Electrical circuit theory lecture notes

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Chapter 1 Basic Electrical Theory and Mathematics Topics 1.0.0 Basic Mathematics 2.0.0 Electrical Terms and Symbols 3.0.0 Electrical Theory 4.0.0 Principles of DC 5.0.0 Principles of AC 6.0.0 Electrical Circuits 7.0.0 Electrical Circuit Computations 8.0.0 Constructing an Electrical Circuit To hear audio, click on the box. Overview The origin of the modern technical and electronic Navy stretches back to the beginning of naval history, when the first navies were no more than small fleets of wooden ships, using wind and oars. The need for technicians then was restricted to a navigator and seamen who could handle the sails. As time passed, the advent of the steam engine and electrical power sources signaled the rise of an energy sources more practical. With this technological advancement, the need for competent technicians increased. Today there is scarcely anyone in the United States Navy who does not use electrical or electronic equipment. This equipment is needed in systems of electric lighting and power, and intercommunications. As a Construction Electrician, your understanding and knowledge of basic electrical theory will able to conduct the Navy’s mission. Basic mathematical skills are used everyday by Construction Electricians. A sound understanding of these basics prepares you for the more complex math skills you’re likely to use on construction projects, ranging from whole numbers, fractions, decimals, ratios, proportions, percentages, and square roots to measurements and calculations using geometric shapes. Safety can be impacted by calculations you make for your project. For example, machinery electrical load requirements require precise calculations to prevent equipment damage and personnel injury or death. NAVEDTRA 14026A 1-1Objectives When you have completed this chapter, you will be able to do the following: 1. Understand basic mathematics. 2. Identify electrical terms and symbols 3. Understand electrical theory. 4. Understand the electrical principles of Direct Current (DC). 5. Understand the electrical principles of Alternating Current (AC). 6. Understand the requirements and configurations of electrical circuits. 7. Understand the requirements of electrical circuit computations. 8. Understand the requirements of constructing an electrical circuit. Prerequisites This course map shows all of the chapters in Construction Electrician Basic. The suggested training order begins at the bottom and proceeds up. Skill levels increase as you advance on the course map. C Test Equipment, Motors, and Controllers E Communications and Lighting Systems Interior Wiring and Lighting Power Distribution Power Generation B Basic Line Construction/Maintenance Vehicle Operations and Maintenance A Pole Climbing and Rescue S Drawings and Specifications I Construction Support C Basic Electrical Theory and Mathematics NAVEDTRA 14026A 1-2Features of this Manual This manual has several features which make it easy to use online. • Figure and table numbers in the text are italicized. The figure or table is either next to or below the text that refers to it. • The first time a glossary term appears in the text, it is bold and italicized. When your cursor crosses over that word or phrase, a popup box displays with the appropriate definition. • Audio and video clips are included in the text, with italicized instructions telling you where to click to activate it. • Review questions that apply to a section are listed under the Test Your Knowledge banner at the end of the section. Select the answer you choose. If the answer is correct, you will be taken to the next section heading. If the answer is incorrect, you will be taken to the area in the chapter where the information is for review. When you have completed your review, select anywhere in that area to return to the review question. Try to answer the question again. • Review questions are included at the end of this chapter. Select the answer you choose. If the answer is correct, you will be taken to the next question. If the answer is incorrect, you will be taken to the area in the chapter where the information is for review. When you have completed your review, select anywhere in that area to return to the review question. Try to answer the question again. NAVEDTRA 14026A 1-31.0.0 BASIC MATHEMATICS 1.1.0 Parts of a Whole Number Whole numbers are made up of digits, which can be any numerical symbol from 0 to 9. Each digit of a whole number represents a place value, as shown in Figure 1-1. The number in this example is read as one million two hundred thirty-four thousand five hundred sixty-seven. 1 2 3 4 5 6 7 Figure 1-1 – Place values in whole numbers. Every digit has a value that depends on its place, or location, in the whole number. In the example above, the place value of the 1 is one million; the place value of the 5 is five hundred. Numbers can be positive or negative. Positive numbers are larger than zero and don’t usually have a positive sign (+) before them. Negative numbers are smaller than zero and always have a negative sign (-) before them. Zero is not positive or negative; it never has a positive or negative sign before it. Any whole number that doesn’t have a negative sign in front of it is a positive number. 1.2.0 Decimals Decimals are numbers in the base 10 system, using any numerical symbol from 0 to 9. There are place values in decimal numbers that are similar to the place values in whole numbers, except that decimal numbers appear to the right of a decimal point and do not use comma separators. Place values in decimal numbers are shown in Figure 1-2. . 1 2 3 4 5 6 Figure 1-2 – Place values in decimal numbers. As in whole numbers, decimal numbers have values that depend on their place in the number. In the example above, the place value of the 1 is one tenth, the place value of the 3 is three thousandths. Operations with decimals are very similar to operations with whole numbers. The only difference is that you have to keep track of the decimal point. NAVEDTRA 14026A 1-4 Decimal Point Millions Hundred Tenths Thousands Hundredths Ten Thousands Thousandths Thousands Ten Hundreds Thousandths Tens Hundred Thousandths Units Millionths 1.2.1 Adding Decimals When you add numbers containing decimals, you need to make sure you keep the decimal points lined up. For example, if you add 12.34 to 5.678, they should look like this when you add them: 12.34 + 5.678 Make sure that you add each column of numbers, starting with the numbers that are farthest right. In this case, the first number has no digit in the thousandths position, so it can be treated as a zero: 12.340 +5.678 8 As you move left adding the columns, make sure to carry any numbers greater than 10. When you add 4 and 7 in the hundredths column, the sum is 11. Record a 1 in the hundredths column and carry a 1 to the tenths column as shown below: 1 12.340 +5.678 18 When you add the tenths column, you have to add 3 and 6, and the 1 you carried from the sum in the hundredths column. This will give you a sum of 10, so record the 0 in the tenths column and carry a 1 to the units’ column as shown below: 1 12.340 +5.678 .018 Add the remaining numbers as you would any whole number. Remember to place the decimal point between the units’ column and the tenths column, as shown below: 12.340 +5.678 18.018 1.2.2 Subtracting Decimals Subtracting decimals is very similar to adding decimals. You need to line up the decimal points as in addition. Subtracting 5.678 from 12.34 looks like this: 12.37 - 1.248 NAVEDTRA 14026A 1-5Since there are only 2 decimal points after the whole number in 12.34, we need to add a zero at the end so we can subtract the three decimal points in 5.678. 12.370 - 1.248 You subtract columns the same way as you add them, starting with the farthest right column. In this case, you can’t subtract 8 from 0, so you need to borrow from the hundredths column to be able to subtract from 10, as shown below: 6 1 12.37 0 - 1.24 8 2 You now have 6 to subtract 4 from, since you borrowed 1 from 7. The rest of the numbers subtract normally, as shown below: 6 1 12.37 0 - 1.24 8 11.122 1.2.3 Multiplying Decimals When you multiply numbers with decimals, there is a two step process. First you multiply the numbers as if they were whole numbers. Then you place the decimal point in the correct location. The example below shows the product of 1.2 and 3.4, before the decimal is placed. 1.2 x 3.4 408 To get the correct location for the decimal point, count the number of decimal places in each number and add the number of decimal places. In this case, each number has one decimal place, so the product will have two decimal places. The product of the equation is 4.08. 1.2.4 Dividing Decimals When you divide numbers with decimals, there is a four step process. 1. Convert the divisor to a whole number. A divisor of .1 becomes 2. 2. Convert the dividend by the same number of decimal places as the divisor. In this case, 2.34 becomes 23.4. NAVEDTRA 14026A 1-63. Divide the two numbers as shown below. 11.7 2 23.4 2 3 2 14 14 0 4. Place the decimal according to the number of decimal places in the dividend. 11.7 2 23.4 1.3.0 Fractions A fraction is a part of a whole. Fractions are usually written as two numbers separated by a slash, such as 1/2. The slash means the same thing as the division sign (÷), so 1/2 = 1 ÷ 2. Figure 1-3 shows a whole triangle shaded blue and a triangle with one half (1/2) shaded blue. Figure 1-3 – Whole and half triangles. The bottom number of a fraction is called the denominator and tells how many parts the whole is being divided into. The top number of a fraction is called the numerator and tells how many of the parts are being used. In the example of 1/2, 2 is the denominator and 1 is the numerator. The denominator and numerator are also known as the terms of the fraction. Equivalent fractions are different fractions which mean the same amount. For example, 1/2 is an equivalent fraction to 2/4, 10/20, and 25/50. 1.3.1 Reducing Fractions to Their Lowest Terms Fractions shown with different numbers can have the same value. Fractions are easier to work with when they are at the lowest terms possible. For example, it is easier to work with the fraction 1/2 than it is to work with the equivalent fraction 17/34. To reduce a fraction to its lowest terms, there are three steps. 1. Determine what the largest number is that will divide evenly into both the numerator and the denominator. If the only number that will divide evenly into both numbers is 1, the fraction is at its lowest terms. NAVEDTRA 14026A 1-72. Divide both the numerator and denominator by the number you determined in Step 1. 3. For the fraction 8/32, the largest number that evenly divides both the numerator 8 and the denominator 32 is 8. Reducing the fraction to its lowest terms looks like this: 8/32 ÷ 8/8 = ¼ 1.3.2 Comparing Fractions and Finding the Lowest Common Denominator Comparing fractions is simple if the two fractions have the same denominator. In this case, the fraction with the larger numerator is larger than the fraction with the smaller numerator. Most fractions that you need to compare won’t have the same denominator. You need to convert them to the same denominator to compare them. The simplest way to convert fractions to the same denominator is to multiply their denominators to get a common denominator, and then convert each fraction to the resulting denominator. For example, if you are comparing 3/4 to 5/7, you would convert and compare them as shown below. 1. Find the common denominator. 4 x 7 = 28 2. Convert each fraction to the common denominator. 3/4 x 7/7 = 21/28 5/7 x 4/4 = 20/28 3. Compare the fractions. You find that 3/4 is larger than 5/7. Just as you can find the lowest terms for single fractions, you can find the lowest common denominator for multiple fractions. 1. Reduce both fractions to their lowest terms. 2. Determine the lowest common multiple for the denominators. You may find that one denominator is a multiple of the other. For example, if you are comparing 1/4 and 3/8, the denominator 8 is a multiple of the denominator 4. 3. Convert the fractions to equivalent fractions with the common denominator. 1/4 x 2/2 = 2/8 3/8 x 1/1 = 3/8 4. Compare the fractions. You find that 1/4 is smaller than 3/8. 1.3.3 Adding Fractions Sometimes you calculate a number where the numerator is larger than the denominator. This is called an improper fraction. You can convert an improper fraction to a whole number and a fraction, which is known as a mixed number. Start by adding the fractions as you would normally. To add 5/7 to 3/4: 1. Find the common denominator. 7 x 4 = 28 2. Convert each fraction to the common denominator. NAVEDTRA 14026A 1-8 5/7 x 4/4 = 20/28 3/4 x 7/7 = 21/28 3. Add the numerators of the fractions, and place the sum over the common denominator. Do NOT add the denominators. 20/28 + 21/28 = 41/28 4. Convert the improper fraction to a mixed number. 41 ÷ 28 = 1 with a remainder of 13 or 1 13/28 The remainder becomes the numerator for the fraction portion of the mixed number. The resulting mixed number is 1 13/28. 1.3.4 Subtracting Fractions When you need to subtract measurements that include fractions on construction projects, it is very similar to adding fractions. If the denominators of the fractions are the same, subtract the numerators, place the result over the denominator, and reduce the resulting fraction to its lowest terms. If the denominators are not the same, follow these steps. 1. Write out the equation. 3/4 – 1/8 = x 2. Determine the common denominator for the fractions you need to subtract. For the fractions 3/4 and 1/8, the common denominator is 4 x 8 = 32 3. Convert the fractions to equivalent fractions with the common denominator. 3/4 x 8/8 = 24/32 1/8 x 4/4 = 4/32 4. Subtract the numerators of the fractions, and place the result over the common denominator. Do NOT subtract the denominators. 24/32 – 4/32 = 20/32 5. Reduce the resulting fraction ot its lowest terms. 20/32 ÷ 4/4 = 5/8 Sometimes you need to subtract a fraction from a whole number. To do this you need to convert the whole number to an equivalent fraction, and then make your subtraction. In this example we’ll subtract 5/8 from 1. 1. Write out the equation. - 5/8 = x 2. Convert the whole number to an equivalent fraction. 1 x 8/8 = 8/8 3. Subtract the numerators of the fractions, and place the result over the common denominator. Do NOT subtract the denominators. 8/8 – 5/8 = 3/8 NAVEDTRA 14026A 1-94. Reduce the resulting fraction to its lowest terms. In this case the result is already in its lowest terms. 1.3.5 Multiplying Fractions Multiplying fractions is fairly simple, since you don’t need to worry about finding a common denominator. When you read or hear that you need to find a part of a number, such as 3/8 of 5/6, it means you need to multiply the numbers using the steps below. 1. Write out the equation. 3/8 x 5/6 = x 2. Multiply the numerators. 3 x 5 = 15 3. Multiply the denominators. 8 x 6 = 48 4. Reduce the resulting fraction to its lowest terms. 15/48 ÷ 3/3 = 5/16 In this case, 3 is the largest number that can be evenly divided into both the numerator and the denominator. You may find it easier to work with the fractions if you reduce them to their lowest terms before you multiply them. 1.3.6 Dividing Fractions Dividing fractions is very similar to multiplying fractions, except that you invert or flip the fraction you are dividing by. Use the following steps to divide 7/8 by 1/4. 1. Write out the equation. 7/8 ÷ 1/4 = x 2. Invert the fraction you are dividing by. 1/4 becomes 4/1 3. Convert the division sign (÷) to a multiplication sign (x) and write the new equation. 7/8 ÷ 1/4 becomes 7/8 x 4/1 4. Multiply the numerators. 7 x 4 = 28 5. Multiply the denominators. 8 x 1 = 8 6. Reduce the resulting fraction to its lowest terms. 28/8 ÷ 4/4 = 7/2 7. Convert the improper fraction to a mixed number. 3 1/2 NAVEDTRA 14026A 1-101.4.0 Conversions – Fractions and Decimals There will be times when you need to convert numbers so that all of the numbers you are working with are in the same format. The most common conversions you will work with are from fractions to decimals and from decimals to fractions. 1.4.1 Converting Fractions to Decimals To convert a number from a fraction to a decimal, divide the numerator by the denominator. 1.4.2 Converting Decimals to Fractions There are three steps to convert a decimal to a fraction. The decimal .125 can be converted to a fraction as follows: 1. Place the number to the right of the decimal point in the numerator. 125/ 2. Count the number of decimal places in the number. Place this number of zeros following a 1 in the denominator. 125/1000 3. Reduce the fraction to its lowest terms. 125/1000 ÷ 125/125 = 1/8 1.4.3 Converting Inches to Decimal Equivalents in Feet Sometimes you will need to convert measurements in inches to decimal equivalents in feet. .1÷ 2=.05 The results are that 6 inches converts to 0.5 foot. 1.5.0 Ratios and Proportions 1.5.1 Ratios A ratio is a comparison of two numbers, which can be expressed in three ways. A comparison of the numbers 1 and 2 can be expressed as follows: 1:2 1/2 1 to 2 One place where ratios come into play for Builders is Rule 42 for concrete mixes. This rule specifies a ratio of 1:2:4 for cement, sand, and aggregates. Ratios can be used to calculate the quantities of materials needed for a project. If your specifications call for a 1:2:4 concrete mix with 2-inch coarse aggregates, you use Rule 42 to figure the material amounts. Add 1:2:4, which gives you 7. Then compute your material requirements as follows: 42 cu ft ÷ 7 = 6 cu ft 1 x 6 = 6 cu ft of cement 2 x 6 = 12 cu ft of sand NAVEDTRA 14026A 1-11 4 x 6 = 24 cu ft of coarse aggregates 1.5.2 Proportions A proportion is an equation showing a ratio on each side. The equation shows that the two ratios are equal, as shown below: 1:2 = 2:4 You will usually work with proportions to figure an unknown number on one side of the equation. If you have a ratio of 1:2 and need to figure the equivalent ratio of n:8, there are three steps. 1. Write out the proportion. 2:4 = n:8 OR 2/4 =n/8 2. Use the cross product. 4 x n = 2 x 8 4n = 16 3. Solve the proportion. n= 16/4 n = 4 1.6.0 Percentages A percentage is a number expressed as a fraction of 100. You will usually see percentages with the percent sign, as in 35%. You can calculate the percentage of a material that has been used in two steps. 1. Divide the used amount by the initial amount. 2. Multiply the result by 100. If you had an initial supply of 300 sheets of plywood and you have used 80 of them, you calculate the percent used as follows: 80/300 = .27 .27 x 100 = 27% If you need to know what percent you have remaining, you subtract the percent used from 100, as follows: 100 – 27 = 73% If you have not calculated the percent used, you can still calculate the percent remaining with two steps. 1. Calculate the amount remaining. 300 – 80 = 220 2. Calculate the percent remaining. 220/300 = .73 .73 x 100 = 73% NAVEDTRA 14026A 1-121.7.0 Conversions – Percentages and Decimals 1.7.1 Converting Percentages to Decimals Convert a decimal to a percentage by multiplying the decimal by 100. If you need the percentage equivalent of .74, perform the following calculation: .74 x 100 = 74% 1.7.2 Converting Decimals to Percentages Convert a decimal to a percentage by multiplying the decimal by 100. If you need the percentage equivalent of .74, perform the following calculation: .74 x 100 = 74% 1.8.0 Square Roots The square root of a number is a value that, multiplied by itself, gives the original 2 number. In other words, if you have a value x, the square root r is a number such that r = x. A simple example is the square root of 9, which is 3. There is a table in Appendix I of NAVEDTRA 14139 Mathematics, Basic Math, and Algebra called Squares, Cubes, Square Roots, Cube Roots, Logarithms, and Reciprocals of Numbers that you can use to look up a square root. If that resource or a calculator with a square root function is not available, there are several methods of calculating a square root. The simplest of these methods is called the Babylonian Method, which is repeated until you get as close to the square root as you need to. This example will calculate the square root of 8. 1. Estimate a number that you think is close to the square root. For this example, use 3 as the estimate. 2. Divide the number you are trying to calculate the sqyare root of by your estimate. 8/3 = 2.67 3. Add that number to your estimate. 3 + 2.67 = 5.67 4. Divide the sum by 2. 5.67/2 = 2.835 5. Test your result by multiplying the number by itself. If the result is accurate enough, great Stop here. 6. If the number is not accurate enough, use the result as your new estimate. In our example, when 2.835 is squared, the result is 8.037225. Using a second round brings us to a possible square root of 2.828, with a result of 7.997584. 7. Repeat these steps until you have as accurate a result as you need. 1.9.0 Metric System The metric system is a decimal-based system of units. We will focus on units of weight, length, volume, and temperature. NAVEDTRA 14026A 1-131.9.1 Units of Weight The standard metric unit of mass is the gram. Table 1-1 shows units of mass, their equivalents in grams, and the abbreviations for the units of mass. Table 1-1 – Metric Units of Mass. Unit of Mass Equivalent in Grams Abbreviation 1 milligram 0.001 gram mg 1 centigram 0.01 gram cg 1 decigram 0.1 gram dg 1 gram 1 gram g 1 kilogram 1000 grams kg 1.9.2 Units of Length The standard metric unit of length is the meter. Table 1-2 shows units of length, their equivalents in meters, and the abbreviations for the units of length. Table 1-2 – Metric Units of Length. Unit of Length Equivalent in Meters Abbreviation 1 millimeter 0.001 meter mm 1 centimeter 0.01 meter cm 1 decimeter 0.1 meter dm 1 meter 1 meter m 1 kilometer 1000 meters km 1.9.3 Units of Volume The standard metric unit of volume is the liter. Table 1-3 shows units of volume, their equivalents in liters, and the abbreviations for units of volume. Table 1-3 – Metric Units of Volume. Unit of Volume Equivalent in Liters Abbreviation 1 milliliter 0.001 liter ml 1 centiliter 0.01 liter cl 1 deciliter 0.1 liter dl 1 liter 1 liter l 1 kiloliter 1000 liters kl NAVEDTRA 14026A 1-141.9.4 Units of Temperature The standard metric unit of temperature is the degree Celsius. The boiling point of water at sea level is 100°Celsius, or 100°C. The freezing point of water at sea level is 0°Celsius, or 0°C. A day with a temperature of 30°C is considered hot. 1.9.5 Metric Conversion There will be times when you need to convert to metric equivalents of measurements. Table 1-4 shows conversions for some of the most common measurements. Table 1-4 – Conversion to Metric Equivalents. English When You Know Multiply By To Find Metric Symbol Symbol in inches 25.4 millimeters mm ft feet 0.305 meters m yd yards 0.914 meters m mi miles 1.61 kilometers km in² square inches 645.2 square mm² millimeters ft² square feet 0.0903 square meters m² yd² square yards 0.836 square meters m² ac acres 0.405 hectares ha mi² square miles 2.59 square kilometers km² fl oz fluid ounces 29.57 milliliters mL gal gallons 3.785 liters L ft³ cubic feet 0.028 cubic meters m³ yd³ cubic yards 0.765 cubic meters m³ oz ounces 28.35 grams g lb pounds 0.454 kilograms kg T short tons (2000 0.907 Megagrams Mg (or “t”) lb) (“metric ton”) °F Fahrenheit (F-32) x 5/9 Celsius °C Or (F-32)/1.8 NAVEDTRA 14026A 1-15 TEMP MASS VOLUME AREA LENGTH 1.10.0 Using Measuring Tools Measuring tools are a key part of a Builder’s toolkit. You will most likely use a standard (English) ruler, an architect’s scale, and a metric ruler, as shown in Figure 1-4 There are conversions between standard and metric measurements, but you will have better results if you measure with the appropriate ruler, such as a standard ruler when you are working in the United States. Standard (English) Ruler Architect’s Scale Metric Ruler Figure 1-4 – Types of measurement tools. 1.10.1 Using a Standard Ruler A standard ruler is divided into inches and feet. Inches are divided into fractions of an inch, including halves, fourths, eighths, and sixteenths, as represented in Figure 1-5 There are some rulers that are further divided into thirty-seconds and sixty-fourths of an inch. Figure 1-5 – Inch divided into 16ths. An English or metric ruler is read from left to right. The arrow in Figure 1-6 is at 2 and 5/16 inches. Figure 1-6 – Measuring on an English ruler. NAVEDTRA 14026A 1-161.10.2 Using the Architect’s Scale An architect’s scale is used to read all plans except site plans. It measures interior and exterior dimensions for structures and buildings, including rooms, walls, doors, and windows. Table 1-5 shows scales that are generally grouped in pairs using the same dual-numbered index line. Table 1-5 – Common architect scale groupings. Scale One Scale Two Ratio Ratio Description Abbreviation Description Abbreviation Equivalent Equivalent Three 3” = 1’ 0” 1:4 One and one 1 1/2” = 1’ 0” 1:8 inches to half inches to the foot the foot One inch to 1” = 1’ 0” 1:12 One half inch 1/2” = 1’ 0” 1:24 the foot to the foot Three 3/4” = 1’ 0” 1:16 Three 3/8” = 1’ 0” 1:32 quarters eighths inch inch to the to the foot foot One quarter 1/4” = 1’ 0” 1:48 One eighth 1/8” = 1’ 0” 1:96 inch to the inch to the foot foot Three 3/16” = 1’ 0” 1:64 Three thirty- 3/32” = 1’ 0” 1:128 sixteenths seconds inch inch to the to the foot foot Numbers on architect scales can be read from left to right or right to left, depending on which scale you are using. Unlike standard rulers, the 0 point on an architect’s scale is not at the end of the measuring line. Any numbers below 0 represent fractions of one foot. Determine what scale you need to use from the drawing you are working with. Find the matching scale on one of the ends of the architect’s scale you are using. If the scale you need is shown on the left of the architect’s scale, measure and read from left to right. If the scale you need is shown on the right of the architect’s scale, measure and read from right to left. Figure 1-7 shows measurements on the 1/8” = 1’ 0” and 1/4” = 1’ 0” scales. This diagram shows a reading of 21’ 4” on the one eighth inch to the foot scale, reading from left to right. Notice that the numbers for this scale are the top set, reading 0, 4, 8, 12, etc. The feet are measured to the right of the zero on the scale; the inches are measured to the left of the zero on the scale. The numbers in the bottom set, reading 46, 44, 42, 40, etc. are for the one quarter inch to the foot scale. NAVEDTRA 14026A 1-17 Figure 1-7 – Measuring on an architect’s scale. This diagram shows a reading of 6’ 2” on the one fourth inch to the foot scale, reading from right to left. Notice that the numbers for this scale are the bottom set, reading 0, 2, 4, 6, etc. The feet are measured to the left of the zero on the scale; the inches are measured to the right of the zero on the scale. The numbers in the top set, reading 56, 60, 64, 72, etc. are for the one eighth inch to the foot scale. A metric ruler is divided into millimeters and centimeters, which makes it fairly easy to read, as shown in Figure 1-8. Figure 1-8 – Metric ruler divided into centimeters and millimeters. Centimeters are shown as larger lines with numbers; millimeters are shown as smaller lines. One millimeter is 1/10th of a centimeter. 1.11.0 Construction Geometry Measurements of shapes are a basic part of construction you will use every day. You should be familiar with measuring basic shapes like circles, triangles, squares, and rectangles. 1.11.1 Angles Two straight lines that meet at a common point form an angle. The point where the lines meet to form the angle is called a vertex. Angles are measured with a tool called a protractor, using degrees. There are many different types of angles, as shown in Figure 1-9. 1.11.1.1 Acute Angle An acute angle measures between 0 and 90 degrees. Common acute angles measure Acute Angles 30, 45, and 60 degrees. 1.11.1.2 Right Angle A right angle measures 90 degrees. The two lines that form a right angle are perpendicular to each other. This is the angle that is used most in construction. It Right Angle is indicated in drawings by the symbol . NAVEDTRA 14026A 1-181.11.1.3 Obtuse Angle An obtuse angle measures between 90 and 180 degrees. Obtuse Angles Common obtuse angles are 120, 135, and 150 degrees. 1.11.1.4 Straight Angle Straight Angle A straight angle measures 180 degrees, a flat line. 1.11.1.5 Adjacent Angles Adjacent angles are right next to each; they share a vertex and one side. Adjacent Angles 1.11.1.6 Opposite Angles Opposite angles are formed by two straight lines that cross; they are always equal. Opposite Angles Figure 1-9 – Types of angles. 1.11.2 Shapes Your work in construction involves common geometric shapes. These shapes include rectangles, squares, triangles, and circles, as shown in Figure 1-10. 1.11.2.1 Rectangle A rectangle is a four sided shape with all four angles being right angles. All four angles in a rectangle add up to 360°. A rectangle has two pairs of parallel sides, which makes a rectangle a parallelogram. In a rectangle, the longer sides define the Rectangle length of the rectangle; the shorter sides define the width. 1.11.2.2 Square A square is a special rectangle with four right angles and equal length parallel sides. Each angle in a square is 90°, totaling 360° for all four angles. Square NAVEDTRA 14026A 1-191.11.2.3 Triangle A triangle is a basic shape in geometry, with three sides or edges, also known as line segments. A triangle is a polygon with three corners, or vertices. The three angles of a triangle always add up to 180°. Triangle Types of triangles are classified by the relative lengths of their sides. Right Triangle – A right triangle has one 90°, or right, angle. The longest side of the right triangle is opposite the right angle, and is called the hypotenuse. The other two sides of the right triangle are called the legs. Right Triangle Equilateral Triangle – An equilateral triangle has all three sides of an equal length; this makes it equilinear. It is also equiangular, which means that all three of its internal angles are the same 60°. Equilateral Triangle Isosceles Triangle – An isosceles triangle has two sides of equal length. An isosceles triangle also has two angles equal to each other; the angles opposite the equal sides. Isosceles Triangle Scalene Triangle – A scalene triangle has three sides of different lengths. The angles inside a scalene triangle are also all different. Scalene Triangle NAVEDTRA 14026A 1-20

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