General (1-14)

Computing Surface Area of Three-dimensional Solids
The surface area of a three-dimensional solid is the sum of the areas of the faces of the solid. Surface area is a different concept from that of volume. For example, surface area is the amount of sheet metal needed to build a rectangular fuel tank while volume
is the amount of fuel that the tank can contain. Rectangular Solid The formula for the surface area of a rectangular solid
[Figure 1-24] is given as:

 

Surface Area = 2 × [(Width × Length) + (Width × Height) + (Length × Height)]

                   = 2 × [(W × L) + (W × H) + (L × H)]

 

Cube
The formula for the surface area of a cube [Figure 1‑25] is given as:

Surface Area = 6 × (Side × Side) = 6 × S²

Example: What is the surface area of a cube with a side measure of 8 inches?

Surface Area = 6 × (Side × Side)
                   = 6 × S² = 6 × 8² = 6 × 64
                   = 384 square inches

 

Cylinder
The formula for the surface area of a cylinder [Figure 1-26] is given as:

Surface Area = 2 × π × radius² + π × diameter × height
                   = 2 × π × r² + π × D × H

 

Sphere
The formula for the surface area of a sphere [Figure 1-28] is given as:

Surface Area = 4 × π × radius² = 4 × π × r²

 

 

Cone
The formula for the surface area of a right circular cone [Figure 1-29] is given as:

Surface Area = π × radius × [radius + (radius² + height²)^1⁄2]
                   = π × r × [r + (r² + H²)^1⁄2]

Figure 1-30 summarizes the formulas for computing the volume and surface area of three-dimensional solids.

 

 

 

 

 

Trigonometric Functions
Trigonometry is the study of the relationship between the angles and sides of a triangle. The word trigonometry comes from the Greek trigonon, which means three angles, and metro, which means measure.

 

Right Triangle, Sides and Angles

In Figure 1-31, notice that each angle is labeled with a capital letter. Across from each angle is a corresponding side, each labeled with a lower case letter. This triangle is a right triangle because angle C is a 90° angle. Side a is opposite from angle A, and is sometimes referred to as the “opposite side.” Side b is next to, or adjacent to, angle A and is therefore referred to as the “adjacent side.” Side c is always across from the right angle and is referred to as the “hypotenuse.”

 

Sine, Cosine, and Tangent

The three primary trigonometric functions and their abbreviations are: sine (sin), cosine (cos),
and tangent (tan). These three functions can be found on most scientific calculators. The three trigonometric functions are actually ratios comparing two of the sides of the triangle as follows: 

 

 

 

 

Example: Find the sine of a 30° angle.

Calculator Method:
Using a calculator, select the “sin” feature, enter the number 30, and press “enter.” The calculator should display the answer as 0.5. This means that when angle A equals 30°, then the ratio of the opposite side (a) to the hypotenuse (c) equals 0.5 to 1, so the hypotenuse is twice as long as the opposite side for a 30° angle.
Therefore, sin 30° = 0.5.

Trigonometric Table Method:
When using a trigonometry table, find 30° in the first column. Next, find the value for sin 30° under the second column marked “sine” or “sin.” The value for sin 30° should be 0.5.

 

 

 

 

 

 

Pythagorean Theorem
The Pythagorean Theorem is named after the ancient Greek mathematician, Pythagoras (~500 B.C.). This theorem is used to find the third side of any right triangle when two sides are known. The Pythagorean Theorem states that a² + b² = c². [Figure 1-32] Where c = the hypotenuse of a right triangle, a is one side of the triangle and b is the other side of the triangle.

Example: What is the length of the longest side of a right triangle, given the other sides are 7 inches and 9 inches? The longest side of a right triangle is always side c, the hypotenuse. Use the Pythagorean Theorem to solve for the length of side c as follows:

 

a² + b² = c²
7² + 9² = c²
49 + 81 = c²
     130 = c²

 

 

If c² = 130 then c = √130 = 11.4 inches
Therefore, side c = 11.4 inches.

 

 

 Example: The cargo door opening in a military airplane is a rectangle that is 5 1⁄2 feet tall by 7 feet wide. A section of square steel plate that is 8 feet wide by 8 feet tall by 1 inch thick must fit inside the airplane. Can the square section of steel plate fit through the cargo door? It is obvious that the square steel plate will not fit horizontally through the cargo door. The steel plate
is 8 feet wide and the cargo door is only 7 feet wide.

 

However, if the steel plate is tilted diagonally, will it fit through the cargo door opening?
The diagonal distance across the cargo door opening can be calculated using the Pythagorean Theorem where “a” is the cargo door height, “b” is the cargo door width, and “c” is the diagonal distance across the cargo door opening.

 

 

a² + b² = c²
(5.5 ft)² + (7 ft)² = c²
30.25 + 49 = c²
79.25 = c²

c = 8.9 ft

 

The diagonal distance across the cargo door opening is 8.9 feet, so the 8-foot wide square steel plate will fit diagonally through the cargo door opening and into the airplane.

 

 

 

Measurement Systems
Conventional (U.S. or English) System
Our conventional (U.S. or English) system of measurement is part of our cultural heritage from the days when the thirteen colonies were under British rule. It started as a collection of Anglo-Saxon, Roman, and Norman-French weights and measures. For example, the inch represents the width of the thumb and the foot is from the length of the human foot. Tradition holds that King Henry I decreed that the yard should be the distance from the tip of his nose to the end of his thumb. Since medieval times, commissions appointed by various English monarchs have reduced the chaos of measurement by setting specific standards
for some of the most important units. Some of the conventional units of measure are: inches, feet, yards, miles, ounces, pints, gallons, and pounds. Because the conventional system was not set up systematically, it contains a random collection of conversions. For example, 1 mile = 5,280 feet and 1 foot = 12 inches.

 

Metric System
The metric system, also known as the International System of Units (SI), is the dominant language of measurement used today. Its standardization and decimal features make it well-suited for engineering and aviation work.

The metric system was first envisioned by Gabriel Mouton, Vicar of St. Paul’s Church in Lyons, France. The meter is the unit of length in the metric system, and it is equal to one ten-millionth of the distance from the equator to the North Pole. The liter is the unit of volume and is equal to one cubic decimeter. The gram is the unit of mass and is equal to one cubic centimeter of water.

All of the metric units follow a consistent naming scheme, which consists of attaching a prefix to the unit. For example, since kilo stands for 1,000 one kilometer equals 1,000 meters. Centi is the prefix for one hundredth, so one meter equals one hundred centimeters. Milli is the prefix for one thousandths and one gram equals one thousand milligrams. Refer to Figure 1-33
for the names and definitions of metric prefixes.

 

Measurement Systems and Conversions
The United States primarily uses the conventional (U.S. or English) system, although it is slowly integrating the metric system (SI). A recommendation to transition to the metric system within ten years was initiated in the 1970s. However, this movement lost momentum, and the United States continues to use both measurement systems. Therefore, information to convert between the conventional (U.S., or English) system and the metric (SI) system has been included in Figure 1-37, Applied Mathematics Formula Sheet, at the end of this chapter. Examples of its use are as follows:

 

To convert inches to millimeters, multiply the number of inches by 25.4.

Example: 20 inches = 20 × 25.4 = 508 mm

 

To convert ounces to grams, multiply the number of ounces by 28.35.

Example: 12 ounces = 12 × 28.35 = 340.2 grams