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By the end of this section, you will be able to:
  • Understand linear, square, and cubic measure
  • Use properties of rectangles
  • Use properties of triangles
  • Use properties of trapezoids

Before you get started, take this readiness quiz.

  1. The length of a rectangle is 3 less than the width. Let w represent the width. Write an expression for the length of the rectangle.
    If you missed this problem, review Solve Equations with the Subtraction and Addition Properties of Equality .
  2. Simplify: 1 2 ( 6 h ) .
    If you missed this problem, review Commutative and Associative Properties .
  3. Simplify: 5 2 ( 10.3 7.9 ) .
    If you missed this problem, review Decimals and Fractions .

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 ( [link] ). 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.

A picture of a portion of a tape measure is shown. The top shows the numbers 1 through 5. The portion from the beginning to the 1 has a red circle and an arrow to a picture from 0 to 1 inch, with 1 sixteenth, 1 eighth, 3 eighths, 1 half, and 3 fourths labeled. Above this, it is labeled “Standard Measures.” The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2. The region from the edge to about 3 and a half has a red circle with an arrow pointing to a picture from 0 to 3.5. It is labeled 0, 1 cm, 1.7 cm, 2.3 cm and 3.5 cm. Above this, it is labeled “Metric (S).”
This tape measure measures inches along the top and centimeters along the bottom.

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 ( [link] ).

Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimeter. The larger one has sides labeled 1 inch and is 1 square inch.
Square measures have sides that are each 1 unit in length.

[link] shows a rectangular rug that is 2 feet long by 3 feet wide. Each square is 1 foot wide by 1 foot long, or 1 square foot. The rug is made of 6 squares. The area of the rug is 6 square feet.

A rectangle is shown. It has 3 squares across and 2 squares down, a total of 6 squares.
The rug contains six squares of 1 square foot each, so the total area of the rug is 6 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 ( [link] ).

Two cubes are shown. The smaller one has sides labeled 1 cm and is labeled as 1 cubic centimeter. The larger one has sides labeled 1 inch and is labeled as 1 cubic inch.
Cubic measures have sides that are 1 unit in length.

Suppose the cube in [link] measures 3 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 27 little cubes, with each one measuring one inch on all sides. So each little cube has a volume of 1 cubic inch, and the volume of the big cube is 27 cubic inches.

A cube is shown, comprised of smaller cubes. Each side of the cube has 3 smaller cubes across, for a total of 27 smaller cubes.
A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.
Doing the Manipulative Mathematics activity Visualizing Area and Perimeter will help you develop a better understanding of the difference between the area of a figure and its perimeter.
Practice Key Terms 6

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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