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LessonGrades 8 - 10

Discovering Relationships between Side Length and Area

A 17th-century drawing shows Netherland surveyors measuring dykes and canals. To measure distance, a standing man looks through a device on a vertical stick to “sight on a rod” towards another man with a tall stick some distance away. At his feet lie surveying and drafting instruments, including a surveyor's chain for measuring distances. In the background, a cannon and its target (a tower) illustrate the artillery use of triangles and angles. Other surveying principles and geometric concepts are also suggested, including a globe topped by a mountain to indicate elevation measurements.Surveying uses many geometric concepts.

Through this lesson and its two associated activities, students are introduced to the use of geometry in engineering design, and conclude by making scale models of objects of their choice. The practice of developing scale models is often used in engineering design to analyze the effectiveness of proposed design solutions. In this lesson, students complete fencing (square) and fire pit (circle) word problems on two worksheets—which involves side and radius dimensions, perimeters, circumferences and areas—guiding them to discover the relationships between the side length of a square and its area, and the radius of a circle and its area. They also think of real-world engineering applications of the geometry concepts.

Relationships and patterns are extremely important in engineering. To understand how different pieces of a system work together and relate, engineers routinely test for patterns by changing parameters during experiments and tests. They figure out the relationships between different variables in a system, and then use this understanding to optimize system performance. In vehicle design, engineers employ this strategy to determine the ideal engine air/fuel mixing ratios, while other engineers use the same strategy to meet requirements and minimize costs during construction projects. Engineers also need to understand the concepts of area and volume as they design products. For example, a container that maximizes volume and minimizes surface area reduces costs and can increases production efficiency, which are important consideration for engineers who design product packaging such as beverage cartons and aluminum cans.

After this lesson, students should be able to:

  • State the relationship between the side length and area of a square.
  • State the relationship between the radius and area of a circle.
  • Describe examples of how area, perimeter and circumference are used in engineering.

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