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Algebra Foundations

Students derive the formulas for the area and perimeter of squares and rectangles, and practice using these formulas.

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After completing this tutorial, you will be able to complete the following:

- Derive the formula for the perimeter of rectangles.
- Calculate the perimeter of rectangles.
- Derive the formula for the area of rectangles.
- Calculate the areas of rectangles.
- Derive the formula for the perimeter of squares.
- Calculate the perimeter of squares.
- Derive the formula for the area of squares.
- Calculate the areas of squares.

Recall that the perimeter of a region is the length of the path that surrounds a region (or the sum of the lengths of the sides of the region), and that the area of this region is the number of square units covered by the region.

Consider a rectangle of length 5 units and width 3 units. We can divide this rectangle into unit squares:

To find the sum of the lengths of all of the sides, we would have 3 units + 5 units + 3 units + 5 units = 16 units. Furthermore, by counting all of the unit squares in the rectangle of length 5 units and width 3 units, we find that the area of the rectangle is 15 square units. Clearly, the rectangle contains 3 rows of 5 squares each. The sum of all of these squares is 5 + 5 + 5 or 3(5).

In general, the perimeter of a rectangle, in this case ABCD, is the sum of all of the side lengths: P=|AB|+|BC|+|CD|+|DA| Since this is a rectangle, the lengths of the opposite sides are equal, so |AB|=|CD| and |BC|=|DA|. If we let l denote the length of and w denote the length of CD, the perimeter is

In general, to find the area of rectangle ABCD, we would first note that if we let l denote the length of and w denote the length of CD, and if the lengths of the sides are integers, we can fill the rectangle with w columns, where each column has l unit squares. In this case, the area of rectangle ABCD is A = w . l

If the lengths of the sides are not integers, the unit squares would not fit properly inside the rectangle. If this is the case, the area is still defined to be the product of w and l.

Deriving the formulae for the perimeter and area of a square is straightforward. First, we note that the sides of a square have equal length. We can denote the length of a side as a. Then, since a square is a rectangle, we see that the perimeter of the square is equal to Similarly, both the length and the width of the square is equal to a, so the area of the square is

Notice that in order to discuss the area and perimeter of squares and rectangles, we needed the side lengths. At times, these will not be provided, but we can use other tools, such as the Pythagorean theorem or the distance formula to find the side lengths. For example, consider a rectangle given on the coordinate plane with vertex coordinates A(2,4), B(4,2), C(8,6), and D(6,8)

We can plot the points in the coordinate plane, and then draw the line segments connecting them, as shown above.

We can then calculate the distance between two consecutive points that mark the endpoints of the segment that represents the length of the rectangle. Next, we can calculate the distance between two consecutive points that mark the endpoints of the segment that represents the width of the rectangle, as follows:

Since this is a rectangle, |AB|=|DC| and |AD|=|BC|.

The perimeter of rectangle ABCD is units. The area of the rectangle is

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should know the definitions of area, perimeter, square, and rectangle; be able to explain the relationship between the area and the perimeter of regions given on a plane and between the area and side lengths of regions on a plane; know the properties of squares and rectangles; be able to calculate the distance between two points on the coordinate plane; and be familiar with the Pythagorean theorem. |

Course | Algebra Foundations |

Type of Tutorial | Visual Proof |

Key Vocabulary | area of rectangle, area of square, formula for area of rectangle |