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# Area and Perimeter of Triangles

## Area is half the base times the height while the perimeter is the sum of the sides.

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Practice Area and Perimeter of Triangles
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Find the Dimensions and Area of Triangles

Have you ever seen flowers in a median on a highway? Take a look at this dilemma.

Jessie saw this median as she rode to school. The height of each triangle in the median is 7 feet and the base is 5. Jessie wondered about the area of each triangle. If there are seven triangles in a row, what is the total area of all seven triangles?

To figure this out, you will need to understand area and triangles. Pay attention and you will be able to solve this two-part problem at the end of the Concept.

### Guidance

Area is the amount of two-dimensional space a figure covers.

Do you know how to find the area of triangles using formulas and problem solving? Let’s see how this works by first understanding the formula for the area of triangles.

How do we find the area of a triangle?

Area, as we have said, is the amount of space a figure covers. To find area, we multiply the dimensions, or sides, of the figure. In a triangle, those dimensions are its height, \begin{align*}h\end{align*}, and its base, \begin{align*}b\end{align*}. The area formula for triangles is

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

Write this formula for area of a triangle down in your notebook. Be sure to write “Triangle Area” with it.

The base is the area at the bottom of the triangle opposite the vertex or top point.

When finding the area of triangles, remember that the height of a triangle is always perpendicular to the base.

The height is not necessarily a side of the triangle; this happens only in right triangles, because the two sides joined by a right angle are perpendicular.

You can see in the right triangle that the left side is also the height of the triangle. It is perpendicular to the base. The equilateral triangle has a dotted line to show you the measurement for the height of the triangle.

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.

Remember that we always measure area in square units because we are combining two dimensions.

The area of this triangle is 88 square centimeters.

Finding the area of a triangle is just that simple. We can also use this same formula to figure out a missing dimension of the triangle. This is possible if we are given the area and one dimension to start with. Then we can use the formula, substitute the values and solve for the missing dimension.

Let’s take a look.

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*}.

Remember, when you divide both sides by a fraction, you need to multiply by its reciprocal. To divide by \begin{align*}\frac{1}{2}\end{align*}, 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.

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

#### Example A

Base = 6 inches, Height = 4 inches

Solution:  \begin{align*}12\end{align*} square inches

#### Example B

Base = 3.5 feet, Height = 4 feet

Solution:  \begin{align*}7\end{align*} sq. feet

#### Example C

Base = 8 mm, Height = 9 mm

Solution:  \begin{align*}36\end{align*}sq. mm

Now let's go back to the dilemma from the beginning of the Concept.

To figure this out, first, we have to find the area of one triangle.

Next, we take the 17.5 and multiply it by 7 because there are 7 triangles in the median.

17.5 x 7 = 122.5 square feet.

Here are the two answers to the dilemma.

### Vocabulary

Polygon
a simple closed figure made up of at least three line segments.
Triangle
a three-sided polygon.
Area
the two-dimensional space that a figure occupies.
Height of the triangle
the line perpendicular to the base.
Base of a triangle
the line perpendicular to the height.

### Guided Practice

Here is one for you to try on your own.

Solution

You 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.

### Practice

Directions: Find the area of each triangle described below.

1. \begin{align*}b=10 \ inches, \ h=5 \ inches\end{align*}
2. \begin{align*}b=7 \ inches, \ h=5.5 \ inches\end{align*}
3. \begin{align*}b=8 \ feet, \ height=6 \ feet\end{align*}
4. \begin{align*} b=9 \ feet, \ height=7.5 \ feet\end{align*}
5. \begin{align*} b=12 \ meters, \ h=9 \ meters\end{align*}
6. \begin{align*} b=15 \ feet, \ h=12 \ feet\end{align*}
7. \begin{align*} b=12.5 \ feet, \ h=3.5 \ feet\end{align*}
8. \begin{align*} b=15.25 \ feet, \ h=8.5 \ feet\end{align*}
9. \begin{align*} b=25.75 \ feet, \ h=13.5 \ feet\end{align*}

Directions: Find the missing dimension for each triangle given the area and one other dimension.

1. \begin{align*}A=4.5 \ sq.in, \ b=4.5 \ in, \ h= ?\end{align*}
2. \begin{align*}A=21 \ sq.ft, \ b=7 \ ft, \ h= ?\end{align*}
3. \begin{align*}A=60 \ sq.in, \ h=10 \ in, \ b= ?\end{align*}
4. \begin{align*}A=97.5 \ sq.ft, \ h=13 \ ft, \ b= ?\end{align*}
5. \begin{align*}A=187 \ sq.ft, \ b=22 \ ft, \ h= ?\end{align*}
6. \begin{align*}A=405 \ sq.ft, \ b=30 \ ft, \ h= ?\end{align*}

### Vocabulary Language: English

Area

Area

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

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

Height

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

Perimeter

Perimeter is the distance around a two-dimensional figure.
Perpendicular

Perpendicular

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.
Polygon

Polygon

A polygon is a simple closed figure with at least three straight sides.
Right Angle

Right Angle

A right angle is an angle equal to 90 degrees.
Right Triangle

Right Triangle

A right triangle is a triangle with one 90 degree angle.
Triangle

Triangle

A triangle is a polygon with three sides and three angles.
Area of a Parallelogram

Area of a Parallelogram

The area of a parallelogram is equal to the base multiplied by the height: A = bh. The height of a parallelogram is always perpendicular to the base (the sides are not the height).
Area of a Triangle

Area of a Triangle

The area of a triangle is half the area of a parallelogram. Hence the formula: $A = \frac{1}{2}bh \text{ or } A = \frac{bh}{2}$.