The Pentagon Shape, from recognition to construction.

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.