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Math Foundations

Students derive the formulas for the area and perimeter of a parallelogram, 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 a parallelogram.
- Calculate the perimeter of a parallelogram.
- Derive the formula for the area of a parallelogram by using base length and height.
- Calculate the area of a parallelogram by using base length and height.

A parallelogram is a quadrilateral with two pairs of parallel sides. This apparently simple definition actually imposes several other useful properties that will aid in finding formulas for the area and perimeter of parallelograms. Recall that perimeter is the length of the path surrounding a region and that area is the number of square units covered in a region. The perimeter of a polygon is therefore the sum of its side lengths.

A set of parallel sides are called the bases of the parallelogram. We can show that bases of a parallelogram are congruent using the fact that the other sides of the parallelogram form transversals for the parallel sides. This means that opposite sides of a parallelogram have the same length. So, if the side lengths of a parallelogram are a and b, then the perimeter of the parallelogram is P = a + b + a + b = 2a + 2b = 2(a + b).

Recall that a rectangle is a quadrilateral with four right angles and that the area of a rectangle with side lengths l and w is A = w . l . We will derive a formula for the area of a parallelogram from this formula for the area of a rectangle. First, we need another definition. The height of a parallelogram is the shortest distance between two bases. This means that each height corresponds to specific bases. Now consider the following figure:

Notice that by drawing the height of the parallelogram twice in the figure, we divide the parallelogram into two congruent right triangles and a rectangle. We can move one of the right triangles so that its hypotenuse coincides with the hypotenuse of the other triangle, thus creating another rectangle, one with length a and width h:

This means that the area of the rectangle is A = a . h. Since the area of this rectangle is the same as the area of the parallelogram, our formula for the area of a parallelogram is A = a . h .

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should know the formula for the area of a rectangle, know the perimeter of a polygon, and know the definition of a parallelogram and its properties. |

Course | Math Foundations |

Type of Tutorial | Visual Proof |

Key Vocabulary | area, base, height |