Quick Definitions

Let"s go over a couple of key words for this reason we"re all on the same page. Remember that a polygon is a two-dimensional shape with sides drawn by straight lines (no curves) which together form a closeup of the door area. Each point on a polygon where 2 sides accomplish is dubbed a vertex. At every vertex, over there is one interior angle the the polygon. A square, for example, has four interior angles, each of 90 degrees. If the square stood for your classroom, the interior angles room the four corners that the room.

You are watching: Find the number of sides of a regular polygon if one interior angle is 120 degrees.

Sum the the internal angles

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

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

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Let"s talk about a diagonal for a minute. What is a diagonal anyway? A diagonal line is a line segment connecting two nonconsecutive vertices that the polygon. It"s every the lines between points in a polygon if you don"t counting those the are additionally sides the the polygon. In the picture below, BD is a diagonal. As you deserve to see, line segment BD divides square ABCD into two triangles. The sum of the angle in those triangles (180+180=360) is the exact same as the amount of all the angle actions 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 degree measure the the biggest angle of quadrilateral ABCD.

What carry out we know?

We have four unknown angles, yet information about their connection to every other. Due to the fact that we recognize the sum of all four angles must it is in 360 degrees, we simply need one expression which add to our four unknown angles and also sets them equal to 360. Since they room in a ratio, they must have some usual factor the we have to find, referred to as x.


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


Even though we recognize x = 30 us aren"t excellent yet. Us multiply 30 times 4 to uncover the biggest angle. Due to the fact that 30 times 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. All of its angles have actually the very same measure. It is additionally equilateral. Every one of its sides have the exact same length. A square is a regular polygon, and while a square is a type of rectangle, rectangles which space not squares would certainly not be continual polygons.

Example 2

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

What execute we know?

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

A hexagon has 6 sides, therefore x=6.


Let x = 6 in the formula and also simplify:

A regular polygon is equiangular, which means all angles room the same measure. In the case of a continual hexagon, the sum of 720 levels would be dispersed evenly amongst the six sides.

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

Example 3

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

Reversing the formula

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


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

A polygon v 22 sides has actually 22 angle whose sum is 3600 degrees.

Exterior angle of a Polygon

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


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

Let"s watch at two sample questions.

Example 4

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

Remember the formula because that the amount of the interior angles is S=(x-2)*180. A hexagon has actually 6 sides. Due to the fact that x = 6, the sum S can be found by utilizing S = (x - 2)(180)

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

Example 5

If the measure of each internal angle that a constant polygon is 150, find the number of sides of 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 us only know the measure of each interior angle. We"d have to multiply by the variety of angles to uncover the sum... However the whole problem is that us don"t recognize the number of sides however OR the sum!

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

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Before example 4, us learned that we can additionally calculator the measure up of one exterior edge in a consistent polygon as 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 through 150 level interior angles (and 30 degrees exterior angles) has actually 12 sides.