The Nonagon, from recognition to construction.

The Nonagon shape will be introduced to your student around the fifth grade, if it is introduced at all!  The only purpose of its introduction will be for your child to be able to identify it, and for no other reason. I, however think all kids, should also be taught the connections between the different regular polygons also. 

These connections mainly are for finding the area and angles of a regular polygon.  In a four sided regular polygon (square)the degree value of its internal angles are found by

  • Dividing 360 by 4 = 90(This gives the values of the 4 angles at the center of the square)
  • Subtract 90 from 180 to get the value of the internal angles. 180 - 90 = 90

in a 5 sided regular polygon the degrees are found by

  • Dividing 360 by 5 = 72 (This gives the values of the 5 angles at the center of the square)
  • Subtract 72 from 180 to get the value of the internal angles. 180-72=108

The pattern is simply 360 divided by the number of sides of the regular polygon to get the angles at the center. Then subtract your answer from 180 to get the value of the internal angles.

So even if your child is asked about a 20 sided polygon - they should have by now, developed the skills to know that the angle value will be 180-(360/20).

The steps detailed bellow take you from basic identification all the way through to construction of the Nonagon shape, and more!

These steps also include 'pit stops' to complete fun geometry projects and coloring sheets.

These are nice 'breathers' on the learning curve, but they are excellent ways of reinforcing the new knowledge in ways that your kid can get a real life, hands on approach to understanding the basic geometry concepts included.

Okay, so let's get started ...

Identify - How do we know what we look at is an Nonagon Shape?

A Nonagon is identified by the number of sides it has. 

It has NINE sides.

A Regular Nonagon is identified by a combination of the number of sides to the shape, the length of the sides AND the size of its angles.

A Regular Nonagon has NINE Sides equal in length ... and NINE Angles equal in size - all  angles are 40 degrees at the center (360/9=40), with internal angles of 140 degrees  [180-(360/9) = 140]

Once your child is comfortable with how to recognize this shape, offer them some shape worksheets  to see how they get on with identifying it.

How to calculate the Area and Perimeter of the Regular Nonagon Shape

To find the area of a Regular Nonagon we must know two things:

1: The length of one side and,

2: The perpendicular distance from the center of the Nonagon to one of its sides. (In 'math speak' this perpendicular distance is known as the APOTHEM!)  In my diagram I have included 2 (there is a total of 9 in an nonagon) and these are the blue lines.  You can also see that these Apothem's are in fact the radius of a circle inscribed in the nonagon (constructed in green).

The Area of the nonagon is calculated by first finding the area of one of the Isosceles Triangles created by one side of the octagon, and two lines constructed from the center point to each vertex. Then you multiply this answer by 9, as there are nine of these triangles in a Nonagon.

These construction lines are in Orange in my diagram.

Step 1: Area of Isosceles Triangle

Area of any triangle is half its base multiplied by it perpendicular height.  In this case

1/2 a r, where r (the perpendicular height) is the apothem.

Step 2: Multiply by 9

We have 9 of these triangles in the nonagon, so to get its area,  we must get the area of ALL nine triangles.

9 (1/2 a r)

Tidied up this is equal to 9/2 (ar)

The Perimeter of ANY shape is simply the sum total of all the lengths of the shape - and an Nonagon is no different.

The perimeter of a Regular Nonagon is

9 times the length of one of its sides..

How to construct a Regular Nonagon.

To complete this, you will need a ruler, pencil, protractor and a blank piece of paper!

Step 1:  Draw a straight line lightly using your ruler and pencil on your paper. - This is what we call a construction line.

Step 2: Indicate on this line, one point - this point will be the first of the nine vertices.

Note:We know that the nine internal angles of any regular nonagon are each 140 degrees. 

Step 3:  Using your protractor, find the point at 140 degrees to your first line using the point you indicated in the previous step, and draw another construction line. 

You now have two lines of your nonagon.

Step 4: Using your ruler, measure the length of your nonagon side on BOTH of these lines, and mark with a point.

Step 5: Using these new points, repeat Step 3: until you have completed your nonagon shape.

Step 6:  Your nonagon is the shape contained between the nine lines you have constructed.

Step 7: Using a heavier line connect the nine points to finish your construction.

A quick check to ensure your shape is accurate, is to measure all side lengths with your ruler.  If you have done it correctly all sides will measure the same!

Relationship to 3D Shapes

There are no 3d shapes an elementary math student will study involving this shape.

Geometric Coloring Sheets

The use of coloring sheets allows your child to start experimenting with triangles.  A great first step is to encourage your child to color in triangles adjacent to each other with the same color, until their shape starts to look like 'something'.

Perhaps that something will be a rectangle or a house! By doing this, your child will start to realize the connection between different shapes.  You will find some nice free geometric coloring pages to download and get started with.

Fun Geometry Projects

Coming soon!


Theorems & Proofs

Coming soon!

I have created Free printable shape worksheets for you to offer your child for more practice.  Download, print and give them to your kids.  They're available 24/7!

I am sure you will find all the information and worksheets you need here, however if there is anything you cannot find please don't hesitate to contact me.

I love to hear from my readers, and with a little feedback and a few suggestions I can make this a great resource for parents, teachers and tutors alike.

Be sure to explore everything on this site starting at the home page.

Return from this Nonagon page to our Regular Polygons Section.


Return from this page to K6 Geometric Shapes Home Page, to explore all the other great sections I have to offer.