### Quick Definitions

Let"s go over a few key words so we"re all on the very same page. Remember the a **polygon** is a two-dimensional shape with sides drawn by right lines (no curves) i beg your pardon together type a closed area. Each allude on a polygon where two sides meet is dubbed a **vertex**. At every vertex, over there is one **interior angle** of the polygon. A square, for example, has four interior angles, each of 90 degrees. If the square stood for your classroom, the interior angles are the four corners the the room.

You are watching: The sum of the measures of the interior angles of a polygon is 720°. what type of polygon is it?

### Sum of the inner angles

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

For example, a triangle has 3 angles which include up to 180 degrees. A square has 4 angle which include up to 360 degrees. For every extr side friend add, you have to add *another* 180 levels to the full sum.

Let"s talk around a diagonal because that a minute. What is a **diagonal** anyway? A diagonal is a line segment connecting 2 *nonconsecutive* vertices of the polygon. It"s every the lines in between points in a polygon if friend don"t counting those that are likewise sides that the polygon. In the picture below, BD is a diagonal. Together you can see, heat segment BD divides quadrilateral ABCD right into two triangles. The amount of the angles in those triangles (180+180=360) is the same as the sum of all the angle procedures of the rectangle (360).

## Example 1

Quadrilateral ABCD has, that course, four angles. Those 4 angles are in the ratio 2:3:3:4. Find the level measure the the *biggest* edge of quadrilateral ABCD.

### What carry out we know?

We have 4 unknown angles, however information about their partnership to every other. Because we understand the amount of all four angles *must* be 360 degrees, we just need an expression which to add our four unknown angles and sets them same to 360. Due to the fact that they space in a ratio, lock must have actually some usual factor the we have to find, called x.

### Steps:

add the state 2x + 3x + 3x + 4x Equate the sum of the state to 360 solve for x determine the angle procedures in degrees.### Solve

Even though we recognize x = 30 us aren"t done yet. Us multiply 30 times 4 to uncover the best angle. Due to the fact that 30 times 4 = 120, the greatest angle is 120 degrees. Likewise, the other angles are **3***30=90, **3***30=90, and **2***30 = 60.

### Regular Polygons

A regular polygon is equiangular. Every one of its angles have the very same measure. That is likewise equilateral. All of its sides have the exact same length. A square is a regular polygon, and also while a square is a type of rectangle, rectangles which are *not* squares would certainly not be constant polygons.

## Example 2

Find the amount of the degree measures of the angles of a hexagon. Suspect the hexagon is *regular*, uncover the level measure of each internal angle.

### What carry out we know?

We deserve to use the formula S = (x - 2)(180) to sum the degree 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 way all angles are the exact same measure. In the situation of a regular hexagon, the sum of 720 degrees would be distributed evenly amongst the six sides.

So, 720/6 = 120. There are six angles in a consistent hexagon, each measuring 120 degrees.

## Example 3

If the sum of the angle of a polygon is 3600 degrees, discover the variety of sides of the polygon.

### Reversing the formula

Again, we have the right to use the formula S = (x - 2)(180), yet this time we"re fixing for x rather of S. No huge deal!

### Solve

In this problem, let S = 3600 and also solve because that x.

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

### Exterior angles of a Polygon

At every vertex the a polygon, an exterior angle may be formed by expanding one side of the polygon so that the interior and also exterior angle at that vertex space supplementary (add up to 180). In the snapshot below, angles a, b c and also d are exterior and the amount of their level measures is 360.

If a consistent polygon has actually x sides, climate the degree measure of each exterior angle is 360 split by x.

Let"s look at two sample questions.

## Example 4

Find the degree measure of each interior and exterior edge of a continuous hexagon.

Remember the formula because that the amount of the inner angles is S=(x-2)*180. A hexagon has actually 6 sides. Since x = 6, the sum S deserve to be found by using S = (x - 2)(180)

There are six angles in a hexagon, and also in a consistent hexagon they space all equal. Each is 720/6, or 120 degrees. Us now recognize that interior and also exterior angles are *supplementary* (add as much as 180) at each vertex, so the measure of each exterior edge is 180 - 120 = 60.

## Example 5

If the measure up of each interior angle that a constant polygon is 150, discover the variety of sides that the polygon.

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

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

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Before example 4, us learned that us can likewise calculator the measure up of one exterior angle in a regular polygon together 360/x, whereby x is the number of sides. Now we have actually a way to discover the answer!

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

Our polygon with 150 level interior angle (and 30 levels exterior angles) has actually 12 sides.