Discovering the Area Formula for Trapezoids Lesson 3

  • Activities with Rigor and Coherence (ARCs) / Discovering Area Relationships / Discovering the Area Formula for Trapezoids Lesson 3
  • Discovering the Area Formula for Trapezoids

    Lesson 3 of 4
    6th grade

    45–60 minutes


    Students explore several strategies for calculating the area of a trapezoid while discovering the area formula for trapezoids.



    This lesson is written to focus on trapezoids that have exactly one pair of parallel sides. It may be necessary to review the properties of trapezoids in relation to other quadrilaterals.

    In partners or small groups display or distribute the Trapezoid or Not? Activity Sheet (download from Materials section above), and ask students to identify which quadrilaterals are trapezoids. Discuss responses to address any misconceptions.


    The main portion of this lesson involves the derivation of an area formula for trapezoids.

    Diagram of a Trapezoid

    Divide students into partners or small groups and distribute the Area of Trapezoids Activity Sheet. Have students work together to come up with multiple strategies for determining the area of a trapezoid, using a different strategy for each trapezoid.

    After students have had some time to discuss suggestions in their groups, conduct a whole class discussion.

    The discussion should start with asking students for their ideas about decomposing the trapezoid. Once all ideas have been discussed, the teacher can encourage students to find additional representations by asking guiding questions to suggest alternative strategies.

    area of a trapezoid method 1 area of a trapezoid method 2 area of a trapezoid method 3
    area of a trapezoid method 4 area of a trapezoid method 5


    After highlighting a variety of strategies to discover the area of a trapezoid, be sure to emphasize that all strategies result in the same formula.

    Assessment (optional)

    Students work together to measure the bases and heights and calculate the area of the trapezoids on the Trapezoid Activity Sheet.


    Activity 1 (Technology Option)

    Open the Area Tool using a computer. The trapezoids tab can be used to demonstrate that the midline is the average of the two bases as well as investigating the relationship of the height and the length of the bases to the area.

    Two parallelograms with the same base and height can have the same area.

    Activity 2

    The state of Nevada is relatively close to the shape of a trapezoid. Find the approximate area of the state using the border lengths of the surrounding states. The distance along the Utah border is approximately 554 km, the distance along the Oregon/Idaho border is about 487 km, the distance along the California border is about 330 km on the west side of Nevada and about 635km on the southwest side; the distance on the Arizona border is about 218 km. Using what you know about trapezoids, estimate the area of Nevada. Justify your answer, and provide an explanation of the mathematics used.

    map of the western united states

    Teacher Reflection

    • Which strategy allowed students to visualize the formula for finding the area of a trapezoid?
    • Which strategy do you think resonated with students the most? Why?
    • What misconceptions did your students have and how were they addressed?
    • How did the questions you posed strengthen student understanding?

    Leave your thoughts in the comments below.

    Related Material

    From Teaching Children Mathematics, by Tutita Casa, Ann Spinelli, M. Gavin

    A collection of short, fun math videos explaining math concepts and minds on video.

    Other Lessons in This Activity

    Lesson 1 of 4

    Students develop the area of triangles formula using the area of rectangles and by comparing triangles with equal bases and heights.

    Lesson 2 of 4

    Students use prior knowledge of the area formula for rectangles and triangles to discover the formula for the area of parallelograms.

    Lesson 4 of 4

    Students will estimate the area of irregular shapes and use a process of decomposition to calculate the areas of irregular polygons.

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  • Essential Question(s)

    • How can you use the area of parallelograms or triangles to determine the area of trapezoids?
    • What strategies could you use to develop the area of a trapezoid?


    CCSS, Content Standards to specific grade/standard

    • 6.G.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.

    CCSS, Standards for Mathematical Practices

    • SMP 5 Use appropriate tools strategically.
    • SMP 8 Look for and express regularity in repeated reasoning.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Pose purposeful questions.