Quick Definitions

Let"s walk over a couple of key words for this reason we"re every on the same page. Remember the a polygon is a two-dimensional form with sides attracted by directly lines (no curves) which together type a closeup of the door area. Each allude on a polygon where 2 sides meet is dubbed a vertex. At each vertex, there is an 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 interior angles are the 4 corners of the room.

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Sum the the inner angles

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

For example, a triangle has actually 3 angles which add up to 180 degrees. A square has actually 4 angle which add up come 360 degrees. For every extr side you add, you have to include another 180 levels to the complete 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 the the polygon. It"s all the lines in between points in a polygon if you don"t count those that are additionally sides of the polygon. In the picture below, BD is a diagonal. As you have the right to see, line segment BD divides quadrilateral ABCD right into two triangles. The sum of the angles in those triangles (180+180=360) is the same as the sum of all the angle procedures of the rectangle (360).

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Example 1

Quadrilateral ABCD has, that course, four angles. Those 4 angles are in the proportion 2:3:3:4. Discover the degree measure of the biggest edge of square ABCD.

What carry out we know?

We have 4 unknown angles, however information about their partnership to every other. Since we recognize the sum of all four angles must be 360 degrees, we simply need an expression which to add our 4 unknown angles and also sets them equal to 360. Because they room in a ratio, lock must have some common factor that we must find, dubbed x.

Steps:

include the state 2x + 3x + 3x + 4x Equate the sum of the state to 360 resolve for x determine the angle steps in degrees.

Solve


Even despite we know x = 30 we aren"t done yet. Us multiply 30 times 4 to find the greatest angle. Because 30 time 4 = 120, the greatest angle is 120 degrees. Likewise, the various other angles are 3*30=90, 3*30=90, and also 2*30 = 60.

Regular Polygons

A continuous polygon is equiangular. Every one of its angles have the very same measure. That is likewise equilateral. Every one of its sides have actually the same length. A square is a consistent polygon, and also while a square is a type of rectangle, rectangles which space not squares would certainly not be consistent polygons.

Example 2

Find the amount of the level measures that the angle of a hexagon. Assuming the hexagon is regular, uncover the level measure that each interior angle.

What carry out we know?

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

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

Solve

Let x = 6 in the formula and also simplify:


A regular polygon is equiangular, which means all angles are the same measure. In the situation of a consistent hexagon, the amount of 720 levels would be distributed evenly among the six sides.

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

Example 3

If the amount of the angles of a polygon is 3600 degrees, uncover the number of sides that the polygon.

Reversing the formula

Again, we can use the formula S = (x - 2)(180), however this time we"re addressing for x rather of S. No big deal!

Solve

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


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

Exterior angles of a Polygon

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

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

Let"s look at at two sample questions.

Example 4

Find the degree measure of every interior and exterior angle of a continual hexagon.

Remember the formula because that the sum of the internal angles is S=(x-2)*180. A hexagon has 6 sides. Because x = 6, the amount S can be found by using S = (x - 2)(180)


There are six angles in a hexagon, and also in a continual hexagon they are all equal. Every is 720/6, or 120 degrees. Us now recognize that interior and exterior angles are supplementary (add up to 180) at every vertex, so the measure of each exterior edge is 180 - 120 = 60.

Example 5

If the measure of each inner angle the a consistent polygon is 150, find the variety of sides the the polygon.

Previously we established the number 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 us only know the measure up of each internal angle. We"d need to multiply by the number of angles to discover the sum... Yet the whole difficulty is that we don"t know the number of sides however OR the sum!

But, since the measure up of each inner angle is 150, we also recognize the measure up of an exterior angle attracted at any type of vertex in regards to this polygon is 180 - 150 = 30. That"s due to the fact that they kind supplementary pairs (interior+exterior=180).

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

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

Our polygon v 150 degree interior angles (and 30 levels exterior angles) has actually 12 sides.