## Perimeter, Circumference, and Area

**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. Let l and w denote the length and width of a rectangle. The perimeter, P, is given by P = 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). This can be understood from the following rectangle of base 3 and height 2. There are two rows of three squares each, so there are 3 × 2 = 6 square units.

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.

To find the area of a triangle, draw a parallelogram with base b and perpendicular height h. Draw a diagonal to divide the parallelogram into two congruent triangles. Since the area of the parallelogram is A = bh, the area of each of the two congruent triangles must be A = bh.

The area of each triangle is one

half the area of the parallelogram

**Circumference and Area of Circles**

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).

Small Circle circumference = C _{1}diameter = d _{1} |
and |
Large Circle circumference = C _{2}diameter = d _{2} |

To find the area of a circle, think of the circle as a pie, divided into slices, or sectors. The total area is the sum of the areas of the slices. Each pie slice can be approximated by a triangle with height r—the radius of the circle—and base b. The area of all the slices put together is then approximately (the sum of the bases) × (the radius). But the sum of the bases is approximately the circumference (2πr). So the area of a circle is approximately × 2πr × r = πr^{2}. The greater the number of sectors, the closer the approximation will be to πr^{2}.

**Teaching Model 20.2:** Circumference