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Finding the Area of Trapezoids

  • Lesson
6-8
1
Measurement
Christopher Johnston
location: unknown

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

Prior to this lesson, students will need experience in measuring rectangles and triangles and calculating their areas.

Students will also need to be able to identify trapezoids. It may be beneficial to review the properties of trapezoids in relation to other quadrilaterals. For example, students could be given pictures of various quadrilaterals and identify which ones are trapezoids and which ones are not. Require students to explain their classifications. Be sure that students are given examples of trapezoids in various orientations. It might also be useful to present the following Venn diagram to help students understand how quadrilaterals are related to one another:

1893 traps venn 

As a warm-up, allow students to estimate the areas of trapezoids. Using the Trapezoids Activity Sheet, students working in groups can overlay centimeter grid paper and estimate the total area in square centimeters. (If possible, provide centimeter grid paper on a transparency sheet; alternatively, copy the Centimeter Grid Paper onto thin paper.) Later in the lesson, students will use measurements and a formula to calculate the area, and they will compare their estimates with their calculations.

pdficon  Centimeter Grid Paper 

pdficon  Trapezoids Activity Sheet 


Students should work in groups of three for the main part of the lesson. All students are responsible for the work, but the following tasks could be assigned:

  • Recorder: Records all important information on the record sheet.
  • Measurer: Double checks all measurements and calculations.
  • Reporter: Shares all pertinent information with the class.

These roles are important because they hold each student in the group accountable. Each student needs to be an active participant, and no one student should do all of the work.

Once students are divided into groups, display a standard trapezoid, like the one shown below, on the overhead projector or chalkboard:

1893 trapezoid1 

The main portion of this lesson involves the derivation of an area formula for trapezoids. Prior to that, allow students some exploration time. Ask them to suggest methods for finding the area of the trapezoid shown above. As a prompt, ask, "What other shapes could you use to help you? Are there any shapes for which you already know how to find the area?" After students have had some time to discuss suggestions in their groups, ask the reporter from some groups to share their ideas.

Students may suggest several possibilities. One likely idea is to divide the trapezoid into three pieces—a rectangle and two triangles—as shown below:

1893 trapezoid2 

Using any of the methods that they identify, have students determine the area of a trapezoid with bases of 24 cm and 10 cm and with legs of 15 cm and 13 cm. Allow all groups an opportunity to determine the area, and be sure to get class consensus on what the area is. It is important that all groups determine the area correctly. Later in the lesson, students will compare the area they find here with the area of a rectangle formed by dividing the trapezoid, and it is critical that the answers match.

Another option is to divide the trapezoid as shown below, using a segment that connects the midpoints of the legs. Although the decomposition in the figure above will allow students to calculate the area, the figure below demonstrates how any trapezoid can be transformed into a rectangle, leading students to understand the derivation of the area formula for trapezoids.

1893 fold trapezoid 

When triangles are removed from each corner and rotated, a rectangle will be formed. It’s important for kids to see that the mid-line is equal to the average of the bases. This is the basis for the proof—the mid-line is equal to the base of the newly formed rectangle, and the mid-line can be expressed as ½(b1 + b2), so the proof falls immediately into place. To be sure that students see this relationship, ask, "How is the mid-line related to the two bases?" Students might suggest that the length of the mid-line is "exactly between" the lengths of the two bases; more precisely, some students may indicate that it is equal to the average of the two bases, giving the necessary expression.

Remind students that the area of a rectangle is base × height; for the rectangle formed from the original trapezoid, the base is ½(b1 + b2) and the height is h, so the area of the rectangle (and, consequently, of the trapezoid) is A = ½h(b1 + b2). This is the traditional formula for finding the area of the trapezoid.

After this proof has been demonstrated, students can measure the dimensions of the rectangle and calculate its area. Ask the class, "How does the area of this new rectangle relate to the area of the trapezoid from which it was formed?" When calculating the area of the rectangle, students should get the same answer as obtained above for the area of the trapezoid. More importantly, students should realize that only the shape changed, not the area.

Allow students to work in groups to calculate the area of the trapezoids on the Trapezoids Activity Sheet. Circulate as students work, offering help as necessary. Also note successful approaches that students use to calculate the area of each trapezoid, and be sure to have students share those approaches during a class discussion. After ample time, gather the class together, and discuss the results.

To push the discussion further, you might want to return to the original trapezoid shape and use other decompositions to prove the area formula for a trapezoid. If your students are prepared for the algebra involved in generating such proofs, seeing more than one derivation can be very valuable. For instance, if students suggested that the trapezoid be divided into two triangles and a rectangle, then label the pieces as shown below, and discuss the labels with students:

1893 trapezoid3 

The proof of the area formula then falls into place as follows:

  • The area of the rectangle is b1h.
  • The area of the triangle on the left side is ½xh.
  • The area of the triangle on the right side
    is ½(b2 – b1 – x) × h = ½b2h – ½b1h – ½xh.
  • The combined area of the three pieces, then,
    is b1h + ½xh + ½b2h – ½b1h – ½xh, which simplifies to ½b1h + ½b2h. This further simplifies to ½(b1 + b2h, which is the standard area formula.

You may wish to explore the proof for special trapezoids, such as right and isosceles trapezoids. Although the proofs will be similar, students may benefit from multiple opportunities to verify the formula.

Students can also use the Area Tool for Trapezoids to investigate the relationship of the height and the length of the bases to the area.

appiconArea Tool for Trapezoids 

Return to the warm-up activity during which students estimated the areas of the trapezoids on the Trapezoids Activity Sheet. Have students compare their estimates with the calculated areas. How close did they come? Were students able to predict the area within one square centimeter?

To conclude the lesson and allow for practice, rearrange students into pairs. Each student should create a trapezoid, measure its dimensions, and determine its area. They should then swap trapezoids, measure and calculate the area, and discuss the results.

Assessments 

1. Use the Trapezoid Activity Sheet as a form of assessment.

2. The teacher can circulate throughout the room while students are completing the online activities to assess their understanding.

Extensions 

  1. The area under a speed-time graph is equal to the total distance traveled. The shape of a graph with speed increasing followed by a steady speed, like the graph below, is a trapezoid. Students can calculate the total distance traveled by calculating the area of the trapezoid.

    1893 speed time 

  2. Allow students to use graphing calculators to find the area of trapezoids. See the user guide for instructions.
 

Questions for Students 

1. Today, we discussed several methods for finding the area of trapezoids. Which method did you like the best? Why? Is using the formula easier for you than other methods?

[The formula may not always be easier, but it can always be used to determine the area.]

2. Describe a situation where finding the area of a trapezoid might be necessary.

[When painting a wall, it is estimated that one gallon of paint will cover about 300-400 square feet. If a trapezoidal wall needs to be painted, estimating its area would be necessary to determine how much paint is needed.]

3. Some people say that trapezoids are the "weird" members of the quadrilateral family. Do you agree? Explain your answer.

[Of the four special types of quadrilaterals—trapezoids, parallelograms, rhombuses, and squares—only trapezoids are not required to have two pairs of parallel sides. Therefore, they are a little different from the others.]

Teacher Reflection 

  • As you listened to the students’ conversations, what vocabulary usage did you notice? Were students using the correct terms? Did students seem comfortable with the language and procedures involved in the lesson?
  • Was students’ level of participation high or low? What could be done to foster greater participation in the future?Did you find the online assessments worthwhile? Did they relate to the lesson?
  • Was your lesson appropriately adapted for the diverse learner?
  • Did students favor one method of finding area over another? Which one? Did they demonstrate an understanding of why the formula works?
 
1874icon
Measurement

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.
1882icon
Measurement

Finding the Area of Parallelograms

6-8
Students will use their knowledge of rectangles to discover the area formula for parallelograms.
1902icon
Measurement

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:

  • Use their knowledge of the area formula for rectangles to derive an area formula for trapezoids.
  • Explore alternative methods for determining the area of trapezoids.
  • Calculate the area of trapezoids, given the base and height.

Common Core State Standards – Mathematics

Grade 6, Geometry

  • CCSS.Math.Content.6.G.A.1
    Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.