Area of a rectangle.

1.     Draw a rectangle composed of whole units on graph paper. How would you calculate the area concretely?

 

2.     We call the segment on the bottom of the rectangle the base, b. The top side of the rectangle can also be called the base. Why?

 

3.     What do we call the segment on the side of the rectangle? (altitude or height**) The altitude is the perpendicular distance from one base to the other base.

 

4.      Write a formula for the area, A, of a rectangle in terms of its base, b, and its height, h.

 

**Typically, altitude and height are interchangeable because it is said that the height is just the measure of the altitude.

 

Area of a right triangle.

5.      Draw a rectangle on graph paper. Calculate the area.

 

6.      Draw a diagonal of your rectangle to form two right triangles. What would you guess about these triangles? Based on this, what is the area of each triangle?

 

7.      If you are given a right triangle, can you always form a rectangle by fitting it together with a copy of itself? Illustrate your answer with examples by choosing a right triangle, making an exact copy, and following the suggested procedure.

 

8.      Write a formula for the area, A, of a right triangle in terms of its base, b, and its height, h.

 

 

Area formula for parallelograms.

 

**Note: Those of you copying the activity from the web, label the second point from the left on the top as A.

 

1.  Copy the drawing at right onto graph paper. Draw an altitude in the parallelogram from point A to the base so that a right triangle forms.

  1. Cut out the parallelogram. Cut off the right triangle, translate it to the opposite side, and fit it to the figure.

 

What kind of quadrilateral forms? What is the area of the figure in terms of b and h of the original parallelogram? How does the area of the parallelogram relate to the area of the figure formed by the translation?

 

  1. Write a formula for the area of a parallelogram in terms of its base, b, and its height, h.

 

 

 

How do you know that the triangle will always fit, as in step 2?

 

 

 

How can you use the method from the activity for the area of a right triangle to derive the formula for the area of an obtuse triangle with base b and height h? Work through the procedure using the correct quadrilateral.

 

What can you conclude about the area of any triangle?

 

 

 

 

 

 

Area formula for trapezoids.

 

1.  Make two copies of the trapezoid at right on graph paper and cut them out. The bases of the trapezoid are b1 and b2, and the height h.

 

  1. Find a way to fit the two copies of the trapezoid together to form a parallelogram.

 

  1. Write a formula for the area of the parallelogram that uses an expression based on

    b1 + b2. Use the formula you wrote for the parallelogram to write a formula for the area of the original trapezoid.