9.5: Areas of Triangles
Introduction
Figuring out Home
Miguel loves watching the games when the Wildcats play. He loves when they win, but he is such a fan of baseball that he doesn’t even mind when they lose. Miguel cheers when the players run towards home plate.
He watches them slide in and run across it.
“Home plate is such an interesting shape,” he thinks to himself one day.
He decides to do a little research on the design of home plate. Here is what he finds out.
Home plate is a pentagon which can be divided up. The base of home plate is 17 inches wide. The sides of home plate are each 8.5 inches. The distance from the tip of home plate to the base is also 17 inches.
Miguel thinks that this is very interesting information. He wonders if he can figure out the area of the figure. It is made up of a rectangle and a triangle.
As Miguel works on this, you work on it too. You know how to find the area of a rectangle already, but what about a triangle?
This lesson will teach you how to find the area of a triangle. When finished, you can work on figuring out home.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills:
- Recognize the formula for area of a triangle.
- Find areas of triangles given base and height.
- Find unknown dimensions of triangles given area and an unknown dimension.
- Find areas of combined figures involving triangles.
Teaching Time
I. Recognize the Formula for Area of a Triangle
In the last lesson we looked at how to find the area of a parallelogram. Here is that formula once again.
\begin{align*}A=bh\end{align*}
This lesson focuses on the area of a triangle. Remember that a triangle is a three sided figure made up of line segments with three sides and three angles. Area is the amount of space inside a two-dimensional figure. We can find the area of a triangle just like we found the area of a parallelogram. The really interesting thing is that the area of a triangle is related to the area of a parallelogram. Take a look at this figure.
Notice that the parallelogram has been divided into two triangles.
We know that the formula for finding the area of a parallelogram requires us to multiply the base times the height. Well, if a triangle is one-half of a parallelogram, can you figure out the formula for finding the area of a triangle?
Here it is.
\begin{align*}A= \frac{1}{2} bh\end{align*}
It certainly does. With this formula, the area of a triangle will be snap to figure out!
Take a minute to write this formula down in your notebook.
II. Find Areas of Triangles Given a Base and a Height
Now that you can identify formula for finding the area of a triangle, let’s look at using it in problem solving.
Example
Find the area of the triangle below.
We can see that the base is 11 centimeters and the height is 16 centimeters. We simply put these numbers into the appropriate places in the formula.
\begin{align*}A & = \frac{1}{2} bh\\ A & = \frac{1}{2} 11(16)\\ A & = \frac{1}{2} (176) \\ A & = 88 \ cm^2\end{align*}
Remember that we always measure area in square units because we are combining two dimensions. The area of this triangle is 88 square centimeters. Let’s try another.
Example
What is the area of the triangle below?
Notice that the height is shown by the dashed line. It is perpendicular to the base. We put it and the base into the formula and solve.
\begin{align*}A & = \frac{1}{2} bh \\ A & = \frac{1}{2} 5(17) \\ A & = \frac{1}{2} (85) \\ A & = 42.5 \ cm^2\end{align*}
The area of this triangle is 42.5 sq. cm. Now it’s time for you to try a few on your own.
9I. Lesson Exercises
Find the area of each triangle.
- Base = 12 in, height = 6 inches
- Base = 9 inches, height = 4 inches
- Base = 11 inches, height = 7 inches
Take a few minutes to check your work. Did you label each measurement using square units?
III. Find Unknown Dimensions of Triangles Given Area and Another Dimension
Sometimes a problem will give us the area and ask us to find one of the dimensions of the triangle—either its base or its height. We simply put the information we know in for the appropriate variable in the formula and solve for the unknown measurement. Let’s try an example.
Example
A triangle has an area of \begin{align*}44 \ m^2\end{align*}. The base of the triangle is 8 m. What is its height?
In this problem, we know the area and the base of the triangle. We put these numbers into the formula and solve for the height, \begin{align*}h\end{align*}.
\begin{align*}A & = \frac{1}{2} bh \\ 44 & = \frac{1}{2} 8h \\ 44 \div \frac{1}{2} & = 8h \\ 44(2) & = 8h \\ 88 & = 8h \\ 11 \ m & = h \end{align*}
Remember, when you divide by a fraction, you need to multiply by its reciprocal. To divide by one-half then, we multiply by 2. Keep this in mind when you use the area formula.
By solving for \begin{align*}h\end{align*}, we have found that the height of the triangle is 11 meters. Let’s check our calculation to be sure. We can check by putting the base and height into the formula and solving for area.
\begin{align*}A & = \frac{1}{2} bh \\ A & = \frac{1}{2} 8(11) \\ A & = \frac{1}{2} (88) \\ A & = 44 \ m^2\end{align*}
We know from the problem that the area is \begin{align*}44 \ m^2\end{align*}, so our calculation is correct.
Now try a few of these on your own.
9J. Lesson Exercises
Given the area and one other dimension, find the missing dimension.
- Base = 4 inches, Area = 6 sq. inches, what is the height?
- Base = 5 feet, Area = 7.5 sq. feet, what is the height?
- Base = 7 meters, Area = 17.5 sq. meters, what is the height?
Take a few minutes to go over your responses with a partner. Correct any errors and then move on to the next section.
IV. Find Areas of Combined Figures Involving Triangles
Now that you know how to find the area of a triangle, you can use that information to figure out the area of figures that are made up of more than one shape. Remember that we could divide a parallelogram into two triangles? If we know the area of the triangles, we can add their areas together to find the area of the parallelogram. We can do this for all kinds of figures. If we can divide the figure into triangles, we can find the area of each triangle and add the areas together.
Also, we have seen that we can apply the formula for finding the area of triangles to different kinds of situations. Sometimes we need to solve for the area, but other times we may need to find the height or the base. We can use information given about a larger figure whenever that information corresponds to the height, base, or area of one of the triangles contained within it. Let’s try an example to get a better idea of how this works.
Example
Find the area of the figure below.
We need to find the area of the whole figure. To do so, we can divide it into shapes whose area formulas we know. We can see that one triangle has been drawn already. Can we draw another to divide the figure again?
Now we have divided the figure into a rectangle and two triangles. If we can find the area of each of these, we can add them together to find the area of the whole figure.
Let’s give it a try.
First, let’s calculate the area of the rectangle in the center. We know the formula for the area of rectangles is \begin{align*}A = lw\end{align*}. Do we know the length and width of the rectangle? The width is represented by the dashed line. It is 5 inches. The length of the rectangle is the same as the bottom edge of the figure: 9 inches. Let’s put these numbers into the formula and solve for area.
\begin{align*}A & = lw\\ A & = 9(5)\\ A & = 45 \ in.^2\end{align*}
Now let’s find the area of one of the triangles. We know that the height is 5 inches. What is the base? Look carefully at the figure. We know that its top edge is 22 inches. We also know that the length of the rectangle in the middle is 9 inches. We need to subtract that so we don’t include it as part of the bases of the triangles.
That means there are \begin{align*}22 - 9 = 13\end{align*} inches left of the bottom edge. If we divide this equally, we find that each triangle has a base of 6.5 inches.
Now we have the height and base of each triangle (they have the same height and base), so we can calculate the area.
\begin{align*}A & = \frac{1}{2} bh\\ A & = \frac{1}{2} 6.5(5)\\ A & = \frac{1}{2} (32.5)\\ A & = 16.25 \ in.^2\end{align*}
Great! Now we have the area of each shape within the figure. All we have to do is add these together to find the area of the whole figure.
\begin{align*}& \text{triangle} \ 1 \qquad \qquad \quad \text{rectangle} \ \qquad \qquad \ \text{triangle}\ 2 \qquad \quad \qquad \text{whole figure}\\ & 16.25 \ in.^2 \qquad + \qquad 45 \ in.^2 \qquad + \qquad 16.25 \ in.^2 \qquad = \qquad 77.5 \ in.^2\end{align*}
The area of the whole figure is 77.5 square inches.
Example
What is the area of the figure below?
This time we need to divide the figure into smaller shapes ourselves. Can you draw any lines to make triangles inside the figure?
Now we have two triangles and a square. Let’s see if we have all the measurements we need to use the area formulas for these two shapes.
We know that all of the sides of a square are congruent, so we can fill in the measurements for the other two sides, which must also be 2 inches. This gives us the base of each triangle. Do we know the height? Each is 5 inches. We have all the information we need to solve for the areas of the triangles and the square. Let’s calculate the area of each smaller shape.
\begin{align*}& \text{Square} && \text{Triangle} \ 1 && \text{Triangle} \ 2\\ & A = s^2 && A = \frac{1}{2} bh && A = \frac{1}{2} bh\\ & A = 2^2 && A = \frac{1}{2} 2(5) && A = \frac{1}{2} 2(5)\\ & A = 4 \ in.^2 && A = \frac{1}{2} (10) && A = \frac{1}{2} (10)\\ & && A = 5 \ in.^2 && A = 5 \ in.^2\end{align*}
Great! We have found the area of each smaller shape. Let’s add them together to find the area of the whole figure.
\begin{align*}& \text{square} \qquad \qquad \quad \text{triangle} \ 1 \qquad \qquad \ \text{triangle} \ 2 \qquad \quad \qquad \text{whole figure}\\ & 4 \ in.^2 \qquad + \qquad \quad 5 \ in.^2 \qquad \ + \qquad \quad 5 \ in.^2 \qquad \ = \qquad \quad 14 \ in.^2\end{align*}
The area of the whole figure is 14 square inches.
Now let’s use what we have learned to help Miguel with his dilemma.
Real–Life Example Completed
Figuring Out Home
Here is the original problem once again. Reread it and underline any important information.
Miguel loves watching the games when the Wildcats play. He loves when they win, but he is such a fan of baseball that he doesn’t even mind when they lose. Miguel cheers when the players run towards home plate.
He watches them slide in and run across it.
“Home plate is such an interesting shape,” he thinks to himself one day.
He decides to do a little research on the design of home plate. Here is what he finds out.
Home plate is a pentagon which can be divided up. The base of home plate is 17 inches wide. The sides of home plate are each 8.5 inches. The distance from the tip of home plate to the base is 17 inches.
Miguel thinks that this is very interesting information. He wonders if he can figure out the area of the figure. It is made up of a rectangle and a triangle.
Here is a drawing of home plate with its dimensions.
Now, we can begin by figuring out the area of the rectangle.
The length of the rectangle is 17 inches, the width is 8.5
\begin{align*}A& = 17(8.5)\\ A&=144.5\end{align*}
The area of the rectangle is 144.5 square inches.
Now let’s look at the triangle. We know that the base of the triangle is 17”. The height of the whole plate is 17 inches, but that includes the rectangle width too. We need to subtract that from the total.
17 – 8.5 = 8.5 inches
Let’s use our formula to find the area of the triangle.
\begin{align*}A & = \frac{1}{2} bh \\ A & = \frac{1}{2}(17)(8.5) \\ A & = \frac{1}{2}(144.5) \\ A & = 72.25\end{align*}
Now we add up the two areas.
\begin{align*}144.5 + 72.25 = 216.75 \ square \ inches\end{align*}
This is the approximate measure of home plate.
You can read more about home plate at http://wiki.answers.com/Q/What_are_the_dimensions_of_home_plate
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Triangle
- a figure with three sides and three angles.
- Area
- the space enclosed inside a two-dimensional figure.
- Base
- the bottom part of the triangle
- Height
- the length of the triangle from the base to the vertex
Technology Integration
Khan Academy, Triangle Area Proofs
Time to Practice
Directions: Find the area of each triangle given the base and height.
1. Base = 9 in, height = 4 in
2. Base = 6 in, height = 3 in
3. Base = 7 in, height = 4 in
4. Base = 9 m, height = 7 m
5. Base = 12 ft, height = 10 feet
6. Base = 14 feet, height = 5 feet
7. Base = 14 feet, height = 13 feet
8. Base = 11 meters, height = 8 meters
9. Base = 13 feet, height = 8.5 feet
10. Base = 11.5 meters, height = 9 meters
Directions: Find the missing base or height given the area and one other dimension.
11. Area = 13.5 sq. meters, Base = 9 meters
12. Area = 21 sq. meters, Base = 7 meters
13. Area = 12 sq. meters, Base = 8 meters
14. Area = 33 sq. ft, Base = 11 feet
15. Area = 37.5 sq. ft. Base = 15 feet
16. Area = 60 sq. ft., height = 10 ft.
17. Area = 20.25 sq. in, height = 4.5 in
18. Area = 72 sq. in, height = 8 in
19. Area = 22.5 sq. feet, height = 5 feet
20. Area = 19.25 sq. in, height = 5.5 in
Directions: Solve each problem.
21. Julius drew a triangle that had a base of 15 inches and a height of 11 inches. What is the area of the triangle Julius drew?
22. A triangle has an area of 108 square centimeters. If its height is 9 cm, what is its base?
23. What is the height of a triangle whose base is 36 inches and area is 234 square inches?
24. Tina is painting a triangular sign. The height of the sign is 32 feet. The base is 27 feet. How many square feet will Tracy paint?