## Perimeter, Area, and Volume

**Perimeter**

The perimeter of a polygon is the distance around it, or the sum of the lengths of its sides. Opposite sides of a rectangle are congruent. If l and w denote the length and width of a rectangle, then the perimeter P is given by l + w + l + w, or 2 × l + 2 × w. Since perimeter is a length or distance, the unit of measure may be inches, feet, yards, miles, centimeters, meters, or kilometers.

**Area**

Area is measured in square units. A square unit is a square where 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 of a rectangle is given by the formula l × w (length times width) or b × h (base times height). This can be seen in a rectangle with a base 3 and height 2. There are two rows of three squares each so there are 3 × 2 = 6 unit squares.

For area, the symbols used involve an exponent of 2, such as 4 in.² (4 square inches). The use of abbreviations such as “sq in.” or “sq ft” is discouraged.

**Solid Figures**

At this grade level, only right solid figures are considered. Students learn that prisms have rectangular faces, that the curved surface of a cylinder is perpendicular to the bases, and that the line from the vertex of a pyramid or a cone lies over the center point of the base. A net is a two-dimensional pattern that can be cut and folded to enclose a solid figure. Prisms and pyramids are named according to the shapes of their bases.

The surface area of a solid figure is the sum of the areas of all faces of the figure. By analyzing the faces of solid figures, several formulas can be given for finding the surface area of a solid figure. However, students should realize that they do not need to memorize these formulas since they can find the area of each face and then add to find the surface area of a figure.

A prism has two congruent polygons for bases (top and bottom) and rectangles for sides that join the corresponding edges of the polygonal bases. The volume of a solid figure is a measure of the amount of space the figure occupies. The volume of a prism is the area of its base times its height. If the height is 1, then each square unit in the base will give rise to 1 cubic unit of volume in the prism. In general, each square unit in the base corresponds to a rectangular prism with base area b and height h. Taking the sum of the volumes results in the formula V = B × h, where B is the area of the base and h is the height of the prism. Since the area of the base of a rectangular prism is equal to the length times the width, the formula for the volume of a rectangular prism can also be written as V = l × w × h. Volume is measured in cubic units, such as cubic centimeters, cubic feet, or cubic inches.

**Teaching Model 18.6:** Volume