Math Calculations For HERS Raters 1

Why Worry 2

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Calculating Areas 5

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Other Complex Shapes Insulated Hip Roof 8

Develop a Sequence for Problem Solving 1. Convert Measurements to Decimals: 1 foot 3” = 1.25 feet = 6” etc. 2. Simplify Shapes to:  Rectangles or Squares  Right Triangles (one angle is 90 degrees)  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

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 = inches = inches = inches = inches = inches = Convert Measurements to Decimals

Example 4 ft 8 inches 8 inches = 1/12 = 0.67 Answer 4.67 feet 12 Convert Measurements to Decimal Feet

Example 6.25 feet 0.25 * 12 = 3 inches Answer 6 ft 3 inches 13 Convert Measurements to Feet/Inches

Your Turn- Conversions Convert to Decimal Feet: Convert to Feet/Inches One foot- two inches = = Seven inches = = One foot – five inches = = Two feet – nine inches = = Three feet – ten inches = = 14

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

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 18 Simplify The Shape

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

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

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

Right Triangle- Pythagorean Theorem 90 ° A C B 22

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

90 ° A C B 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 !!!! 24 Right Triangle- Pythagorean Theorem

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

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

90 ° 4’ 3” Raft Length ? 15’ 8” 27 Right Triangle- Sample Calculation 3 inches = 3/12 ft = 0.25 ft 4’ 3” = 4.25 ft 8 inches = 8/12 ft = 0.67 ft 15’ 8’ = ft

90 ° 4’ 3” Raft Length ? 15’ 8” 28 Right Triangle- Sample Calculation A 2 = 4.25 x 4.25 = B 2 = x = C 2 = = C = S = ft

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

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 covers 2,000 then 10,000/2,000 = 5 bags As a Ratio 1 bag = “X” bags Cross multiply 2,000 ft² 10,000 ft² 10,000 ft² x 1 bag = “X” bags x 2,000 ft² “X” bags = 1 bag * 10,000 ft² Divide 2,000 ft² X bags = 5 31

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 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 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 = 1 bag = X bags 2,000 ft² 10,000 ft² 3 teaspoons = X teaspoons 4 people 2 people 34

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

Visualizing Slope Z

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

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

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

Calculate Rise Example: Pitch 4:12 (Ratio) _4_ = Rise 12 10ft Cross multiply & Divide 4 x 10 = Rise x 12 Rise = 10 * 4 = 3.33 ft 12 Convert to feet – inches 3 ft – 4” Rise Run 10ft

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

Roof Pitch Calculations Your Turn 42

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

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

P = width + length + slope length width 45 Calculating Perimeter - Triangle Slope

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

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

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

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

Applying the Calculations Floor Area Wall Area Conditioned Space Volume 50

Area by Component (ft 2 ) 51

Area by Component (ft 2 ) 52 XY Z

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

Area of Triangle “X” (ft 2 ) A X = length x height 2 height length XY 54

Area of Triangle Y (ft 2 ) A Y = length x width 2 width length XY 55

Total Area (ft 2 ) A T = A X + A Y + A Z 56 XY Z

Area by Component (ft 2 ) 57

Area by Component (ft 2 ) 58

Area by Component (ft 2 ) W X YZ 59

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

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

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

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

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

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

Volume – Cathedral Ceiling A B C 66

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

Volume by Component “B”(ft 3 ) 68 A B C

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

Volume by Component “C” (ft 3 ) 70 A B C

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

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

Volume - Kneewall Z 73

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

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

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

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

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

-Your Turn What is the Slope ? 2. What is Height of Peak ? 79

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

Working with a Circular Shape 81

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

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”   = 3.14 D 83 Diameter of a Circle

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

The area of a circle is equal to  times the radius (r) squared. r a =  r² Remember “  ” is a constant = 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. Now multiply  times the product of r² to get the area (a) of the circle. ” 85 Area of a Circle

Area of a Circle (ft 2 ) a =  D 2 4 = 3.14 * Diameter * Diameter 4 or a =  r 2 = 3.14 * radius * radius Diameter radius. 86

Volume of a Cylinder (ft 3 ) v =  D 2 * h 4 = 3.14 * Diameter * Diameter * height 4 or v =  r 2 * L = 3.14 * radius * radius * height 87 h = height of the cylinder

Area of a Semi-Circle (ft 2 ) a =  r 2 2 = 3.14 x radius x radius 2 Or a =  D 8 = 3.14 *Diameter * Diameter 8 Diameter radius Area (a)= “pie” times the length of the radius squared divided by

Volume of 1/2 a Cylinder (ft 3 ) h = height of the cylinder 89 Volume =  r 2 x h 2 = 3.14 x radius x radius x height 2 or using diameter (D) Volume =  D 2 x h 8 = 3.14 x Diameter x Diameter x height 8

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

Semi-Circle Perimeter (ft) Diameter radius C =  x Diameter 2 C = 3.14 x Diameter 2 or C =  x radius C = 3.14 x radius 91

Area by Component (ft 2 ) 92

Area by Component (ft 2 ) Z Y 93

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

Area of the Semi-Circle “Z” (ft 2 ) A Z =  r 2 2 = 3.14 x radius x radius 2 or A Z =  D 2 8 = 3.14 x Diameter x Diameter 8 Z Diameter radius 95

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

Volume (ft3) Know A Y + A Z V Y = A Y x L V Z = A Z x L V T = V Y + V Z Y 97 Z L = Length

Semi-Circle Calculations 98 -Your Turn-

Special Cases Ducts Tray Ceilings 99

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

Volume – Tray Ceiling 102

Volume – Tray Ceiling

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

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

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

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

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

Volume – Tray Ceiling Sloped Corners = Pyramid 109 5

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

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

Area – Tray Ceiling 112

Ceiling Area – Tray Ceiling

Ceiling Area – Tray Ceiling Area

Ceiling Area – Tray Ceiling Area

Ceiling Area – Tray Ceiling Areas 3 & width ? length 116

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

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