Then, as a class, compare answers and discuss the methods for finding the solutions.ĭemonstrate using Geometer's Sketchpad the properties of the side lengths and apothem lengths of regular octagons. Will these methods ever produce different answers?įinally, distribute the octagon area worksheet and have students complete in groups, using either method. What is the area?ĭiscuss with students the two different methods for finding the area of a regular octagon.Īs a class, discuss the advantages and disadvantages for each of the methods and ask the following questions: Again, let s = 3.00 cm and let a = 3.62 cm. Now, we can fill in the values for s and a to find the area of this regular octagon. 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: 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. Have students recall the definition of apothem. First, let's revisit the octagon that was broken into triangles. Ask the students to identify the values of m and s in this example, before computing the area.Īnd now, using the apothem we can find the area in fewer steps. 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. So we have discovered a general formula for the area, using the smaller shapes inside the octagon! Click here for the Geometer's Sketchpad file for the regular octagons. Finally, we have: If necessary, review the algebra steps involved in the simplification process with students. What does our area expression look like now? Ask students to simplify the equation. Then, substitute the value we found for m. Then, we will identify and use the apothem to find the area of our regular octagon and compare both of the methods.īegin by showing the figures below and ask the following questions:Ĭlick here to view the GSP sketch of these images.įirst, 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. Similar to the hexagon and pentagon, first we will divide the regular octagon into parts to find the area. Click here to use the GSP file and watch the animation for the changing perimeter. Demonstrate using Geometer's Sketchpad the properties of the side lengths of regular octagon. 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?Ģ. Is there a way to find the perimeter of a regular octagon without measuring all eight sides? What happens to the perimeter when the side lengths are changed? What do you notice about the side lengths of our octagon? Is this what you expected? Once worksheet is complete and students have compared their answers, ask the class as a whole the following questions: Have students get into pairs and distribute the octagon perimeter worksheet. Collectively recall the definition of perimeter given on review day. Rulers for pairs, octagon perimeter worksheet, octagon area worksheetġ. (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.) Students will use the methods discussed with regular hexagons to find the area of regular octagons. Students will learn the method of finding the perimeter of regular octagons, similar to the previous lessons.
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