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Math Calculations For HERS Raters 1 Math Calculations For HERS Raters 1

Why Worry 2 Why Worry 2

Why Worry 3 Why Worry 3

Why Worry 4 Why Worry 4

Calculating Areas 5 Calculating Areas 5

Calculating Areas 6 Calculating Areas 6

Calculating Areas 7 Calculating Areas 7

Other Complex Shapes Insulated Hip Roof 8 Other Complex Shapes Insulated Hip Roof 8

Develop a Sequence for Problem Solving 1. Convert Measurements to Decimals: 1 foot 3” Develop a Sequence for Problem Solving 1. Convert Measurements to Decimals: 1 foot 3” = 1. 25 feet - - 0. 5 = 6” etc. 2. Simplify Shapes to: v Rectangles or Squares v Right Triangles (one angle is 90 degrees) v Any Shape where the Formula is Known 3. Carefully Evaluate the Known Information 4. Solve the Problem (Answer the Question) 5. Convert your answer to feet & inches OR decimals as the test question requires. 9

Convert Measurements to Decimals Make Calculations in Decimals Convert Inches to Feet by: inches Convert Measurements to Decimals Make Calculations in Decimals Convert Inches to Feet by: inches / 12 = decimal feet Remember: Convert your answer to feet & inches OR decimals as the test question requires. 10

Convert Measurements to Decimals Common Decimals Equivalence I inch = 0. 083 3 inches Convert Measurements to Decimals Common Decimals Equivalence I inch = 0. 083 3 inches = 0. 25 4 inches = 0. 33 6 inches = 0. 50 8 inches = 0. 67 9 inches = 0. 75 11

Convert Measurements to Decimal Feet Example 4 ft 8 inches = 1/12 = 0. Convert Measurements to Decimal Feet Example 4 ft 8 inches = 1/12 = 0. 67 Answer 4. 67 feet 12

Convert Measurements to Feet/Inches Example 6. 25 feet 0. 25 * 12 = 3 Convert Measurements to Feet/Inches Example 6. 25 feet 0. 25 * 12 = 3 inches Answer 6 ft 3 inches 13

Your Turn- Conversions Convert to Decimal Feet: Convert to Feet/Inches One foot- two inches Your Turn- Conversions Convert to Decimal Feet: Convert to Feet/Inches One foot- two inches = 3. 33 = Seven inches = 1. 92 = One foot – five inches = 4. 67 = Two feet – nine inches = 6. 08 = Three feet – ten inches = 5. 50 = 14

Simplify The Shape Hint: Look for Rectangles and Right Triangles 15 Simplify The Shape Hint: Look for Rectangles and Right Triangles 15

Simplify The Shape Hint: Look for Rectangles and Right Triangles 16 Simplify The Shape Hint: Look for Rectangles and Right Triangles 16

Simplify The Shape Hint: Look for Rectangles and Right Triangles 17 Simplify The Shape Hint: Look for Rectangles and Right Triangles 17

Simplify The Shape Hint: Look for Rectangles and Right Triangles 18 Simplify The Shape Hint: Look for Rectangles and Right Triangles 18

Your Turn- Simplify This Shape Hint: Look for Rectangles and Right Triangles 19 Your Turn- Simplify This Shape Hint: Look for Rectangles and Right Triangles 19

Your Turn- Simplify This Shape Hint: Look for Rectangles and Right Triangles 20 Your Turn- Simplify This Shape Hint: Look for Rectangles and Right Triangles 20

Math Calculations Right Triangles • Why Right Triangles –Calculate Length for Rafters 21 Math Calculations Right Triangles • Why Right Triangles –Calculate Length for Rafters 21

Right Triangle- Pythagorean Theorem C A 90° B 22 Right Triangle- Pythagorean Theorem C A 90° B 22

Right Triangle- Pythagorean Theorem A 2 + B 2 = C 2 C A Right Triangle- Pythagorean Theorem A 2 + B 2 = C 2 C A 90 ° B (A 2) 3 X 3 = 9, (B 2) 4 X 4 = 16, (C 2) 9 + 16 = 25 C = √ 25 = 5 23

Right Triangle- Pythagorean Theorem A 2 + B 2 = C 2 Solve for: Right Triangle- Pythagorean Theorem A 2 + B 2 = C 2 Solve for: _______________ A = √ C 2 - B 2 B = √ C 2 - A 2 C = √ A 2 + B 2 _______________ Watch for change in Sign !!!! _______________ C A 90 ° B 24

Right Triangle- Pythagorean Theorem C A A 2 + B 2 = C 2 Right Triangle- Pythagorean Theorem C A A 2 + B 2 = C 2 90° B (A 2) 3 X 3 = 9 (B 2) 4 X 4 = 16 (C 2) 9 + 16 = 25 C = √ 25 = 5 25

Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” 26 Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” 26

Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” 3 inches Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” 3 inches = 3/12 ft = 0. 25 ft 4’ 3” = 4. 25 ft 8 inches = 8/12 ft = 0. 67 ft 15’ 8’ = 15. 67 ft 27

Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” A 2 Right Triangle- Sample Calculation Raft Length ? 4’ 3” 90° 15’ 8” A 2 = 4. 25 x 4. 25 = 18. 06 B 2 = 15. 67 x 15. 67 = 245. 55 C 2 = 18. 06 + 245. 55 = 263. 61 C = S 263. 61 = 16. 24 ft 28

Math Calculations Ratios • Why Ratios –Using Roof Pitch in Calculations 29 Math Calculations Ratios • Why Ratios –Using Roof Pitch in Calculations 29

Everyday Use of Ratio’s • Your going to buy lawn fertilizer – Your lawn Everyday Use of Ratio’s • Your going to buy lawn fertilizer – Your lawn is 10, 000 ft 2 – The fertilizer bag label is: – 1 bag per 2000 ft 2 • How many bags do you buy? 30

Everyday Use of Ratio’s • How many bags do you buy? If 1 bag Everyday Use of Ratio’s • How many bags do you buy? If 1 bag covers 2, 000 then 10, 000/2, 000 = 5 bags As a Ratio 1 bag = “X” bags 2, 000 ft² 10, 000 ft² Cross multiply 10, 000 ft² x 1 bag = “X” bags x 2, 000 ft² “X” bags = 1 bag * 10, 000 ft² 2, 000 ft² X bags = 5 Divide 31

Everyday Use of Ratio’s • Your going to make chili for 2 people – Everyday Use of Ratio’s • Your going to make chili for 2 people – Recipe is of 4 people – The recipe calls for 3 teaspoons of hot pepper • How much hot pepper do you put in? – The right amount not fire engine chili 32

Everyday Use of Ratio’s • How much hot pepper do you put in? If Everyday Use of Ratio’s • How much hot pepper do you put in? If 3 teaspoons is for 4 people then 1 ½ teaspoons is for 2 people As a Ratio 3 teaspoons = “X” teaspoons 4 people 2 people x 3 teaspoons = “X” teaspoons x 4 people X teaspoons = 3 teaspoons x 2 people 4 people X = 1. 5 teaspoons or 1 ½ teaspoons 33

Units of Ratio’s They have to be the same on both sides of the Units of Ratio’s They have to be the same on both sides of the = 1 bag = X bags 2, 000 ft² 10, 000 ft² 3 teaspoons 4 people = X teaspoons 2 people 34

Roof Pitch • Roof slope express as a ratio – 4 : 12 – Roof Pitch • Roof slope express as a ratio – 4 : 12 – 6 : 12 – 12 : 12 • Drawn on a Plan as – 12 4 • In ratio form = _4_ 12 35

Visualizing Slope 12 Z 6 6 12 36 Visualizing Slope 12 Z 6 6 12 36

Calculating Rise or Run Slope = 4 : 12 or Rise : Run 4 Calculating Rise or Run Slope = 4 : 12 or Rise : Run 4 12 Z Rise Run X On Blueprints, Slope = “X” : 12 12 ”x” = Rise 12 Run 37

Roof Terms Roof Run and Roof Span = 2 * Roof Run Roof Span Roof Terms Roof Run and Roof Span = 2 * Roof Run Roof Span is double the Roof Run. or Roof Run is half of the Roof Span. Roof Run = Roof Span 2 12 Z Roof Rise (Pitch) 6 6 12 Roof Run Roof Span 38

Calculate Run Example: Pitch 8 : 12 Ratio _8 _ = 16 ft 12 Calculate Run Example: Pitch 8 : 12 Ratio _8 _ = 16 ft 12 Run 12 8 Cross Multiply & Divide Rise Z 16 ft 8 Run x 8 = 16 x 12 8 Run = 16 x 12 = 24 ft 8 What is the Span ? Hint: Run is ½ Span 2 x 24 = 48 ft Run 39

Calculate Rise Example: Pitch 4: 12 (Ratio) _4_ = Rise 12 10 ft Cross Calculate Rise Example: Pitch 4: 12 (Ratio) _4_ = Rise 12 10 ft Cross multiply & Divide 12 4 x 10 = Rise x 12 Rise = 10 * 4 = 3. 33 ft 12 Convert to feet – inches 3 ft – 4” 4 Rise Run 10 ft 40

Calculate Pitch Example: Pitch “X” : 12 Ratio “X” = 15 ft 12 18 Calculate Pitch Example: Pitch “X” : 12 Ratio “X” = 15 ft 12 18 ft Cross Multiply & Divide “X” x 18 = 15 x 12 12 “X” Rise 15 ft Z “X” = 12 x 15 = 10 18 Pitch 10 : 12 Run 18 ft 41

Roof Pitch Calculations Your Turn 42 Roof Pitch Calculations Your Turn 42

Calculating Perimeter, Area and Volume Two Most Common Shapes: • Rectangles • Triangles 43 Calculating Perimeter, Area and Volume Two Most Common Shapes: • Rectangles • Triangles 43

Calculating Perimeter - Rectangle Perimeter = Distance around the outside edge P = 2 Calculating Perimeter - Rectangle Perimeter = Distance around the outside edge P = 2 x length + 2 x width length width 44

Calculating Perimeter - Triangle P = width + length + slope pe o Sl Calculating Perimeter - Triangle P = width + length + slope pe o Sl length width 45

Calculating Area - Rectangle For a Rectangle Area equal the length times the width Calculating Area - Rectangle For a Rectangle Area equal the length times the width A = length x width length width 46

Calculating Area - Triangle Area = ½ width times length A = length x Calculating Area - Triangle Area = ½ width times length A = length x width 2 length width 47

Calculating Volume - Rectangle Volume = length x width x height width length 48 Calculating Volume - Rectangle Volume = length x width x height width length 48

Volume - Triangle Volume = ½ of Length times Width times Height V = Volume - Triangle Volume = ½ of Length times Width times Height V = length x width x height 2 height width length 49

Applying the Calculations • Floor Area • Wall Area • Conditioned Space Volume 50 Applying the Calculations • Floor Area • Wall Area • Conditioned Space Volume 50

Area by Component 2) (ft 51 Area by Component 2) (ft 51

Area by Component (ft 2) X Y Z 52 Area by Component (ft 2) X Y Z 52

Area of a Rectangle Z (ft 2) Area of “Z” = length x width Area of a Rectangle Z (ft 2) Area of “Z” = length x width length width Z 53

Area of Triangle “X” (ft 2) AX = length x height 2 height X Area of Triangle “X” (ft 2) AX = length x height 2 height X Y length 54

Area of Triangle Y (ft 2) AY = length x width 2 X Y Area of Triangle Y (ft 2) AY = length x width 2 X Y width length 55

Total Area 2) (ft AT = A X + A Y + AZ X Total Area 2) (ft AT = A X + A Y + AZ X Y Z 56

Area by Component (ft 2) 57 Area by Component (ft 2) 57

Area by Component (ft 2) 58 Area by Component (ft 2) 58

Area by Component (ft 2) Y X Z W 59 Area by Component (ft 2) Y X Z W 59

Area by Component “W”(ft 2) AW = length x width W Length 60 Area by Component “W”(ft 2) AW = length x width W Length 60

Area by Component “X”(ft 2) AX= length x width length X 61 Area by Component “X”(ft 2) AX= length x width length X 61

Area by Component “Y”(ft 2) AY = length x width 2 width length Y Area by Component “Y”(ft 2) AY = length x width 2 width length Y Length 62

Area by Component “Z”(ft 2) AZ = length x width 2 length Z width Area by Component “Z”(ft 2) AZ = length x width 2 length Z width 63

Area by Component (ft 2) AT = A W + A X + A Area by Component (ft 2) AT = A W + A X + A Y + A Z Y X Z W 64

3) (ft Calculating Volume A Room with a Cathedral Ceiling 65 3) (ft Calculating Volume A Room with a Cathedral Ceiling 65

Volume – Cathedral Ceiling B C A 66 Volume – Cathedral Ceiling B C A 66

Volume by Component “A”(ft 3) Va = length x width x height A width Volume by Component “A”(ft 3) Va = length x width x height A width length 67

Volume by Component “B” (ft 3) BC A Vb = Rise x Run x Volume by Component “B” (ft 3) BC A Vb = Rise x Run x length 2 Rise (height) B Run (width) length 69

Volume by Component “C” (ft 3) BC A Vc = Rise x Run x Volume by Component “C” (ft 3) BC A Vc = Rise x Run x length 2 Rise (height) C Run (width) length 71

Cathedral Ceiling Volume by Component (ft 3) BC A Vt = Va + Vb Cathedral Ceiling Volume by Component (ft 3) BC A Vt = Va + Vb + Vc B C A 72

Volume - Kneewall Z 73 Volume - Kneewall Z 73

Volume - Kneewall Added a Small Cube - D Vt = Va + Vb Volume - Kneewall Added a Small Cube - D Vt = Va + Vb + Vc + Vd B D B Z C A 74

Perimeter (ft) P = 2 x length + 2 x width length width 75 Perimeter (ft) P = 2 x length + 2 x width length width 75

Perimeter (ft) C = ? ? D C E B F A 76 Perimeter (ft) C = ? ? D C E B F A 76

Perimeter (ft) X Y C length = e √ X 2 + Y 2 Perimeter (ft) X Y C length = e √ X 2 + Y 2 77

Perimeter (ft) P=A+B+C+D+E+F D C E B F A 78 Perimeter (ft) P=A+B+C+D+E+F D C E B F A 78

-Your Turn 1. What is the Slope ? 2. What is Height of Peak -Your Turn 1. What is the Slope ? 2. What is Height of Peak ? 6 20 8 36 79

6’-8” 5’-0” 10’-0” 6’-1 1/2” Building is 40’ long 23’-4” 9’-4 1/2” -Your Turn- 6’-8” 5’-0” 10’-0” 6’-1 1/2” Building is 40’ long 23’-4” 9’-4 1/2” -Your Turn- Calculate: 1. 2. 3. 4. 5. Floor Area Wall Area Roof Area Volume Perimeter 80

Working with a Circular Shape 81 Working with a Circular Shape 81

Circles Circumference (c)= Distance around the outside edge of the circle 82 Circles Circumference (c)= Distance around the outside edge of the circle 82

Diameter of a Circle Diameter = Distance across a circle (D) If you divide Diameter of a Circle Diameter = Distance across a circle (D) If you divide the distance around the circle (circumference – c ) by the diameter the answer will ALWAYS be = 3. 14 It is a constant called “pie” D = 3. 14 83

Radius of a Circle Radius = Distance from the center of a circle to Radius of a Circle Radius = Distance from the center of a circle to the edge (r) “r” = ½ diameter r 84

Area of a Circle Remember “ ” is a constant = 3. 14. The Area of a Circle Remember “ ” is a constant = 3. 14. The area of a circle is equal to times the radius (r) squared. The length of “r” is one half of the diameter (the distance across the circle. ) Take “r” and multiply it by itself to get r². r a = r² Now multiply times the product of r² to get the area (a) of the circle. ” 85

Area of a Circle (ft 2) radius . a = D 2 4 = Area of a Circle (ft 2) radius . a = D 2 4 = 3. 14 * Diameter 4 or a = r 2 = 3. 14 * radius Diameter 86

Volume of a Cylinder (ft 3) v = D 2 * h h = Volume of a Cylinder (ft 3) v = D 2 * h h = height of the cylinder 4 = 3. 14 * Diameter * height 4 or v = r 2 * L = 3. 14 * radius * height 87

Area of a Semi-Circle (ft 2) Area (a)= “pie” times the length of the Area of a Semi-Circle (ft 2) Area (a)= “pie” times the length of the radius squared divided by 2 a = r 2 2 = 3. 14 x radius 2 Or 2 a= D 8 = 3. 14 *Diameter * Diameter 8 radius Diameter 88

Volume of 1/2 a Cylinder 3) (ft Volume = r 2 x h h Volume of 1/2 a Cylinder 3) (ft Volume = r 2 x h h = height of the cylinder 2 = 3. 14 x radius x height 2 or using diameter (D) Volume = D 2 x h 8 = 3. 14 x Diameter x height 8 89

Perimeter of a Semi-Circle (ft) C = ? ? C B D A 90 Perimeter of a Semi-Circle (ft) C = ? ? C B D A 90

Semi-Circle Perimeter (ft) C = x Diameter 2 C = 3. 14 x Diameter Semi-Circle Perimeter (ft) C = x Diameter 2 C = 3. 14 x Diameter 2 or Diameter radius C = x radius C = 3. 14 x radius 91

Area by Component (ft 2) 92 Area by Component (ft 2) 92

Area by Component (ft 2) Z Y 93 Area by Component (ft 2) Z Y 93

Area of the Rectangle “Y” (ft 2) AY = length x width Y length Area of the Rectangle “Y” (ft 2) AY = length x width Y length 94

Area of the Semi-Circle “Z” (ft 2) AZ = r 2 2 = 3. Area of the Semi-Circle “Z” (ft 2) AZ = r 2 2 = 3. 14 x radius 2 or A Z = D 2 Diameter Z radius 8 = 3. 14 x Diameter 8 95

Total Area (ft 2) AT = A Y + A Z Z Y 96 Total Area (ft 2) AT = A Y + A Z Z Y 96

Volume (ft 3) Know AY + AZ L = Length VY = A Y Volume (ft 3) Know AY + AZ L = Length VY = A Y x L VZ = A Z x L VT = V Y + V Z Z Y 97

Semi-Circle Calculations -Your Turn- 98 Semi-Circle Calculations -Your Turn- 98

Special Cases • Ducts • Tray Ceilings 99 Special Cases • Ducts • Tray Ceilings 99

Duct Surface Area Rectangular Duct: Surface Area = 2 x (height + width) x Duct Surface Area Rectangular Duct: Surface Area = 2 x (height + width) x length Round Duct: Surface Area = 3. 14 x Duct Diameter x length 100

Special Case – Tray Ceiling 101 Special Case – Tray Ceiling 101

Volume – Tray Ceiling 102 Volume – Tray Ceiling 102

Volume – Tray Ceiling 2 4 3 1 5 103 Volume – Tray Ceiling 2 4 3 1 5 103

Volume – Tray Ceiling length width height 1 V 1 = length x width Volume – Tray Ceiling length width height 1 V 1 = length x width x height 104

Volume – Tray Ceiling width length height 2 V 2 = length x width Volume – Tray Ceiling width length height 2 V 2 = length x width x height 105

Volume – Tray Ceiling 2 Sloped Sides V 3 = Rise x Run x Volume – Tray Ceiling 2 Sloped Sides V 3 = Rise x Run x length 3 length Rise Run 106

Volume – Tray Ceiling 4 Rise length Run 2 Sloped Sides V 4 = Volume – Tray Ceiling 4 Rise length Run 2 Sloped Sides V 4 = Rise x Run x length 107

Area – Pyramid 5 height width length 4 Sloped Corners (Pyramid) a = 2 Area – Pyramid 5 height width length 4 Sloped Corners (Pyramid) a = 2 x length x width x height 108

Volume – Tray Ceiling 5 Sloped Corners = Pyramid 109 Volume – Tray Ceiling 5 Sloped Corners = Pyramid 109

Volume – Pyramid height width length Pyramid V 5 = 1/3 x length x Volume – Pyramid height width length Pyramid V 5 = 1/3 x length x width x height 110

Volume – Tray Ceiling VT = V 1 + V 2 + V 3 Volume – Tray Ceiling VT = V 1 + V 2 + V 3 + V 4 + V 5 2 4 3 1 5 111

Area – Tray Ceiling 112 Area – Tray Ceiling 112

Ceiling Area – Tray Ceiling 1 4 2 3 5 113 Ceiling Area – Tray Ceiling 1 4 2 3 5 113

Ceiling Area – Tray Ceiling Area 1 1 4 2 3 5 114 Ceiling Area – Tray Ceiling Area 1 1 4 2 3 5 114

Ceiling Area – Tray Ceiling Area 2 2 1 4 2 3 5 115 Ceiling Area – Tray Ceiling Area 2 2 1 4 2 3 5 115

Ceiling Area – Tray Ceiling Areas 3 & 4 width ? 1 4 2 Ceiling Area – Tray Ceiling Areas 3 & 4 width ? 1 4 2 3 length 5 116

Ceiling Area – Tray Ceiling Areas 3 & 4 width ? X 1 4 Ceiling Area – Tray Ceiling Areas 3 & 4 width ? X 1 4 2 Y 3 width = e X 2 + Y 2 5 117

Area – Tray Ceiling height width 1 4 length 4 Sloped Corners (Pyramid) 2 Area – Tray Ceiling height width 1 4 length 4 Sloped Corners (Pyramid) 2 3 A 4 = 2 x length x width x height 5 118