Area and Perimeter of Regular Polygons

Octagon

Content Objective:

Students will learn the method of finding the perimeter of regular octagons, similar to the previous lessons.

Students will use the methods discussed with regular hexagons to find the area of regular octagons. (The use of two different methods to find the area will help students understand the concept of using multiple problem solving techniques to produce the same result.)

Materials:

Rulers for pairs, octagon perimeter worksheet, octagon area worksheet

Procedure:

Perimeter

1. Collectively recall the definition of perimeter given on review day. Have students get into pairs and distribute the octagon 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 octagon? 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 octagon without measuring all eight sides?

Recalling the general expressions for the perimeter of regular hexagon? Do you notice a pattern? What can we expect when we discover the perimeter of a regular octagon?

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

Area

Similar to the hexagon and pentagon, first we will divide the regular octagon into parts to find the area. Then, we will identify and use the apothem to find the area of our regular octagon and compare both of the methods.

METHOD 1:

Begin by showing the figures below and ask the following questions:

Click here to view the GSP sketch of these images.

First, how can we break this figure into familiar shapes?

Are these triangles regular? How can you tell?

How else can we break up the regular octagon into at least one regular shapes?

How do you know this square is regular? What are its side lengths?

Are the other shapes regular? Why not?

To find the area of this regular octagon we could sum all the areas of the interior shapes.

Knowing the triangles are not regular, how would we find their side length based on the length s?

Recall the general side lengths of a special right triangle with an angle of 45 degrees:

So each leg m will have a length of

So, what is the area of ONE of the four triangles?

Finally, what about the area of ONE of the four rectangles?

Now that we have an side length of m, we can find the area of the entire regular octagon.

Have students generate the area expression below by summing the smaller areas within the octagon.

Then, substitute the value we found for m. What does our area expression look like now?

Ask students to simplify the equation. Finally, we have:

If necessary, review the algebra steps involved in the simplification process with students. Click to view the algebra steps. Click here for the Geometer's Sketchpad file for the regular octagons.

So we have discovered a general formula for the area, using the smaller shapes inside the octagon!

Example 1:

Use the above area expression to calculate the area of an octagon with side length of s = 3.00 cm for comparison with method 2 later. Ask the students to identify the values of m and s in this example, before computing the area.

METHOD 2:

And now, using the apothem we can find the area in fewer steps. First, let's revisit the octagon that was broken into triangles.

Have students recall the definition of apothem. Can you identify an apothem of this regular octagon?

Notice the apothem is the same as the height of the interior triangles of our polygon, similar to the hexagon.

Recall the expression using the apothem length for a regular polygon. How can we express this formula for our regular octagon?

How can this expression be simplified now that we know the perimeter of a regular octagon?

Example 2:

Now, we can fill in the values for s and a to find the area of this regular octagon. Again, let s = 3.00 cm and let a = 3.62 cm. What is the area?

COMPARING METHODS:

Discuss with students the two different methods for finding the area of a regular octagon.

As a class, discuss the advantages and disadvantages for each of the methods and ask the following questions:

Which method is easier to formulate?

Which method produces quicker results?

Will these methods ever produce different answers?

Finally, distribute the octagon 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 octagons. Click here to use the GSP file and watch the animation for the changing area.

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