## Volume and Surface Area

**Solid Figures**

At this grade level, only right solid figures are considered. This means 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 to the base is perpendicular to the base. A net is a two-dimensional pattern that can be cut and folded to make 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 surfaces 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. The formulas can be used as a convenient shortcut. These formulas are:

Surface Area of a rectangular prism: SA = 2wh + 2lw + 2lh

Surface Area of a square pyramid: SA = s^{2} + 4(bh)

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 prism is the area of the base times the 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 1 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.

The volume of a pyramid is V = Bh, where B is the area of the base and h is the height. The factor arises in this formula in much the same way that does in the formula for the area of a triangle. A unit square can be cut into two congruent right triangles. Each of these right triangles has an area equal to the area of the rectangle. Similarly, a unit cube can be cut into three congruent pyramids, each with a square base. Therefore, each of these pyramids must have volume the volume of the cube.

Similarly, the volume of a cylinder is V = Bh, where B is the area of the base and h is the height. Since B =πr^{2}, the formula can also be written as V = πr^{2}h. The volume of a cone is the volume of a cylinder with the same base and height; therefore, the formula is V =πr^{2}h.

**Teaching Model 21.4:** Volume of Rectangular Prisms