Unit 1, Lesson 8
Formula for the Area of a Triangle
Accelerated 6
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In this activity, students think about the meaning of base and height in a triangle by studying examples and non-examples. Then, they examine some statements about bases and heights and determine if the statements are true. The goal is for students to see that in a triangle:
Monitor for students who notice similarities between the bases and heights in a triangle and those in a parallelogram. Ask them to share their observations later.
As students justify how they know whether the given statements are true and consider others’ justifications, they practice constructing logical reasoning and critiquing the reasoning of others (MP3).
Students will have many opportunities to make sense of bases and heights in this lesson and an upcoming one, so they do not need to know how to draw a height correctly at this point.
Remind students that recently they looked at bases and heights of parallelograms. Tell students that they will now examine bases and heights of triangles.
Display the examples and non-examples of bases and heights for all to see. Read aloud the first paragraph of the activity and the description of each set of images. Give students a minute to observe the images. Then tell students to use the examples and non-examples to determine what is true about bases and heights in a triangle.
Arrange students in groups of 2. Give them 2–3 minutes of quiet think time and then a minute to discuss their responses with a partner.
Here are six copies of a triangle. In each copy, one side is labeled base.
In the first three drawings, the dashed segments represent heights of the triangle.
In the next three drawings, the dashed segments do not represent heights of the triangle.
Select all the statements that are true about bases and heights in a triangle.
If students are unsure how to interpret the diagrams, ask them to point out parts of the diagrams that might be unclear. Clarify as needed.
Students may not remember from their experience with parallelograms that a height needs to be perpendicular to a base. Consider posting a diagram of a parallelogram—with its base and height labeled—in a visible place in the room so that it can serve as a reference.
For each statement, ask students to indicate whether they think it is true. For each statement, ask one or two students to explain how they know. Encourage students to use the examples and counterexamples to support their argument. (Ror instance, “The last statement is not true because the examples show dashed segments or heights that do not go through a vertex.”) Make sure that students agree about each statement before moving on. Display the true statements for all to see.
Students should see that only Statements 1 and 4 are true—that any side of a triangle can be a base, and a segment for the corresponding height must be drawn at a right angle to the base. What is missing—an important gap to fill during discussion—is the length of any segment representing a height.
Ask students, “How long should a segment that shows a height be? If we draw a perpendicular line from the base, where do we stop?” Solicit some ideas from students. Then, highlight the following:
If any students noticed connections between the bases and heights in a triangle with those in a related parallelogram, invite them to share their observations. Otherwise, draw students’ attention to it. Consider duplicating a triangle that shows a base and a height (by tracing on patty paper or creating a paper cutout). Use the original and the copy to compose a parallelogram. Ask students: “Suppose we choose the same side as the base of both the parallelogram and the triangle. What do you notice about the height of each shape?” (The two shapes have the same height.)
MLR8: Discussion Supports
To Gather
Geometry toolkits
In this activity, students generalize a formula for the area of triangles. They build on their observations about the relationship between the area of a triangle and the area of a parallelogram with the same base and height.
Students first find the areas of several triangles given base and height measurements. They notice regularity in repeated reasoning and arrive at an expression for finding the area of any triangle (MP8). Students might write
At the end of the activity, consider giving students a chance to reason more abstractly and think about why the expression
Arrange students in groups of 2–3. Explain that they will now find the area of some triangles using what they know about base-height pairs in triangles and the relationships between triangles and parallelograms.
Give students 5 minutes to complete the activity. Provide access to geometry toolkits, especially tracing paper. Ask them to find the area of at least two triangles independently before discussing with their partner(s).
If needed, remind students how they reasoned about the area of triangles in the previous lesson (such as by composing a parallelogram, enclosing a triangle with one or more rectangles, and so on.). Encourage them to refer to their previous work and to use tracing paper as needed.
For each triangle:
| triangle | base (units) | height (units) | area (square units) |
|---|---|---|---|
| A | |||
| B | |||
| C | |||
| D | |||
| any triangle |
In the last row, write an expression for the area of any triangle, using
Select a few students to share their expression for finding the area of any triangle. Record each expression for all to see.
To give students a chance to reason logically and deductively about their expression, ask, “Can you explain why this expression is true for any triangle?”
Display the following diagrams for all to see. Give students a minute to observe the diagrams. Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression
When dealing only with the variables
If time permits, select students who used different diagrams to share their explanation, starting with the most commonly used diagram (most likely Jada's). Ask other students to support, refine, or disagree with their arguments. If time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.
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In this activity, students apply the expression they previously generated to find the areas of various triangles. Each diagram is labeled with two or three measurements. Before calculating, students think about which lengths can be used to find the area of each triangle.
As students work, monitor for students who choose different bases for Triangles B and D. Later, invite them to contribute to the discussion about finding the areas of right triangles.
Explain to students that they will now practice using their expression to find the area of triangles without a grid. For each triangle, ask students to be prepared to explain which measurement they chose for the base and which one for the corresponding height and why.
Keep students in groups of 2–4. Give students 3 minutes of quiet think time and 1 minute to discuss their responses with their group.
For each triangle, circle a base measurement that you can use to find the area of the triangle. Then, find the area of any three triangles. Show your reasoning.
A triangle labeled A. Triangle A has one side length of 5 with a perpendicular length of 6 from that side to the opposite vertex.
A triangle labeled B. Triangle B has sides of length 4, 4, and unknown with a 90-degree angle between the two known sides.
A triangle labeled C. Triangle C has side lengths 7, 3.5, and unknown. The perpendicular length from the side of length 7 to the opposite vertex is 3.
A triangle labeled D. Triangle D has side lengths 8, 3.5, and 8.73. There is a 90-degree angle between the sides of length 8 and 3.5.
A triangle labeled E. Triangle E has sides 10, 6, and unknown. The perpendicular length from the side of length 6 to the opposite vertex is 5.
The aim of this whole-class discussion is to deepen students’ awareness of the base and height of triangles. Discuss questions such as:
Optional
MLR8: Discussion Supports
This activity allows students to practice identifying the base and height of triangles and using them to find areas.
Because there are no directions on which base or height to use, and because not all sides would enable them to calculate area easily, students need to think structurally and choose strategically. All triangles in the problems have either a vertical or a horizontal side. Choosing such a side as the base makes it easier to identify the corresponding height.
In some cases, students may opt to use a combination of area-reasoning strategies rather than finding the base and height of the shaded triangles and applying the formula. For instance, they may enclose a shaded triangle with a rectangle and subtract the areas of extra triangles (with or without using the formula on those extra triangles). Notice students who use such strategies so they can share later.
Keep students in groups of 2. Explain that they will now practice locating or drawing heights and using them to find the area of triangles. Give students 8–10 minutes of quiet think time and time to share their responses with a partner afterward. Provide access to their geometry toolkits (especially index cards).
If time is limited, consider asking students to find the area of two or three triangles instead of all four.
For each triangle, identify and label a base and height. If needed, draw a line segment to show the height.
Then find the area of the triangle. Show your reasoning. (The side length of each square on the grid is 1 unit.)
Focus the whole-class discussion on how students went about identifying bases and heights. Discuss:
Here are three pairs of bases and heights for the same triangle. The dashed segments in the diagrams represent heights.
A segment showing a height must be drawn at a right angle to the base, but it can be drawn in more than one place. It does not have to go through the opposite vertex, as long as it connects the base and a line that is parallel to the base and goes through the opposite vertex, as shown here.
The base-height pairs in a triangle are closely related to those in a parallelogram. Recall that two copies of a triangle can be composed into one or more parallelograms. Each parallelogram composed of the triangle and its copy shares at least one base with the triangle.
Two identical triangles, each with a copy composing the triangle into two different parallelograms. In each parallelogram has the bottom side labeled “base” and dashed lines at right angles to the base indicating the height of the parallelogram.
For any base that they share, the corresponding height is also shared, as shown by the dashed segments.
We can use the base-height measurements and our knowledge of parallelograms to find the area of any triangle.
The area of Triangle A is 15 square units because
The area of Triangle B is 4.5 square units because
The area of Triangle C is 24 square units because
In each case, one side of the triangle is the base but neither of the other sides is the height. This is because the angle between them is not a right angle.
In right triangles, however, the two sides that are perpendicular can be a base and a height.
The area of this triangle is 18 square units whether we use 4 units or 9 units for the base.
Help us improve by sharing suggestions or reporting issues.
Students may not be inclined to write an expression using the variables
The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide which measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the Warm-up activity.
Some students may omit the step of dividing by two when finding the area of a triangle. Remind them of the idea that a triangle takes up half as much space as does a parallelogram with the same base and height, post images for reference, or point out the importance of the
Students may think that a vertical side of a triangle is the height regardless of the segment used as the base. If this happens, have them use an index card as a straightedge to check if the two segments they are using as base and height are perpendicular.
Some students may not immediately see that choosing a side that is either vertical or horizontal would enable them to find the corresponding height very easily. They may choose a non-vertical or non-horizontal side and not take advantage of the grid. Ask if a different side might make it easier to determine the base-height lengths without having to measure.
Building Toward
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