# Math Models and Geometry

## Use Properties of Rectangles, Triangles, and Trapezoids

In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids.

### Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.
When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side ().
 shows a rectangular rug that is  feet long by  feet wide. Each square is  foot wide by  foot long, or  square foot. The rug is made of  squares. The area of the rug is square feet.
When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side ().
Suppose the cube in  measures  inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find  little cubes, with each one measuring one inch on all sides. So each little cube has a volume of   cubic inch, and the volume of the big cube is  cubic inches.
Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.
Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.
Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.
 shows a square tile that is  inch on each side. If an ant walked around the edge of the tile, it would walk  inches. This distance is the perimeter of the tile.
Since the tile is a square that is  inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

### Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length,  and the adjacent side as the width,  See .
The perimeter,  of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk  units, or two lengths and two widths. The perimeter then is
What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was  feet long by  feet wide, and its area was  square feet. See . Since  we see that the area,  is the length,  times the width,  so the area of a rectangle is 
For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here.
In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

### Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in , we’ve labeled the length  and the width  so it’s area is 
We can divide this rectangle into two congruent triangles (). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or  This example helps us see why the formula for the area of a triangle is 
The formula for the area of a triangle is  where  is the base and  is the height.
To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a  angle with the base.   shows three triangles with the base and height of each marked.

### Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle.  shows both types of triangles.

### Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base  and the length of the bigger base  The height,  of a trapezoid is the distance between the two bases as shown in .
The formula for the area of a trapezoid is:
Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See .
The height of the trapezoid is also the height of each of the two triangles. See .
The formula for the area of a trapezoid is
If we distribute, we get,

### Key Concepts

1. Properties of Rectangles
     
2. Triangle Properties
     


### Practice Makes Perfect

Understand Linear, Square, and Cubic Measure 
In the following exercises, determine whether you would measure each item using linear, square, or cubic units.
In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is  cm.
Use the Properties of Rectangles 
In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle.
In the following exercises, solve.
Use the Properties of Triangles 
In the following exercises, solve using the properties of triangles.
Use the Properties of Trapezoids 
In the following exercises, solve using the properties of trapezoids.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.


ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
