Introduction of Rectangular Solids: Rectangular solid is same as rectangular prism and it is a three dimensional figure. It is a rectangular prism in which it has six rectangular faces, twelve edges and eight vertices. Rectangular solid includes volume and surface area formula. The rectangular solid has three dimensions they are length width and height.
The Lesson: We ask in practical terms two questions about 3 dimensional objects. First, how much paint does it take to paint the object (we would have to know the surface area). Second, how much water can the object hold (we would have to know the volume). We show how to calculate surface area and volume.
A rectangular solid is a 3 dimensional object with six sides, all of which are rectangles. We first examine a cube, in which all six sides are squares. In the diagram below, a square of side 2 inches is used to form a cube with six square sides. The square clearly has an area of 4 square inches. Therefore the cube has a surface area of 48 square inches because its sides are composed of six of these squares. The cube is composed of 8 smaller cubes with a side of 1 inch. The volume of this cube is 8 cubic inches.
We note that a square has its name because its area is the square of the length of a side. The area 4 =2² . The cube has its name because the volume of a cube of the length of a side. The volume is 8 =2³. We can generalize this result for cubes by saying that the volume of a cube of side a is a³. Since the area of one side is the length of the side squared, the entire surface area of the cube is 6 x 4 = 24. This can also be generalized for any cube of side a. The surface area is 6 x a²= 6a².
In the diagram above, we show a rectangular solid at right with dimensions 5 x 2 x 3 inches. These are the measures of the length l, the width w and the height h. The area of the Front/Back rectangles is 15 square inches. The area of the Sides is 6 square inches, and the area of the Top/Bottom rectangles is 10 square inches. Adding these we get the surface area which is 62 square inches. The volume is found by multiplying the lengths of the sides as we did with the cube. The volume is 30 cubic inches.
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