## Discovering the Area Formula for Triangles

• Lesson
6-8
1

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.

Prior to this lesson, students need experience in measuring squares and rectangles that are not squares and calculating their areas. As a warm-up, have students measure at least square and one rectangle found within the classroom, record its dimensions, and calculate the area of each. For example, they could measure floor tiles, windows, chalkboards, bulletin boards, desktops, shelves, and so forth. Challenge them to measure as many different shapes as possible and report back to the class.

Divide students into groups of three. All three are responsible for the work, but the following tasks could be assigned:

• Recorder: Keeps a record of all important information.
• Measurement Verifier: Confirms all measurements and calculations.
• Reporter: Shares all pertinent information with the class.

Distribute the Squares and Rectangles activity sheet. Each member of the group should measure the dimensions of each shape on the sheet and calculate its area. Allow students to share within their groups, and check the measurements and calculations before allowing them to continue. If necessary, review the area formula for rectangles: A = b × h.

 Squares and Rectangles Activity Sheet

Using rulers, students should draw one diagonal in each of the shapes A, B, and C, and then cut each shape along the diagonal into two parts. In their groups, have students estimate the area of each triangle formed by dividing shapes A, B, and C in half along the diagonal.

Students can estimate the areas using any methods they choose. One method is to simply count the number of squares, half-squares, and partial squares that are formed when the shapes are divided. Another method is to realize that each shape has an area equal to half the area of the original shape. (Students can see this by placing one half over the other.)

Discuss the results with the class as a whole.

Similarly, have students estimate the area of the largest triangle formed when shape D is divided into three parts as shown below. As with A, B, and C, students can estimate the area by counting squares, or they can fit the two smaller triangles together to form the larger triangle.

Within each group, students will likely have chosen several different points along the top of the rectangle. However, each student should notice that the area of the largest triangle is equal to half the area of the original rectangle. More importantly, students within each group should realize that this is the case, regardless of the point chosen along the top.

To allow students further opportunity to explore this idea, you may ask them to use the Area Tool for Triangles.

 Area Tool for Triangles

Students should realize that though the shape of the triangle may change, the base, height, and area do not. To emphasize the point, ask them to drag point B so that point D lies directly on top of point A; this forms a right triangle with the right angle at A, as shown below. Then, ask students to drag point B so that point D lies directly on top of point C; this forms a right triangle with the right angle at C. Students quickly recognize that these triangles are congruent, so they must have the same area.

Discuss the results with the class. Ask students how the area of each triangle relates to the area of the original shape. Students should realize that, in each case, the area of the triangle is equal to one-half the area of the rectangle. (At this point, you may be tempted to give students the formula A = ½bh, but it will be more valuable to let them formalize the rule on their own, through the next activity and the ensuing discussion.)

Distribute the Unknown Triangle activity sheet, in which the dimensions of two triangles are known, and ask students to determine the area of both triangles. Allow them to use any method they like, but encourage them to use what they just discovered to find the area. Students will likely realize that the first shape is a right triangle, so it is congruent to one-half a rectangle that was divided along the diagonal. However, they may have more difficulty realizing that the area of the second triangle is equal to one-half the area of a 3 × 4 rectangle. As students work, circulate and ask questions to lead them to this conclusion.

 Unknown Triangles Activity Sheet

Ask students to create a formula for determining the area of a triangle. Have them explain their reasoning and prove that their formula works. To prompt discussion, you may need to ask leading questions, such as, "How is the area of a triangle related to the area of a rectangle?" and "What is the formula for finding the area of a rectangle?" Be careful not to ask too many questions too soon in the discussion, as learning will be more effective if students generate the formula on their own.

Assessments

1. The Bermuda Triangle is the triangular region defined by San Juan, Puerto Rico; Miami, Florida; and Bermuda. Using a map, students should determine the dimensions of the Bermuda Triangle, measure the distances using the scale on the map, and calculate the area of the triangle. You can use the Bermuda Triangle Map to display this region for students.

2. In pairs, allow each student to cut a triangle for their partner to measure and calculate the area of. Each student should check the other student’s results and work together to resolve any disagreements.

Extensions

1. Using the Internet, students should research the history of the Bermuda Triangle to determine its dimensions. Students can report back to the class with their findings. Some questions to ask the students include:
• Is the Bermuda Triangle truly a triangle? If not, what shape is it? Why? If it’s not a triangle, are you able to approximate the total area covered by the Bermuda Triangle?
• Do you think there is a "center" to the Bermuda Triangle? How would you find it?

2. Allow students to research other famous triangles and calculate their areas. Some possibilities include the Sunni Triangle (which extends northwest from Baghdad, Iraq); Research Triangle Park (connecting Raleigh, Durham, and Chapel Hill, NC); and Point State Park (along the Allegheny and Monongahela Rivers in Pittsburgh, PA). In pairs, students can quiz each other and check their calculations.

Questions for Students

1. Do two triangles with the same height have the same area? Why or why not? Give some examples.

[Two triangles with the same height have the same area only if they also have the same base. Consider two triangles, both of which have a height of 4 inches. If one of the triangles has a base of 3 inches, its area is A = ½ × 3 × 4 = 6 in.2. If the other triangle has a base of 5 inches, its area is A = ½ × 5 × 4 = 10 in.2. The areas are clearly not equal.

On the other hand, the two triangles below have the same area because the base and height have equal measures.

It doesn't matter that the shapes of the triangles are different.]

2. Explain how other shapes (besides rectangles and squares) could been have used to derive the area formula for triangles.

[The formula for finding the area of a parallelogram is A = bh, the same formula used for rectangles. By dividing a parallelogram along the diagonal, two congruent triangles are formed, which would lead to the same conclusion; namely, that the area formula for a triangle is A = ½bh.]

Teacher Reflection

• Did students come up with alternative methods for finding the areas of their triangles? If so, how did you react to their explanations?
• What were some of the ways that the students illustrated that they were actively engaged in the learning process?
• Did students demonstrate an understanding of how and why we use the formula A = ½bh?
• Did you notice any positive or negative effects of asking students to check each others’ work?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### 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:
• Determine the areas of rectangles and squares
• Derive an area formula for triangles
• Use the area formula to calculate the area of a triangle or to find one of the dimensions