Connections between the properties of rectangles, trapezoids, and parallelograms should be made so that students will better understand the formulas used to determine the areas of these quadrilaterals.

• Discuss the definition of a rectangle and a parallelogram. State that all rectangles are parallelograms, but only those parallelograms with right angles are rectangles.
  - Parallelograms are quadrilaterals with opposite sides parallel.
  - Rectangles are parallelograms with a right angle.

• Demonstrate and discuss how the formula for the area of a parallelogram (A=bh) will also apply to a rectangle because a rectangle is always a parallelogram.

• Demonstrate how a parallelogram can be recomposed to form a rectangle. Have the students draw a parallelogram on graph paper. Then have the students cut one end of the parallelogram off by making a vertical cut from a vertex and remove the triangle at the end. Place it at the other end of the figure to form a rectangle.

  

• Demonstrate how an isosceles trapezoid can be recomposed to form a rectangle. The area for an isosceles triangle can now be thought of as A = bh. The base in this case is the one formed by subtracting the b₂ from b₁.

  

• If the trapezoid is not isosceles, create a copy of the trapezoid, flip it, and join it to the end of the original trapezoid.

  

  This forms a parallelogram with the same height as the original trapezoid and parallel bases of length b₁+b₂. The area of a parallelogram is A = bh, so the combined area of the two trapezoids is A = (b₁+b₂)h. Because we only want half this area, the formula for the area of a trapezoid is:

  

• Demonstrate how any trapezoid can be recomposed to form a rectangle and a triangle.

  

  Translate the base with length b₁ to the left so that the bases and the altitude are perpendicular. Does this trapezoid have the same area as the original? [Yes. The bases and the altitude have the same length as the original.]

  

  Determine the area of the trapezoid by showing it as the composition of a rectangle and a triangle.

  

  The rectangle has a base with length b₁ and a height of h. The triangle has sides of length h and b₂ - b₁.

• Model several area problems using the formulas. Remember for trapezoids, the bases are the sides that are parallel and it makes no difference which one has length b₁ or b₂.

  Example 1:

  Determine the area of the trapezoid with the formula.

  

  Determine the area of the trapezoid by showing it as the composition of a rectangle and a triangle.

  

  The area of the rectangle is A = bh = (6) (8) = 48 square inches. The area of the triangle is square inches. The combined area is 48 + 39 = 87 square inches.

  Example 2:

  Determine the area of the parallelogram.