The Septagon, from recognition to construction.
The Septagon (also known as the heptagon) will be introduced to your student around the thrid grade. More for recognition purposes than anything else. In general after the Pentagon, Regular Polygons with an ODD number of sides are not studied in too much detail by the elementary math student.
The connection that will be required between the Septagon, and the previously studied pentagon and hexagon, is in calculating the area. Your child should start to see the pattern that is evolving in the formula for ALL regular polygons.
The steps detailed bellow take you from basic identification all the way through to construction of the septagon, 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 a Septagon?A Septagon is identified by the number of sides it has.
It has SEVEN sides.
A Regular Septagon 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 Septagon has Seven Sides equal in length ... and Seven Angles equal in size (all are 51.43 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 Septagon.
To find the area of a Regular Septagon we must know two things:
1: The length of one side and, 2: The perpendicular distance from the center of the septagon 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 7 in a septagon) 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 septagon (constructed in green)
The Area of the septagon is calculated by first finding the area of one of the Isoscoles Triangles created by one side of the septagon, and two lines constructed from the center point to each vertex. Then you multiply this answer by 7, as there are seven of these triangles in a septagon. 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 7We have 7 of these triangles in the septagon, so to get its area, we must get the area of ALL seven triangles.
7 1/2 a r
7/2 a r
These questions can be presented in many different ways, depending on the grade level of your child. For example, if your child is just learning about area (perhaps second or third grade) they could be given the area of one of these trianles. To solve this, they would simply multiply the area given by 7. If on the other hand, your child is in 4th grade, they will probably be given a question which has a solution similar to the explanation above. But what about a 6th grader? These students could possibly be given information that will require a more complicated solution. And complicated solutions demand an understanding of what is happening, not just a 'plug and play' system with a formula. For example, they could be given the above question, but instead of being given the value of 'r', perhaps they are given the vlaue of the orange construction lines! What then? Would your student see the 14 right angled triangles, rather than the 7 isosoceles? Would they make the connection to the Theorem of Pythagoras to find the length of 'r'?
These are good examples of how a shape can be studied all the way through 7 grades, and still get more and more difficult. The difficult isn't 'tricks' put out there to make a childs life miserable, they are there to ensure a solid understanding. The only way any student can gain this level of understanding is through practice.
Ther Perimeter of ANY shape is simply the sum total of all the lengths of the shape - and a Septagon shape is no different.
The perimeter of a septagon is 7 times the length of one of its sides..
How to construct a Septagon 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 seven vertices.
Note:We know that the seven angles of any regualr septagon are equal in size and add to 360 degrees. So each angle is 1/7 of 360 degrees which is 51.43 degrees.
Step 3: Using your protractor, find the point at 51.43 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 septagon.
Step 4:Using your ruler, measure the length of your septagon 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 septagon shape.
Step 5: Your septagon is the shape contained between the seven points of intersection of these seven lines.
Step 6:Using a heavier line connect the seven points to finish your construction.
A quick check to ensure your septagon 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 Septagonal Pyramid and the Septagonal Prism. However a point to note here, is that it is not likely that these three dimensional shapes will be studied at all. This is probably due to the awkwardness of the angle of measurement and long decimal calculations.
It is far more likeley that the pyramids and prisms studied will be the pentagonal, hexagonal and/or octagonal.
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 septagon and other shapes (especially triangles). You will find some nice free geometric coloring pages to download here.
Fun Geometry ProjectsCOMMING SOON!
Theorems & ProofsCOMMING SOON!