Teaching Models

Perimeter, Area, and Circumference

Perimeter and Area
The perimeter of a polygon is the distance around it, or the sum of the lengths of its sides. For a simple figure like a rectangle, opposite sides are congruent. If l and w denote the length and width of a rectangle, then the perimeter P is given by 2l + 2w.

Area is measured in square units. A square unit is a square in which each side has a length of one unit. The area of an object is the number of square units it takes to cover the object without any overlap. The area, A, of a rectangle is given by the formula l × w (length times width) or b × h (base times height).

The area formula for a parallelogram is the same as for a rectangle, A = bh. To show that this is true, students can cut and reassemble the parallelogram to form a rectangle with the same base and height.

area formula for parallelogram is A=bh

By drawing a diagonal of a parallelogram, it can be shown that the area of a triangle is one-half the area of the parallelogram. It can further be shown that every right triangle can be fitted with a congruent copy of itself to form a rectangle. So the area of the right triangle is one half the area of the rectangle, or A = one-halfbh where b is the base and h is the height of the triangle. Any non-right triangle can be written as the sum or difference of right triangles. By adding or subtracting the areas of right triangles, it can be shown that the formula A = one-halfbh applies to all triangles.

Triangle

The perimeter of a circle is called the circumference. All circles are similar. Therefore, for any two circles, the ratio of the perimeters (circumferences) is the same as the ratio of the diameters. The ratio of the circumference to the diameter in each circle is the same. This constant ratio of circumference to diameter is denoted by the Greek letter π (pi). So C = πd or C = 2πr.

small circle

Small Circle

circumference = C1
diameter = d1
c equals d


and
large circle

Large Circle

circumference = C1
diameter = d1
c = d

Teaching Model 16.3: Area of a Parallelogram


Houghton Mifflin Math Grade 5