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# Triangle Area

## Half the product of base and height

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Triangle Area
License: CC BY-NC 3.0

Three kids form the Tri-County Dance Team. They are painting a triangular shaped logo on their T-shirts and want the base to be 6 inches long and the height 4 inches. How many square inches of fabric paint do they need for three shirts?

In this concept, you will learn how to find the area of any triangle when the base and height are given.

### Finding the Area of a Triangle

A triangle is a three sided figure made up of three sides and three angles. The area of a triangle is related to the area of a parallelogram. The formula for finding the area of a parallelogram, \begin{align*}A = bh\end{align*}.

License: CC BY-NC 3.0

Notice that the parallelogram above has been divided into two triangles. Each triangle equals one half of the parallelogram. The formula for the area of a triangle is one half the area of a parallelogram.

The formula can be written: \begin{align*} A=\frac{1}{2} \ bh \ \text{or} \ A=\frac{bh}{2}\end{align*}

Let's look at an example. Find the area of the triangle.

License: CC BY-NC 3.0

First, write the formula.

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

Next, insert the given values.

\begin{align*}A=\frac{1}{2}(11)(16)\end{align*}

Then express all terms as fractions and multiply.

\begin{align*}\begin{array}{rcl} A &=& \left( \frac{1}{2} \right)\left( \frac{11}{1} \right) \left( \frac{16}{1} \right) \\ A &=& 88 \ sq \ cm \end{array}\end{align*}

The answer is 88 sq. cm.

### Examples

#### Example 1

Earlier, you were given a problem about the Tri-County Dance Team and their triangular logo.

They need to buy enough fabric paint for three triangles that measure 6 inches at the base and are 4 inches high. The paint that they buy should cover how many square feet?

First, write the formula.

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

Next, fill in the values that you are given.

\begin{align*}A=\frac{(6)(4)}{2}\end{align*}

Then, do the arithmetic.

\begin{align*}A=12 \ sq \ in\end{align*}

Remember, there are three shirts, so multiply by 3.

\begin{align*}12 \times 3=36\end{align*}

The answer is 36 sq. in. The team needs enough paint to cover 36 square inches.

#### Example 2

What is the area of the triangle below?

License: CC BY-NC 3.0

First, write the formula.

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

Next, insert the given values.

\begin{align*}A=\frac{1}{2}(17)(5)\end{align*}

Then, express all terms as fractions and multiply.

\begin{align*}\begin{array}{rcl} A &=& \left( \frac{1}{2} \right)\left( \frac{17}{1} \right) \left( \frac{5}{1} \right) \\ A &=& 42.5 \ sq \ cm \end{array}\end{align*}

The answer is 42.5 sq. cm.

#### Example 3

A triangle has a base of \begin{align*}9^{{\prime}{\prime}}\end{align*} and a height of \begin{align*}4^{{\prime}{\prime}}\end{align*}. What is its area?

First, write the formula.

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

Next, fill in the values that you are given.

\begin{align*}A=\frac{(9)(4)}{2}\end{align*}

Then, do the arithmetic.

\begin{align*}A=18 \ sq \ in\end{align*}

The answer is 18 sq. in.

#### Example 4

Find the area of a triangle that has a base = 11 inches and a height = 7 inches.

First, write the formula.

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

Next, fill in the values that you are given.

\begin{align*}A=\frac{(11)(7)}{2}\end{align*}

Then, do the arithmetic.

\begin{align*}A=38.5 \ sq \ in\end{align*}

The answer is 38.5 sq. in.

### Review

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

To see the Review answers, open this PDF file and look for section 9.12.

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### Vocabulary Language: English

Area

Area is the space within the perimeter of a two-dimensional figure.

Base

The side of a triangle parallel with the bottom edge of the paper or screen is commonly called the base. The base of an isosceles triangle is the non-congruent side in the triangle.

Height

The height of a triangle is the perpendicular distance from the base of the triangle to the opposite vertex of the triangle.

Triangle

A triangle is a polygon with three sides and three angles.