Topic: Area of Parallelograms and Triangles

Learning Objectives:

  • The Student Will derive area formulas for parallelograms and triangles
  • The Student Will apply and extend basic area formulas.
Prior Knowledge:.Axioms of area; basic facts about parallelograms and triangles.

Motivational Problem: Draw a rectangle and a series of "knocked-over rectangles" (parallelograms). Compare and contrast the areas? On what does the area of the parallelogram depend?

Overview

The key techniques for this topic are applying the basic postulates of area to finding the areas of parallelograms and triangles. By "cutting and pasting" the parallelogram, and applying the Area Congruence Postulate and the Area Addition Postulate, you can see that the area depends on base and height. Then, by noting that any triangle is really half a parallelogram, the formula for the area of a triangle follows immediately. These proofs offer an excellent opportunity to integrate simple, narrative proofs from the early part of curriculum into a new topic that students should already have familiarity with.

Immediate applications include finding the formula for the area of a rhombus, finding the areas of right triangles and equilateral triangles, and developing strategies for finding the areas of isosceles triangles.

Once you have the formula for the area of a triangle, you can find the review the techniques of finding the area of of arbitrary polygons in the plane by "completing the rectangle" and subtracting off right triangles.

Key Questions: How does the area of parallelogram relate to the area of rectangle? Why is the area of a triangle half base times height? What are the strategies for find the areas of various triangles?

Looking Ahead: Decomposing figures into triangles will be key in finding areas of trapezoids and general polygons.

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