## Finding the Area of Parallelograms

• Lesson
6-8
1

Students will use their knowledge of rectangles to discover the area formula for parallelograms.

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.

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.

 Rectangles and Parallelograms Activity Sheet

With rectangle A, have students cut from the lower left corner to a point on the top edgt 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, as shown below, 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 are identical.

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:

• Has the shape changed? [Yes.]
• Have the dimensions changed? [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.

Assessments

1. Have students determine the area of the state of Tennessee.
2. Allow students to work in pairs. Each student should create a parallelogram, measure the dimensions, and calculate the area. Then, they should tell each other the area of their parallelogram and one of the dimensions (either height or base). Their partner should determine the unknown dimension. To ensure that students have experience using various sizes, distribute blank pieces of 8½" × 11" paper.
3. Provide problems for students to complete individually. For example, a backyard shaped like a parallelogram with a base of 7.9 yards and a height of 2.3 yards. What is the area of this backyard? [18.17 yards2.] In addition, provide problems for which the area and either the base or height are known, and have students find the missing factor.

Extensions

1. Students can create parallelograms by giving coordinates on the coordinate plane; another student can draw the parallelogram and determine its area. Students can use the distance formula to calculate the base and height if the parallelogram is not in a typical orientation.
2. Students can create parallelograms of varying sizes using geoboards. In pairs, one student can create a parallelogram, and the other student can determine its area using the area formula. To verify the result, students can use Pick’s Theorem: I + ½P - 1, where I is the number of points in the interior of the polygon and P is the number of points on the perimeter of the polygon.
3. Annabella Milbanke (aka, Anne Isabelle Milbanke, Lady Byron) was known as the "Princess of Parallelograms." Students can research her on the Internet. One possible site is The MacTutor History of Mathematics Archive; although this page is dedicated to her daughter, Augusta Ada Byron Lovelace, it contains significant information about Milbanke. Students can report to the class on why Milbanke was known as the Princess of Parallelograms.

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 formula?

[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.]

Teacher Reflection

• What alternative methods did students use to calculate the area of parallelograms? Will they always work? Did students clearly explain these methods?
• 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?

### Discovering the Area Formula for Triangles

6-8
In this lesson, students develop the area formula for a triangle. Students find the area of rectangles and squares, and compare them to the areas of triangles derived from the original shape.

### Finding the Area of Parallelograms

6-8
Students will use their knowledge of rectangles to discover the area formula for parallelograms.

### Finding the Area of Trapezoids

6-8
Students discover the area formula for trapezoids, as well as explore alternative methods for calculating the area of a trapezoid.

### Finding the Area of Irregular Figures

6-8
Students will estimate the areas of highly irregular shapes and will use a process of decomposition to calculate the areas of irregular polygons.

### Learning Objectives

Students will:

• Measure rectangles, using appropriate units of measure, and determine their areas
• Use their knowledge of rectangle formulas to derive an area formula for parallelograms
• Explore alternative methods for determining the area of parallelograms
• Use the formula they have derived to calculate the area of a parallelogram (given the base and the height) and to calculate one of the dimensions (given the area and the other dimension)