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 now start developing the ability of 'connecting the dots' between these and the octagon and higer 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 is 360 divided by 4; in a 5 sided regular polygon the degrees are 360/5; in a six sided the degrees are 360/6. The pattern is simply 360 divided by the number of sides of the regular polygon. 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 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 ...
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).
Once your child is comfortable with how to recognize this shape, offer them a shape worksheest to see how they get on with identifying it.
Calculate Area and Perimeter
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 Isoscoles 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 Isoscoles Triangle Area of any triangle is half its base multiplied by it perpendicular height. In this case
1/2 a r
Stpe 2: Multiply by 8We 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
Ther 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 an Octagon ShapeTo complete this, you will need a ruler, pencil, protractor, and a blank piece of paper!
Approach 1: Using a protractor
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 angles of any regualr octagon are equal in size and add to 360 degrees. So each angle is 1/8 of 360 degrees which is 45 degrees.
Step 3: Using your protractor, find the point at 45 degrees to your first line using the point you indicated as the base, mark it, 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 points of intersection of these eight lines.
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 ShapesThe 3d Figures a Kindergarten through sixth grade student is most likely to deal with are the octagonal prism and the octagonal pyramid.
Geometric Coloring SheetsThe use of coloring sheets allows your child to start experimenting with septagons. A great first step is to encourage your child to color in shapes adjacent to eachother with the same color, until their shape starts to look like 'something'. By doing this, your child will start to realize the connection between the octagon and other shapes (especially triangles). You will find some nice free geometric coloring pages to download here.
Fun Geometry ProjectsCOMMING SOON!
Theorems & ProofsCOMMING SOON!