Have you ever tried to figure out the area of a triangle?

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 Concept to learn about finding the area of a triangle!**

### Guidance

Think back to the dilemma you just read. 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.

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.

Now put this information into practice. Take the area of the following parallelograms and find the area of one of the triangles inside the parallelogram.

#### Example A

Area of a rectangle is 12 sq. inches

**Solution: 6 sq. inches**

#### Example B

Area of a parallelogram is 24 sq. feet

**Solution: 12 sq. feet**

#### Example C

Area of a parallelogram is 18 sq. feet

**Solution: 9 sq. feet**

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.

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

- 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.

### Guided Practice

Here is one for you to try on your own.

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. Here are a few problems that use these skills.

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?

**Answer**

\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.**

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.**

### Video Review

James Sousa, Example of the Area of a Triangle

### Practice

Directions: Find the area of each triangle.

1.

Base = 10 in, Height = 4 in

2.

Base = 16 meters, Height = 10 meters

3.

Base = 8 in, Height = 6.5 in

4.

Base = 10 cm, Height = 7 cm

5.

Base = 5 ft, Height = 8.5 feet

Directions: Find the area of each triangle given the base and height.

6. Base = 4 in

Height = 5 in

7. Base = 6 in

Height = 4 in

8. Base = 8 ft

Height = 7 ft

9. Base = 10 meters

Height = 8 meters

10. Base = 10 meters

Height = 5 meters

11. Base = 12 feet

Height = 14 feet

12. Base = 11 feet

Height = 6 feet

13. Base = 14 inches

Height = 8 inches

14. Base = 22 feet

Height = 19 feet

15. Base = 30 cm

Height = 28 cm

16. Base = 18 inches

Height = 16 inches

17. Base = 13 meters

Height = 10 meters

18. Base = 18 meters

Height = 5.5 meters

19. Base = 12.5 feet

Height = 2.5 feet

20. Base = 13.75 inches

Height = 1.5 inches