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9.12: Triangle Area

Difficulty Level: At Grade Created by: CK-12
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Have you ever seen gardening designs in the median of a street?

Miguel saw gardening designs on Center Street. As he was walking, he noticed a series of triangles that line the median. 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?

This Concept will teach you how to find the area of any triangle given the base and the height.

Guidance

In the last Concept, we looked at how to find the area of a parallelogram. Here is that formula once again.

A=bh\begin{align*}A=bh\end{align*}

This Concept focuses on the area of a triangle. Remember that 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.

A=12bh\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.

Now that you can identify formula for finding the area of a triangle, let’s look at using it in problem solving.

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.

AAAA=12bh=1211(16)=12(176)=88 cm2\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.

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.

AAAA=12bh=125(17)=12(85)=42.5 cm2\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.

Find the area of each triangle.

Example A

Base = 12 in, height = 6 inches

Solution: 36 in2\begin{align*}36 \ in^2\end{align*}

Example B

Base = 9 inches, height = 4 inches

Solution: 18 in2\begin{align*}18 \ in^2\end{align*}

Example C

Base = 11 inches, height = 7 inches

Solution: 38.5 in2\begin{align*}38.5 \ in^2\end{align*}

Here is the original problem once again.

Miguel saw gardening designs on Center Street. As he was walking, he noticed a series of triangles that line the median. 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?

To figure this out, we can start with the formula for finding the area of a triangle.

AAAA=12bh=12(6)(5)=12(30)=15 square feet\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.

Vocabulary

Here are the vocabulary words in this Concept.

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

Guided Practice

Here is one for you to try on your own.

What is the area of a triangle with a base of 4.5 ft\begin{align*}4.5 \ ft\end{align*} and a height of 7 ft\begin{align*}7 \ ft\end{align*}?

To figure this out, we can use the formula for finding the area of a triangle.

A=12bh\begin{align*}A = \frac{1}{2}bh\end{align*}

Now we substitute in the given values.

A=12(4.5)(7)\begin{align*}A = \frac{1}{2}(4.5)(7)\end{align*}

A=12(31.5)\begin{align*}A = \frac{1}{2}(31.5)\end{align*}

A=15.75 ft2\begin{align*}A = 15.75 \ ft^2\end{align*}

This is our answer.

Video Review

Here is a video for review.

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

11. Base = 18 meters, height = 15 meters

12. Base = 21 feet, height = 15.5 feet

13. Base = 18 feet, height = 11 feet

14. Base = 20.5 meters, height = 15.5 meters

15. Base = 40 feet, height = 22 feet

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