variety of sides in a polygon by taking the amount of the angles and using the S=(x-2)*180 formula come solve. But, this time we only understand the measure up of each inner angle. We"d have to multiply through the number of angles to uncover the sum... Yet the whole trouble is that we don"t understand the number of sides yet OR the sum!

But, due to the fact that the measure up of each internal angle is 150, us *also* know the measure of one exterior angle drawn at any type of vertex in regards to this polygon is 180 - 150 = 30. That"s because they form supplementary pairs (interior+exterior=180).

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Before example 4, we learned that we can additionally calculator the measure of an exterior angle in a constant polygon as 360/x, where x is the variety of sides. Currently we have a way to discover the answer!

30 = 360/x 30x = 360 x = 360/30 x = 12

Our polygon with 150 degree interior angle (and 30 degrees exterior angles) has 12 sides.