In this lesson, students will use the area formula for rectangles to
discover the area formula for parallelograms. As necessary, review the
concept of area as well as the area formula, A = bh.
It may also be beneficial to review the properties of parallelograms
as they relate to other quadrilaterals. You might ask students
questions such as, "Is every square a parallelogram?" or "Is any
parallelogram also a rhombus?" To help students answer these questions,
you may want to use the diagram below; or, you may want to take some
time to create the diagram with your students.
To begin the lesson, have students look at a U.S. map. Ask
students, "What state is in the shape of a parallelogram?" Although not
exactly, Tennessee is roughly a parallelogram. Working in groups of
three, students should discuss methods that could be used to determine
the area of Tennessee. (At the end of the lesson, students will return
to this problem and discuss their findings with the class.)
Distribute the Rectangles and Parallelograms and the Quadrilateral Area Record Activity Sheets. Give students time to determine the area of
shapes A-E, and have them record the information on the record sheet.
Rectangles and Parallelograms Activity Sheet
Quadrilateral Area Record Activity Sheet
For each rectangle, students can simply count the number of
squares, or they can multiply the length by the width. Alternatively,
they could practice either metric or customary measurements by
measuring the length and width using a ruler, and then multiplying to
find the area. For each parallelogram, students will likely need to
count the squares to determine the area; they will need to combine
partial squares to form full squares when making their estimates.
Then, have student cut out shapes A, B, and C.
With rectangle A, have students cut from the lower left corner
to a point on the top edge that is three units in from the upper left
vertex; this cut will form a 45-degree angle, which divides each of the
squares through which it passes exactly in half. Then, for rectangles B
and C on the activity sheet, have students remove a triangle by cutting
from the lower left corner diagonally to any point along the top edge.
One such cut is shown below. (It might be helpful for students to first
draw a straight line with a ruler. You should also encourage them to
choose a point along the top where the edge and one of the grid lines
meet.) As shown below, students should then place the removed triangle
at the other end of the rectangle. Encourage students to make a
different cut than other members of their group.
Students should then determine the area of the resulting parallelograms and record the results on the record sheet.
Working in the other direction, have students cut out shapes D
and E. From these shapes, have students remove the right triangle on
either the right or left side, and move it to the other
side. Students should realize that this modification changed the
parallelogram to a rectangle with the same area. As above, students
should determine the area of the resulting rectangle and record the
information on the record sheet.
As students complete the record sheet, verify the accuracy of
their measurements and calculations. Without correct values, students
will not discover that the formulas for rectangles and parallelograms
To explore other shapes, students can use the Shape Cutter tool.
Shape Cutter Tool
Students can create either a rectangle or parallelogram, make an
appropriate cut, and then rearrange the pieces. Further, students could
even make other cuts to show that two non-rectangular parallelograms
with the same base and height have the same area, as shown below. (If
the red piece were moved to the other side, notice that a different
parallelogram would be formed.)
After the explorations, discuss what happened as a class, and pose the following questions to students:
- How was the shape changed? [The rectangle was made into a parallelogram. Or, the parallelogram was made into a rectangle.]
- Have the dimensions changed? [The side lengths have changed, but the base and height have not changed.]
- Has the area changed? [No.]
- Did other students in your group make a different cut? If so,
did they get a different area than you did? [The area should be the
same, regardless of the cut that was made.]
At the conclusion of this discussion, students should realize that the area formula for a rectangle, A = bh, is also the area formula for parallelograms.
As a final piece, students should also recognize the
relationship between the area formulas for trapezoids and
parallelograms. The formula for a trapezoid is A = ½(b1 + b2) × h. For a parallelogram, the bases are equal, so b1 = b2. Therefore, using the trapezoid formula to calculate the area of a parallelogram results in the following:
A = ½(b1 + b2) × h = ½(b1 + b1) × h = b1h
As in previous lessons in this unit, students can also use the Area Tool for Parallelograms to investigate the relationship of the height and the length of the base to the area of a parallelogram.
Area Tool for Parallelograms
At the end of the lesson, return to the motivating problem: What
is the area of the state of Tennessee? Students should now use
measurements from the map to determine the height and base, and then
they should use the formula to find the area.
Questions for Students
1. What relationship is shared by parallelograms and rectangles that allow the same formula to be used to find the area of each?
[A rectangle and parallelogram with the same base and height have the
same area. When a triangle is removed from a rectangle and reattached
to form a parallelogram, the base, height, and area remain the same.]
2. Other than using a formula, what methods could you use to determine
the area of a parallelogram? Give an example and show step-by-step how
you found the area. Which is easier — your method, or using the
[A parallelogram could be divided into several parts.
One possible division is dividing it into a rectangle surrounded by two
triangles, as shown below. The area of each piece could then be
calculated, and the areas could be added together.
Although this method will yield the correct answer, it requires
using an area formula three times instead of just once. Therefore, it
is probably easier to use the parallelogram formula.]
3. When finding the area of a parallelogram, why multiply base times height, instead of base times side?
[The height is the perpendicular distance from the base
to the top. The length of the side can change, depending on the
orientation of the parallelogram, but the height never changes.]
4. Can the area formula for parallelograms be extended to rhombuses? Why or why not?
[Yes, because a rhombus is a parallelogram with four congruent sides.]
- What alternative methods did students use to calculate the area of
parallelograms? Will they always work? Did students clearly explain
- Did you find the online assessments worthwhile? Did they
relate to the lesson? Did they challenge students? If not, how could
you change the online assessments so that they were more challenging?
- Were students involved throughout the lesson?
- How does this lesson address the needs of the diverse learner?