Area Activities and Questions

A.    Rectangle Area

1. Rectangle Area Example (Answer These Questions by Writing or Drawing on this Page)

·        What is the area of each rectangle?  Show how you computed the areas.

·        Suggest a way to help a student understand how to the compute the areas. Add something to the drawing to show this idea.

·        Why do you think more than one rectangle is included on this page?

B. Parallelogram Area

The corner cut dissection (*)

One method to deduce the area of a parallelogram is by dissection, i.e., cutting up the parallelogram and reassembling it as a rectangle.  Here is a picture of how this works.

·        The usual way for computing the area of a parallelogram is a formula in terms of BASE and HEIGHT.  Give a definition of base and height and write an area formula using base and height.

C. Triangle Area by "doubling the triangle"

The area of a triangle can be related to the (known) area of a parallelogram by the following bit of cutting and pasting.

Start with any triangle.  Make an exact second copy of the triangle.

Turn one triangle over and attach to the other to form a parallelogram.

·        How can you convince a friendly skeptic that this is indeed a parallelogram?

Therefore, the area of the triangle is one-half of the area of the parallelogram.

I.                    How can you compute the area of a triangle if you know the base BC and the altitude (the height or altitude of the triangle is the height of this parallelogram)?

5. Shearing Triangles (Write Answer on This Sheet)

Start with a triangle ABC. Think of AB as the base.  Then move C along a line parallel to AB.  This produces triangles such as ABC, ABD, ABE in the figure.

·        Explain why each of these 3 triangles has the same area.

6. An Alternative Area Dissection

Cut out any triangle ABC.  Mark the midpoints D of side BC and F of side CA and then midpoint E of DF.  Make the cuts DF and CE. 

·        Now reassemble the 3 pieces as a parallelogram with base AB. 

·        How do you know your figure is a parallelogram?

8. Comparison of Areas

Start with any triangle ABC and arrange 4 exact copies of the triangle in the following figure to make a bigger triangle.  Suppose that triangle ABC has area = T.  Figure out the areas of the polygons listed below. (The answers should be in terms of T, such as 3T or T/2, etc.)

·        What is the area of triangle BEC?

·        What is the area of triangle ADF?

·        What is the area of the quadrilateral BDEC? 

·        What is the area of the triangle AEF? 

·        What is the area of triangle ADG? (G is the point on CE with CG/CE = 1/4.)

9. Areas in a Rectangle

Start with any rectangle ABCD.  Draw the diagonals AC and BD. 

·        If point E is the intersection of the diagonals, which of the triangles ABE, BCE, CDE, DAE have the largest area? 

·        Which triangles have the largest perimeter?

·        How does this change if BC is less than AB? 

·        How can you convince the friendly skeptic that your answer is correct?

14. Can One Always Cut Corners?

One important consequence of the corner cutting and base and height formula for area is that if two parallelograms have the same base and the same height, they have the same area.  But is there a hole in our reasoning?  Look at this example.

In this case, the parallelogram ABEF has the base AB but the construction of a point A' on the base so that FA' is perpendicular to AB fails, if we want A' to be on the segment AB.

How does one measure height from the base AB in this case?  Does base times height equal area in this case? 

Is the formula still true? 

If so, what reason can you give? 

(Optional Challenge) Can you cut apart this parallelogram and reassemble it as a rectangle with the same base and height?

Challenge Problems

I. Two Squares into One

• Draw two squares of the same size.  Cut the squares into pieces and reassemble the pieces into a single square.
• If the side length of the original squares is 1 unit, what is the side length of the new single square? Explain how you know this because of area. 
• Draw or tape your solution on a page and turn it in.

II. Dividing a Square Cake into 5 Pieces (Optional Challenge*)

This problem will be left open for a while for you to think about.

Suppose you have made a lemon cake that has a square base with 8-inch sides and that is 2 inches tall.  This cake has chocolate icing on top and lemon icing on the sides.  Explain how you can divide this cake into 5 pieces so that each piece has the same amount of cake and the same amount of each kind of icing.