Tonight’s Homework – Octagon Patios

I’ll admit, I thought tonight’s homework problem wouldn’t be so hard after tackling the Tennis Court problem and Geoff Kraal’s Pizza Casbah Challenge problem. But I’m getting a lot of stressed e-mails from confused students (who have not learned to embrace confusion I guess), so we’ll see how it goes.

Here it is. Still stressing the idea of scaling area, but adding in a wrinkle of a regular polygon: we’ve never discussed their area, but they should be able to make rectangles out of them, I think, or maybe approximate them as a circle.

The following images are from Home Depot’s web site. The first kit allows you to create an octagon-shaped patio (a regular octagon) and the second kit allows you to expand it to make a larger octagon. The diameters of both octagons and the side length of the larger one are shown.

  1. Find the scale factor from the smaller octagon to the expanded octagon and use it to label the side length of the smaller octagon. 
  2. Using what you’ve learned about scaling factors and areas, do you think it is reasonable for the company to charge the same price for the expansion as they do for the original kit? Why or why not? Your explanation should include mathematical work. 
  3. Find a way to estimate the approximate (or exact) area of the octagons, then use that data to support (or confuse) your explanation in problem 2. 
  4. After answering all of the questions above, either look in your book on page 441 for the formula for area of a regular polygon OR ask Mr. Griswold to give/show it to you. Use this formula to calculate the areas of each of the octagons. Did your method work? Is your answer to #2 the same?

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