The Pentagon Shape, from recognition to construction.
The Pentagon Shape is one of the shapes your Kindergartner learns to identify as part of their math curriculum. But what catches most elementary students out at some point are the two terms - Pentagon and Regular Pentagon. I feel the reason for this is that the shape above is a Regualr Pentagon, but most kids learn to identify it as simply a Pentagon. Is this wrong? No. It is a Pentagon. But a Pentagon is ANY shape composed of 5 intersecting lines. A regular pentagon is a 5 sided shape where ALL lines are the same length and ALL angles are eequal in size. You can investigate Regular Polygons in a little more detail another time.
The seven steps detailed bellow take you from basic identification all the way through to taking a detailed look at the theorems involving the pentagon shape. 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 Pentagon Shape?A Pentagon Shape is identified by the number of sides it has.
It has FIVE sides.
The regular Pentagon is the Pentagon Shape that is studied by students at the elementary level.
A Regular Pentagon 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 Pentagon has Five Sides equal in length ... and Five Angles equal in size (all are 72 degrees).
Once your child is comfortable with how to recognize the pentagon shape, offer them a pentagon shape worksheet or two, to see how they get on with identifying it.
Calculate Area and Perimeter
How to calculate the Area and Perimeter of the Regular Pentagon Shape.
To find the area of a Regular Pentagon we must know two things:
1: The length of one side 2: The perpendicular distance from the center of the pentagon 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 5 in a pentagon) 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 pentagon (constructed in green)
The Area of the pentagon is calculated by first finding the area of one of the Isoscoles Triangles created by one side of the pentagon, and two lines constructed from the center point to each vertex. Then you multiply this answer by 5, as there are five of these triangles in a pentagon. These construction lines are in Orange in my diagram.
Step 1: Area of Isoscoles Triangle Area of any triangle is half its bas multiplied by it perpendicular height. In this case
1/2 a r
Stpe 2: Multiply by 5We have 5 of these triangles in the Pentagon, so to get the area of the pentagon, we must get the area of ALL five triangles.
5 1/2 a r
5/2 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 5. 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 10 right angled triangles, rather than the 5 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 pentagon shape is no different.
The perimeter of a pentagon is 5 times its length.
How to construct a Pentagon 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 five vertices.
Note:We know that the five angles of any regualr pentagon are equal and add to 360 degrees. So each angle is 1/5 of 360 degrees which is 72 degrees.
Step 3: Using your protractor, find the point at 72 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 pentagon.
Step 4:Using your ruler, measure the length of your pentagon 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 pentagon shape.
Step 5: Your pentagon is the shape contained between the five points of intersection of these five lines.
Step 6:Using a heavier line connect the five points to finish your construction.
A quick check to ensure your pentagon 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 Pentagonal Pyramid, and Pentagonal Prism.
They will also be introduced to the Dodecahedron, one of the five platonic solids, which is a three dimensional shape constructed using only regular pentagons. Please be aware, that your student will only be introduced to these, but will NOT have to have the ability to perform any calculations (except perhaps surface area)
Geometric Coloring SheetsThe use of coloring sheets allows your child to start experimenting with pentagons. 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 pentagon and other shapes. You will find some nice free geometric coloring pages to download here.
Fun Geometry ProjectsCOMMING SOON!
Theorems & ProofsCOMMING SOON!