Real-World Relevance
By: Tarun Sontam
In Unit 5, we learned about the properties of a polygon, but we mainly focused on quadrilaterals, especially the parallelogram family. We also grasped the concepts behind an apothem and radius of a polygon, and we learned how changing the dimensions of a polygon affects its area and perimeter. Here are four examples that show how these abstractions are present in our lives, even when we don't notice it.
1. Different Size Coke Cans Are Actually Similar Figures
1. Different Size Coke Cans Are Actually Similar Figures
In Unit 5, we learned about how changing the dimensions of a shape by a scale factor has a unique change on the relationships between the areas and perimeters the shapes. In our world, the soda industry has developed so much, we now see numerous ways of getting these beverages. However, if we look past the cold drink and look at the actual can, many people don't realize that these canisters are actually similar in shape and dimensions. For example, smaller can in the image above has dimensions of 2 inches in base length and 4 inches in height. The bigger can has a base length of 2.5 inches and a height of 5 inches. If we find the scale factor from 2 to 2.5 and 4 to 5, you will notice that the scale factor is 1.25 for both the height and base length. Because both scale factors are the same, that means both of the cylinder cans are similar figures. The ratio of the small can's dimensions to the big can's is 4:5. Therefore, from the concepts we learned from Lesson 9, the ratio of the perimeters of the cans (if we see the cans as rectangles) is also 4:5, and the ratio of the areas of the cans is 16:25, because we have to square the 4 and 5 in 4:5. Similar figures can be found everywhere throughout the world, and surely enough, they will contain the properties we discussed in Unit 5 Lesson 9.
Citation: https://emergentmath.files.wordpress.com/2013/04/both_oz_large_calorie3.png
2. Old Fashion Popcorn Bags Contain Isosceles Trapezoids
Citation: https://emergentmath.files.wordpress.com/2013/04/both_oz_large_calorie3.png
2. Old Fashion Popcorn Bags Contain Isosceles Trapezoids
Nowadays, popcorn is sold in buckets and and paper bags, but back before the development of cinema and snacks, there were old fashion popcorn bags, like the one shown above. Like numerous objects, there is some math associated with this bag: all of its side faces are isosceles trapezoids, which means popcorn bags connect to Unit 5 Lesson 5. If you remove the arc design on the opening of the bag, it will be very apparent that the faces are indeed isosceles trapezoids. If the faces are isosceles trapezoids, that means that the legs, or in this case the side edges of the bag, are congruent, the diagonals of each face are congruent, and the bases, which in this case are the top and bottom edges, are parallel. To find the midsegment of each face, you can use the formula M=(B1+B2)/2, where B1 equals the length of one base, B2 equals the length of the other base. To find the area of each face, you can use the formula A=((B1+B2)H)/2, where H is the height of the trapezoid, or you can use the formula A=MH, where M is the midsegment length. Isosceles trapezoids are found all around us, we just have to be more alert to realize it.
Citation: http://38ccda.medialib.glogster.com/media/caa41d8d23ea5f7031b9f5a0d6c0f5b7cc99b1aa9acec46d5b191cf3f430bd4f/trapezoid-real-life.jpg
3. The New Microsoft Logo Contains Four Squares
Citation: http://38ccda.medialib.glogster.com/media/caa41d8d23ea5f7031b9f5a0d6c0f5b7cc99b1aa9acec46d5b191cf3f430bd4f/trapezoid-real-life.jpg
3. The New Microsoft Logo Contains Four Squares
Microsoft is one of the biggest companies globally for its products and software, and almost every being can recognize its world-famous logo. Before the revamped logo, Microsoft had its four wavy squares, but after the new logo released, it was simply just four squares. This however, connects to what we learned in Unit 5 Lesson 4, when we learned about the rectangle, rhombus, and square. A square is a rhombus, parallelogram, and rectangle, so that means that the Microsoft logo is technically four rhombi, four rectangles, and four parallelograms as well. If the four polygons in the Microsoft logo are squares, that means that the opposite sides of each "window" are parallel, all four sides are congruent, all four angles are 90 degrees and congruent, and the diagonals of each window are perpendicular, congruent, and bisect each other. To find the area of each square, you can use the formula A=S^2, where S is the side length, and you can use the formula P=4S to find the perimeter of each square. Microsoft uses math all the time in their work, even in their logo!
Citation: http://dri1.img.digitalrivercontent.net/Storefront/Site/msusa/images/promo/en-US/msstore_400x400.jpg
4. Regular Polygons Are All Around Us
Citation: http://dri1.img.digitalrivercontent.net/Storefront/Site/msusa/images/promo/en-US/msstore_400x400.jpg
4. Regular Polygons Are All Around Us
Regular polygons were the first things we learned about in Unit 5, and they have congruent angles and sides. There are many regular polygons in our world, and we see some in our lives everyday. For example, a regular stop sign is a regular octagon, meaning it has eight congruent sides and angles. Another example of a regular polygon is the patches on a regular soccer ball, and they are regular hexagons, meaning the patches have six congruent sides and angles. In Unit 5 Lesson 1, we learned the formulas 180(N-2), 180(N-2)/N, and 360/N, which can be used to find the sum of the interior angles of a polygon, the value of each interior angle in a regular polygon, and the value of each exterior angle in a regular polygon. In Unit 5 Lesson 8, we learned that the apothem was the distance from the center of a regular polygon to the midpoint of a side in a polygon, and it forms a 90 degrees angle, meaning that the apothem is the perpendicular bisector of each side in a regular polygon. We also learned that the radius of a regular polygon is is the distance from the center to any vertex. We also learned a universal formula for finding the area of any regular polygon is A=aP/2, where a is the length of the apothem, and P is the perimeter of the regular polygon. Regular polygons are unique shapes, and we can find them anywhere!
Citation for Stop Sign: http://all-free-download.com/free-vector/vector-clip-art/stop_sign_clip_art_12913.html
Citation for Soccer Ball: http://pollywall-e.tripod.com/sitebuildercontent/sitebuilderpictures/polygons-border.jpg
All FORMATTING FOR PICTURES MADE ON GOOGLE DOCS
Citation for Stop Sign: http://all-free-download.com/free-vector/vector-clip-art/stop_sign_clip_art_12913.html
Citation for Soccer Ball: http://pollywall-e.tripod.com/sitebuildercontent/sitebuilderpictures/polygons-border.jpg
All FORMATTING FOR PICTURES MADE ON GOOGLE DOCS