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Generalizing Area of Polygons

Posted by Angie Seltzer on 4/2/2018 to 6-8 Geometry
Generalizing Area of Polygons
Why are there so many different formulas for area? Many students become dependent on applying formulas, but they forget the formulas soon after the chapter test. You can help students by focusing on the general formula that area is length by width.

Start with rectangles. 
Area is a measurement of square units. When a rectangle is drawn on a grid, the square units form equal rows. So, multiply the length of a row by the number of rows. Because the length and width must be perpendicular to each other to create squares, some books call these the base and height measurements. However, it's usually better to think of two-dimensional measurements as length and width.
Area of Rectangles
What about area of parallelograms? 
If you draw any line segment across a parallogram, parallel to a base, that segment will always be the same length as the base. And the width is measured perpendicular to the base. So, the area is still the length by width. 
Area of a Parallelogram
What about trapezoids? 
Think of the "average" distance across a trapezoid as the length of a segment midway between the two parallel bases. One way to find the average is to add the two lengths and divide by 2. To find the area, multiply this average length by the width.

Area of a Trapezoid

Does this work for triangles?
Yes. For triangles the average width is the distance across the middle, which is half of the base. Again, multiply by the width, measured from the base (or an extension of the base) to the opposite vertex.

Area of Triangles

What about other polygons? 
Students should be encouraged to divide other polygons into parts and add the areas. Also, students will be able to estimate any area by multiplying average length by the width. 

How will these ideas help students? 
  • Students won't need to memorize as many formulas. 
  • They will be able to apply the area concept to new shapes more easily. 
  • The concept of volume as length by width by height will be a natural extension. 
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Happy Spring!

Angie Seltzer