Quick Definitions

Let"s go over a couple of key words for this reason we"re every on the exact same page. Remember that a polygon is a two-dimensional form with sides attracted by directly lines (no curves) which together kind a closeup of the door area. Each allude on a polygon where two sides satisfy is dubbed a vertex. At each vertex, there is one interior angle the the polygon. A square, for example, has four interior angles, every of 90 degrees. If the square stood for your classroom, the inner angles are the four corners of the room.

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Sum the the inner angles

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

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

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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 between points in a polygon if friend don"t count those that are likewise sides of the polygon. In the snapshot below, BD is a diagonal. As you deserve to see, line segment BD divides quadrilateral ABCD right into two triangles. The amount of the angles in those triangles (180+180=360) is the very same as the amount of all the angle procedures of the rectangle (360).

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Example 1

Quadrilateral ABCD has, of course, 4 angles. Those four angles space in the ratio 2:3:3:4. Discover the degree measure of the biggest edge of quadrilateral ABCD.

What perform we know?

We have 4 unknown angles, but information about their connection to every other. Since we understand the amount of all four angles must be 360 degrees, we simply need an expression which add to our four unknown angles and also sets them equal to 360. Since they space in a ratio, lock must have some usual factor that we must find, referred to as x.

Steps:

add the state 2x + 3x + 3x + 4x Equate the amount of the state to 360 deal with for x identify the angle procedures in degrees.

Solve


Even despite we understand x = 30 we aren"t excellent yet. We multiply 30 times 4 to discover the biggest angle. Because 30 time 4 = 120, the greatest angle is 120 degrees. Likewise, the various other angles space 3*30=90, 3*30=90, and 2*30 = 60.

Regular Polygons

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

Example 2

Find the sum of the level measures the the angle of a hexagon. Suspect the hexagon is regular, discover the degree measure the each interior angle.

What perform we know?

We can use the formula S = (x - 2)(180) to amount the level measure of any kind of polygon.

A hexagon has actually 6 sides, therefore x=6.

Solve

Let x = 6 in the formula and simplify:


A regular polygon is equiangular, which means all angles room the exact same measure. In the instance of a continual hexagon, the sum of 720 degrees would be distributed evenly among the 6 sides.

So, 720/6 = 120. There are 6 angles in a constant hexagon, every measuring 120 degrees.

Example 3

If the sum of the angle of a polygon is 3600 degrees, find the number 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 solving for x rather of S. No big deal!

Solve

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


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

Exterior angle of a Polygon

At every vertex of a polygon, an exterior angle may be developed by extending one next of the polygon so the the interior and also exterior angles at that vertex are supplementary (add as much as 180). In the snapshot below, angle a, b c and also d room exterior and also the amount of their level measures is 360.

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If a continuous polygon has x sides, then the level measure of every exterior edge is 360 separated by x.

Let"s look at two sample questions.

Example 4

Find the level measure of each interior and exterior angle of a constant hexagon.

Remember the formula because that the amount of the interior angles is S=(x-2)*180. A hexagon has 6 sides. Due to the fact that x = 6, the sum S have the right to be uncovered by making use of S = (x - 2)(180)


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

Example 5

If the measure of each internal angle the a consistent polygon is 150, discover the number of sides of the polygon.

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

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

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Before example 4, us learned that we can likewise calculator the measure up of an exterior angle in a consistent polygon as 360/x, where x is the number of sides. Now we have actually a means to uncover the answer!

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

Our polygon v 150 level interior angles (and 30 degrees exterior angles) has 12 sides.