Lesson 9Formula for the Area of a Triangle

Let’s write and use a formula to find the area of a triangle.

Learning Targets:

  • I can use the area formula to find the area of any triangle.
  • I can write and explain the formula for the area of a triangle.
  • I know what the terms “base” and “height” refer to in a triangle.

9.1 Bases and Heights of a Triangle

Study the examples and non-examples of bases and heights in a triangle. Answer the questions that follow.

  • These dashed segments represent heights of the triangle.
Three copies of a triangle. Each copy shows a different base and height pair.
  • These dashed segments do not represent heights of the triangle.
Three copies of a triangle. Each copy has a different side labeled base and a line from the base that is not the height.

Select all the statements that are true about bases and heights in a triangle.

  1. Any side of a triangle can be a base.
  2. There is only one possible height.
  3. A height is always one of the sides of a triangle.
  4. A height that corresponds to a base must be drawn at an acute angle to the base.
  1. A height that corresponds to a base must be drawn at a right angle to the base.
  2. Once we choose a base, there is only one segment that represents the corresponding height.
  3. A segment representing a height must go through a vertex.

9.2 Finding the Formula for Area of a Triangle

  • For each triangle, label a side that can be used as the base and a segment showing its corresponding height.
  • Record the measurements for the base and height in the table, and find the area of the triangle. (The side length of each square on the grid is 1 unit.)
  • In the last row, write an expression for the area of any triangle using b and h .
Four triangles labeled A--D on a grid.
triangle base (units) height (units) area (square units)
A
B
C
D
any triangle b h

9.3 Applying the Formula for Area of Triangles

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.

Five triangles, all measurements in centimeters: Triangle A has one side length of 5 with a perpendicular length of 6 from that side to the opposite vertex. Triangle B has sides of length 4, 4, and unknown with a 90-degree angle between the two known sides. 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. 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. Triangle E has sides 10, 6, and unknown. The perpendicular length from the side of length 6 to the opposite vertex is 5

Lesson 9 Summary

  • We can choose any of the three sides of a triangle to call the base. The term “base” refers to both the side and its length (the measurement).
  • The corresponding height is the length of a perpendicular segment from the base to the vertex opposite of it. The opposite vertex is the vertex that is not an endpoint of the base.

Here are three pairs of bases and heights for the same triangle. The dashed segments in the diagrams represent heights.

Three images of a triangle, each with a different side labeled “base” and an accompanying dashed line perpendicular to the base indicating the height.

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 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 formula for the area of a parallelogram with base b and height h is b \boldcdot h .
  • A triangle takes up half of the area of a parallelogram with the same base and height. We can therefore express the area A of a triangle as: A = \frac12 \boldcdot b \boldcdot h
Three figures: triangle A with base 5 and height 6; triangle B with base 3 and height 3; triangle C with base 12 and height 4.
  • The area of Triangle A is 15 square units because \frac12 \boldcdot 5 \boldcdot 6=15 .

  • The area of Triangle B is 4.5 square units because \frac12 \boldcdot 3 \boldcdot 3 = 4.5

  • The area of Triangle C is 24 square units because \frac12 \boldcdot 12 \boldcdot 4 = 24 .

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.

A right triangle with legs of length 4 and 9.

Glossary Terms

opposite vertex

For each side of a triangle, there is one vertex that is not on that side. This is the opposite vertex.

For example, point A is the opposite vertex to side BC .

Lesson 9 Practice Problems

  1. Select all drawings in which a corresponding height h for a given base b is correctly identified.

    Six images of the same triangle, labeled A, B, C, D, E, and F. On triangle A, the top side is labeled “b” and a dashed line extending straight down from the right vertex islabeled “h”. On triangle B the top side is labeled “b” and a dashed line extends from the center of the top side to the opposite vertex labeled “h”. On triangle C, the right side is labeled “b” and a dashed line extends from the right top vertex straight down to the level of the bottom vertex. On triangle D the left side is labeled “b” and a perpendicular line labeled “h” extends to the opposite vertex. On triangle E, the right side is labeled “b” and a dashed line labeled “h” extends out from the bottom vertex at a right angle to the left side. On triangle F, the right side is labeled “b” and a perpendicular dashed line labeled “h” extends from the side labeled “b” and extends to the opposite vertex.
  2. For each triangle, a base and its corresponding height are labeled.

    a. Find the area of each triangle.

    b. How is the area related to the base and its corresponding height?

  3. Here is a right triangle. Name a corresponding height for each base.

    A triangle with sides labeled d, e, and f. The angle opposite side D is a right angle. A segment labeled g is perpendicular to side d and extends to the opposite vertex.
    1. Side d
    2. Side e
    3. Side f
  4. Find the area of the shaded triangle. Show your reasoning.

    A square with a shaded triangle contained inside it. The left and bottom sides of the square are labeled six, and the right side is labeled 2 above the point where vertex of the shaded triangle meets the side, and 4 below the point where the vertex meets the side.
  5. Andre drew a line connecting two opposite corners of a parallelogram. Select all true statements about the triangles created by the line Andre drew.

    A parallelogram with a line connecting two opposite corners. The parallelogram has a base of 3 units and a height of 9 units.
    1. Each triangle has two sides that are 3 units long.
    2. Each triangle has a side that is the same length as the diagonal line.
    3. Each triangle has one side that is 3 units long.
    4. When one triangle is placed on top of the other and their sides are aligned, we will see that one triangle is larger than the other.
    5. The two triangles have the same area as each other.
  6. Here is an octagon.

    An octagon with straight sides that are 4 inches long, and angled sides that are both 3 inches high and 3 inches wide.
    1. While estimating the area of the octagon, Lin reasoned that it must be less than 100 square inches. Do you agree? Explain your reasoning.
    2. Find the exact area of the octagon. Show your reasoning.