10.2: Area of Triangles
Introduction
The Triangle in the Quilt Block
Now that Jillian has figured out the parallelograms, she is on to the triangles. There are right triangles in the quilt block that she is working on with her grandmother. Jillian needs to figure out the area of each triangle so that she can calculate the size of the triangle and the amount of fabric that she will need to make them all.
Here is the quilt block once again.
There are 16 right triangles in the quilt block. The good news for Jillian is that they all have the same dimensions. Here are the dimensions of the triangle.
Each side of the triangle is three inches. Given these measurements, what is the area of one of the triangles? What is the area for 16 triangles?
Jillian is puzzled. She just figured out how to find the area of a parallelogram and now she is on to triangles. Jillian knows that the triangle and the square are related, she just isn’t sure how.
Use the information in this lesson to learn about finding the area of a triangle!
What You Will Learn
By the end of this lesson you will learn the following skills:
- Recognize the area of a triangle as half the area of a related parallelogram.
- Find the area of triangles given base and height.
- Find unknown dimensions of triangles given area and another dimension.
- Find areas of combined figures involving triangles.
Teaching Time
I. Recognize the Area of a Triangle as Half the Area of a Related Parallelogram
When Jillian looked at the triangles, she could see that they were related to squares. In fact, triangles are related to parallelograms, and a square is a type of parallelogram.
How is a triangle related to a parallelogram?
Let’s look at a parallelogram and see if we can figure out the connection.
Here is a parallelogram. If you look at it carefully, you will notice that we can divide the parallelogram into two triangles.
A rectangle is a type of parallelogram. We can divide a rectangle into two triangles also.
Notice that a rectangle is divided into two right triangles.
A square is a type of rectangle. We can divide a square into two triangles also.
We have two right triangles here too.
If a parallelogram can be divided into two triangles, then what can we say about the area of a triangle?
Based on this information, we could say that the area of a triangle is one-half the area of a parallelogram. Let’s look at how this works.
Example
What is the area of this parallelogram?
We find the area of the parallelogram by multiplying the base times the height.
\begin{align*}A & = bh\\ A & = 2(5)\\ A & = 10 \ sq. \ inches.\end{align*}
A parallelogram can be divided into two triangles.
We can divide the area of the parallelogram in half and that will give us the area of one of the triangles.
10 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} 5 sq. inches
Based on this information, we can write the following formula for finding the area of a triangle.
\begin{align*}A = \frac{1}{2}bh\end{align*}
If you think about this it makes perfect sense. A triangle is one-half of a parallelogram, so the formula for the parallelogram multiplied by one-half is the formula for finding the area of a triangle. Said another way, the area of the parallelogram is divided in half to find the area of a triangle.
Put this concept into practice. Take the area of the following parallelograms and find the area of one of the triangles inside the parallelogram.
- Area of a rectangle is 12 sq. inches
- Area of a parallelogram is 24 sq. feet
- Area of a parallelogram is 18 sq. feet
Take a few minutes to check your work with a friend. Did you divide the area in half? Did you label it as square inches or feet?
II. Find the Area of Triangles Given Base and Height
Now that you know the formula for finding the area of a triangle, you can apply it to finding the area of specific triangles.
Let’s apply this formula to an example.
Example
Let’s look at this triangle to figure out the parts for the formula.
The base is the bottom part of the triangle. In this example, the base is 8 inches.
Because this is a right triangle, the height is the left side of the triangle.
Next, we apply the formula.
\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(6)(8)\\ A & = \frac{1}{2}(48)\\ A & = 24 \ sq. \ inches\end{align*}
The area of this triangle is 24 square inches.
What about when we have a triangle that is not a right triangle?
Let’s look at an example.
Example
The base of this triangle is the bottom part of the triangle. The base is 5 feet.
The height of the triangle is the distance from the base to the top vertex. In this triangle, the height is 4 feet.
Next, we can substitute these given values into the formula.
\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(4)(5)\\ A & = \frac{1}{2}(20)\\ A & = 10 \ square \ feet\end{align*}
The area of this triangle is 10 square feet.
Try a few of these on your own. Find the area of the following triangles.
1. Base = 6 in
Height = 5 in
2. Base = 9 ft.
Height = 4 ft.
3. Base = 9 m
Height = 7 m
Take a few minutes to check your work. Did you get a decimal for the third answer?
III. Find Unknown Dimensions of Triangles Given Area and Another Dimension
What happens if you have been given the area and one other dimension?
When this happens, you have to be a detective and figure out the missing dimension. For example, if you have been given the area and the base, then you have to figure out the height. If you have been given the area and the height then you have to figure out the base.
You have to be a detective!
Now let’s use the formula for finding the area of a triangle to solve this problem. Remember, you will need your detective skills.
Example
A triangle has an area of 36 square inches and a height of 6 inches. What is the measure of the base?
To figure this out, we start by looking at the formula for finding the area of a triangle.
\begin{align*}A = \frac{1}{2}bh\end{align*}
Next, we fill in the given information.
\begin{align*}36 = \frac{1}{2}(6)b\end{align*}
To solve this problem, we need to first multiply one-half by six.
\begin{align*}36 = 3b\end{align*}
Next, we need to solve for the base. We do this by thinking about what number times three is equal to thirty-six. You could also think of it as dividing thirty-six by 3.
Our answer is 12. The base of this triangle is 12 inches.
Try a few of these on your own.
- A triangle has an area of 42 sq. ft. If the base is 12 feet, what is the measure of the height?
- A triangle has an area of 16 sq. cm. If the height of the triangle is 4 cm, what is the measure of the base?
Take a few minutes to check your answer with a friend. Is your work accurate and labeled correctly?
IV. Find Areas of Combined Figures Involving Triangles
Sometimes, you will have a figure that uses both triangles and another figure like a parallelogram. We can find the area of the individual parts, add them together and find the total area of the entire figure.
How does this work?
Surveyors use this strategy all the time. When they are figuring out the area of a plot of land, they may divide it up into different figures. Then they find the area of each individual section or figure and add up their answers. The sum is the total area of the figure.
Let’s look at an example.
Example
Here are a series of triangles that line the center median of a city street. The triangles are overlapping.
Let’s say that that the first triangle has a base of 6 feet and a height of 5 feet. What is the area of the triangle?
\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(6)(5)\\ A & = \frac{1}{2}(30)\\ A & = 15 \ square \ feet\end{align*}
Let’s say that there are eight triangles in this strip of median. We can take the area of one of the triangles and multiply it by 8.
15(8) = 120 square feet
This is the area of the median. We were able to use the area of each triangle to find the total area of the median.
Let’s look at another example.
Example
Here we have a rectangle and a triangle together. If we want to find the total area of the figure, we need to find the area of the rectangle and the area of the triangle and then find the total sum.
Area of the rectangle
\begin{align*}A & = lw\\ A & = 3(5)\\ A & = 15 \ sq. \ meters\end{align*}
Area of the triangle
\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(3)(1.5)\\ A & = 2.25 \ sq. \ meters\end{align*}
Now we add the two areas together.
Area of rectangle + area of triangle = total area of figure
15 sq. meters + 2.25 sq. meters = 17.25 sq. meters
The total area is 17.25 square meters.
Real Life Example Completed
The Triangle in the Quilt Block
Here is the original problem once again. Use what you learned about finding the area of a triangle to help Jillian with her quilt block. Reread the problem and underline any important information.
Now that Jillian has figured out the parallelograms, she is on to the triangles. There are right triangles in the quilt block that she is working on with her grandmother. Jillian needs to figure out the area of each triangle so that she can calculate the size of the triangle and the amount of fabric that she will need to make them all.
Here is the quilt block once again.
There are 16 right triangles in the quilt block. The good news for Jillian is that they all have the same dimensions. Here are the dimensions of the triangle.
Each side of the triangle is three inches. Given these measurements, what is the area of one of the triangles? What is the area for 16 triangles?
Jillian is puzzled. She just figured out how to find the area of a parallelogram and now she is on to triangles. Jillian knows that the triangle and the square are related, she just isn’t sure how.
First, Jillian needs to find the area for one of the triangles. To do this, she can use the formula for finding the area of a triangle.
\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(3)(3)\\ A & = 4.5 \ square \ inches\end{align*}
Wow! Jillian is a bit nervous about every triangle having an area of 4.5 inches. That might be hard to manage. However, Jillian has another idea. If each triangle has an area of 4.5 inches, then each square has an area of 9 inches.
Think about this-a nine inch square will be easier to cut in half and get two equal triangles.
If Jillian needs 16 triangles, then she can cut 8 nine inch squares, then she will have enough because each square can be cut into two triangles.
How much material will she need?
If each square is 3 \begin{align*}\times\end{align*} 3 or has an area of 9” and Jillian needs eight squares, then the area of material is 9 \begin{align*}\times\end{align*} 8.
Jillian will need 72 square inches of material for the triangles.
Vocabulary
Here are the vocabulary words that can be found in this lesson.
- Triangle
- a polygon with three sides.
- Parallelogram
- a four sided figure with opposite sides congruent and parallel.
- Rectangle
- a parallelogram with opposite sides congruent and parallel and with four right angles.
- Square
- a parallelogram with four congruent, parallel sides and four congruent right angles.
Technology Integration
James Sousa, Example of the Area of a Triangle
Other Videos:
- http://www.mathplayground.com/howto_area_triangle.html – This video is a tutorial on how to find the area of a triangle. It also shows the relationship between triangles and parallelograms.
Time to Practice
Directions: Find the area of each triangle given the base and height.
1. Base = 4 in
Height = 5 in
2. Base = 6 in
Height = 4 in
3. Base = 8 ft
Height = 7 ft
4. Base = 10 meters
Height = 8 meters
5. Base = 10 meters
Height = 5 meters
6. Base = 12 feet
Height = 14 feet
7. Base = 11 feet
Height = 6 feet
8. Base = 14 inches
Height = 8 inches
9. Base = 22 feet
Height = 19 feet
10. Base = 30 cm
Height = 28 cm
11. Base = 18 inches
Height = 16 inches
12. Base = 13 meters
Height = 10 meters
Directions: Find the missing dimension of the triangle given an area and a base or height.
13. Area = 15 sq. in, Base = 10 in, what is the height?
14. Area = 40 sq. in, Base = 20 in, what is the height?
15. Area = 24 sq. ft, Base = 8 ft, what is the height?
16. Area = 16 sq. m, Base = 8 m, what is the height?
17. Area = 25 sq. in, height = 5 in, what is the base?
18. Area = 36 sq. ft, Height = 6 ft, what is the base?
19. Area = 54 sq. cm, Height = 9 cm, what is the base?
20. Area = 80 sq. meters, Base = 16 meters, what is the height?
Directions: Find the area of each triangle.
21.
Base = 10 in, Height = 4 in
22.
Base = 16 meters, Height = 10 meters
23.
Base = 8 in, Height = 6.5 in
24.
Base = 10 cm, Height = 7 cm
25.
Base = 5 ft, Height = 8.5 feet
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