Area and Perimeter of Regular Polygons Hexagon

Area and Perimeter of Regular Polygons

Hexagon

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Content Objective:
Students will learn the method of finding the perimeter of regular hexagons.
Students will discover two different methods for finding the area of regular hexagons as related to regular triangles, and the apothem.
Materials:
Rulers for pairs, hexagon perimeter worksheet, hexagon area worksheet
Procedure:
Perimeter

1. Again, collectively recall the definition of perimeter given on review day and for regular triangles and squares. Have students get into pairs and distribute the hexagon perimeter worksheet. Once worksheet is complete and students have compared their answers, ask the class as a whole the following questions:

What do you notice about the side lengths of our hexagon? Is this what you expected?
What happens to the perimeter when the side lengths are changed?
Is there a way to find the perimeter of a regular hexagon without measuring all six sides?
Recalling the general expressions for the perimeter of regular triangles and squares. Do you notice a pattern? What can we expect when we discover the perimeter of a hexagon?

2. Demonstrate using Geometer's Sketchpad the properties of the side lengths of regular hexagon. Click here to use the GSP file and watch the animation for the changing perimeter.

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Area

We will show two ways of finding the area of a regular hexagon. The first method uses the students' previous knowledge of equilateral triangles.
METHOD 1:
Begin by showing the figures below and ask the following questions:
Click here to use the GSP sketch of these images.


First, how can we break this figure into familiar shapes?What special property do these six triangles have?Given the side of our hexagon is length s, what is the side length of each of the triangles?Recall the method for finding the area of a regular triangle. What is the area of ONE of the triangles in our hexagon?What is a quick way to find the area for the entire hexagon?So we have discovered a general formula for the area, using the smaller triangles inside the hexagon!
Example 1:

Use the area expression above to calculate the area of a hexagon with side length of s = 3.00cm and a height of h = 2.60cm for comparison with method 2 later.

METHOD 2:
Recall the formula for perimeter of our regular hexagon. How can we simplify the expression we found for area?
We will call the perimeter p. So now we have,Now, we can look at the general method for finding the area of a regular polygon. Define apothem for students to begin the second method for finding the area of a regular hexagon.

Apothem - the distance of the line segment from the center of a regular polygon perpendicular to a side (i.e. when a regular polygon is broken into triangles, the apothem is the height of one of the triangles whose base is a side of the regular polygon).

Discuss the concept of apothem with students and ask the following questions:

What does the apothem of a regular hexagon look like?
How is the apothem related to the height we found in the regular hexagons above?
How many apothems does a regular hexagon have? Are the apothems all the same length?
How can we rewrite our formula for area, using the apothem a for the regular hexagon instead of the height h of the regular triangle?

Example 2:

Now we can fill in the values for p and a to find the area of this regular hexagon. Again, let s = 3.00cm and let a = 2.60cm. What is the area?

COMPARING METHODS:
Discuss with students the two different methods for finding the area of a regular hexagon.
As a class, discuss the advantages and disadvantages for each of the methods and ask the following questions:

Which method is easier to formulate?
Will these methods ever produce different answers?

Finally, distribute the hexagon area worksheet and have students complete in groups, using either method. Then, as a class, compare answers and discuss the methods for finding the solutions.
Demonstrate using Geometer's Sketchpad the properties of the side lengths and apothem lengths of regular hexagons. Click here to use the GSP file and watch the animation for the changing area.