Quick Definitions

Let"s walk over a few key words for this reason we"re all on the exact same page. Remember the a polygon is a two-dimensional form with sides drawn by right lines (no curves) which together type a closeup of the door area. Each allude on a polygon where 2 sides satisfy is referred to as a vertex. At every vertex, there is an interior angle that the polygon. A square, because that example, has four interior angles, each of 90 degrees. If the square stood for your classroom, the inner angles space the four corners of the room.

You are watching: Find the number of sides of a regular polygon if one interior angle is 60 degrees.

Sum of the internal angles

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

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

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

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Example 1

Quadrilateral ABCD has, that course, 4 angles. Those four angles are in the proportion 2:3:3:4. Find the degree measure of the biggest angle of quadrilateral ABCD.

What do we know?

We have four unknown angles, but information around their connection to each other. Since we recognize the amount of all four angles must be 360 degrees, we just need an expression which add to our four unknown angles and also sets them equal to 360. Since they are in a ratio, castle must have some usual factor the we have to find, referred to as x.

Steps:

include the terms 2x + 3x + 3x + 4x Equate the amount of the state to 360 fix for x identify the angle steps in degrees.

Solve


Even despite we recognize x = 30 we aren"t excellent yet. We multiply 30 times 4 to discover the greatest angle. Since 30 time 4 = 120, the biggest angle is 120 degrees. Likewise, the various other angles are 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. That is likewise equilateral. Every one of its sides have the same length. A square is a continuous polygon, and also while a square is a kind of rectangle, rectangles which are not squares would not be continuous polygons.

Example 2

Find the sum of the level measures that the angles of a hexagon. Assuming the hexagon is regular, uncover the degree measure of each inner angle.

What do we know?

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

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

Solve

Let x = 6 in the formula and simplify:


A regular polygon is equiangular, which way all angles room the very same measure. In the instance of a consistent hexagon, the sum of 720 levels would be spread evenly among the six sides.

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

Example 3

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

Reversing the formula

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

Solve

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


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

Exterior angle of a Polygon

At each vertex the a polygon, one exterior angle might be created by prolonging one side of the polygon so the the interior and exterior angles at the vertex are supplementary (add up to 180). In the picture below, angle a, b c and also d are exterior and the amount of their level measures is 360.

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

Let"s look at two sample questions.

Example 4

Find the level measure of each interior and also 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 actually 6 sides. Since x = 6, the amount S have the right to be discovered by using S = (x - 2)(180)


There are 6 angles in a hexagon, and in a continual hexagon they space all equal. Each 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 of every exterior angle is 180 - 120 = 60.

Example 5

If the measure of each inner angle of a regular polygon is 150, find the number of sides that the polygon.

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

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

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Before instance 4, we learned that we can also calculator the measure of an exterior angle in a regular polygon as 360/x, wherein x is the variety of sides. Now we have a method to uncover the answer!

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

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