The Octagon Shape, from recognition to construction.

The Octagon shape will be introduced to your student around the thrid grade.  The main reason will be to ensure your child is making the connection of regularity amongst all the Regular Polygons.  

Once the square, pentagon and hexagon have been studied, your child should start to develop the ability of 'connecting the dots' between these and the octagon and higher degrees of regular polygons.  

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 Octagon 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 Octagon Shape?

An Octagon is identified by the number of sides it has. It has EIGHT sides.

A Regular Octagon Shape 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 Octagon has Eight Sides equal in length ... and Eight Angles equal in size - all are 45 degrees at the center (360/8=45), with internal angles of 135 degrees  [180-(360/8) = 135]

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 Octagon Shape

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

1: The length of one side and,

2: The perpendicular distance from the center of the Octagon 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 8 in an octagon) 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 octagon (constructed in green).

The Area of the octagon 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 8, as there are eight of these triangles in a Octagon.

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 8

We have 8 of these triangles in the octagon, so to get its area,  we must get the area of ALL eight triangles.

8 (1/2 a r)

Tidied up this is equal to 4ar

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

The perimeter of a Regular Octagon Shape is

8 times the length of one of its sides..

How to construct a Regular Octagon Shape

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 eight vertices.

Note:We know that the eight internal angles of any regular octagon are each 135 degrees. 

Step 3:  Using your protractor, find the point at 135 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 octagon.

Step 4: Using your ruler, measure the length of your octagon 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 octagon shape.

Step 6:  Your octagon is the shape contained between the eight lines you have constructed.

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

A quick check to ensure your octagon 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

The 3d Figures a Kindergarten through sixth grade student is most likely to deal with are the octagonal prism and the  octagonal pyramid.

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!

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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 or simply visit the MathMomentumCommunity and join the conversation!

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

Return from this Octagon Shape 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.