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
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.
Centimeter Grid Paper
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
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
Students may suggest several possibilities. One likely idea is
to divide the trapezoid into three pieces—a rectangle and two
triangles—as shown below:
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.
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:
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 + b2) h, 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.
Area 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.
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.
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.
- 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.]
- 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
- 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