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Rectangular Solids
Introduction: We examine 3 dimensional objects that are rectangular in shape, measuring their volume, surface area, and the length of a diagonal.

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 is the 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 below, 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.


We generalize this result. The volume of a rectangular solid is lwh. The surface area is found by adding the areas of the sides of the solid.

We can also calculate the length of a diagonal in this rectangular solid. We extend the Pythagorean Theorem and find the length of the diagonal (dotted line in the diagram below) to be .

In general, for a rectangular solid we have the length of the diagonal as .

Let's Practice:
  1. A cube has a side of 3 meters. What are the measures of the surface area, the diagonal, and the volume?
The volume of a cube is the length of the side cubed.
The volume is 33 = 27 cubic meters.
There are six sides which are squares of side 3, each having an area of 9 square meters.
The total surface area is 54 square meters.

The diagonal is .
  1. A rectangular solid has dimensions 3 x 6 x 7. What are the measures of the surface area, the diagonal, and the volume?
The volume is found by multiplying the measures of the sides.
The volume is 126.
There are two sides with dimensions 3 x 6 = 18, two sides with dimensions 3 x 7 = 21 and two sides with dimensions 6 x 7 = 42.
The total surface area is 2(18) + 2(21) + 2(42) = 162.

The diagonal is .

Examples
Example A rectangular solid has dimensions 4 x 5 x a. If the volume is 100, what is the measure of a, what is the surface area, and what is the length of the diagonal?
What is your answer?
 
Example A cube has a surface area of 150. What is the measure of a side of this cube and what is the volume of this cube?
What is your answer?
 



M Ransom

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