Math Calculations For HERS Raters 1 Why Worry 2

Math Calculations For HERS Raters

1

Why Worry

2

Why Worry

3

Why Worry

4

Calculating Areas

5

6

Calculating Areas

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Calculating Areas

Other Complex Shapes

Insulated Hip Roof

8

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:

Rectangles or Squares Right Triangles (one angle is 90 degrees) Any Shape where the Formula is Known

3. Carefully Evaluate the Known Information4. Solve the Problem (Answer the Question)5. Convert your answer to feet & inches OR

decimals as the test question requires.

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Make Calculations in DecimalsConvert Inches to Feet by:inches / 12 = decimal feet

Remember: Convert your answer to feet & inches OR decimals as the test

question requires.

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Convert Measurements to Decimals

Common Decimals EquivalenceI inch = 0.0833 inches = 0.254 inches = 0.336 inches = 0.508 inches = 0.679 inches = 0.75

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Convert Measurements to Decimals

Example4 ft 8 inches

8 inches = 1/12 = 0.67

Answer4.67 feet

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Convert Measurements to Decimal Feet

Example6.25 feet

0.25 * 12 = 3 inches

Answer6 ft 3 inches

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Convert Measurements to Feet/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 =

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Simplify The Shape

Hint: Look for Rectangles and Right Triangles

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Hint: Look for Rectangles and Right Triangles

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Simplify The Shape

Hint: Look for Rectangles and Right Triangles

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Simplify The Shape

Hint: Look for Rectangles and Right Triangles

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Simplify The Shape

Your Turn- Simplify This Shape

Hint: Look for Rectangles and Right Triangles

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Hint: Look for Rectangles and Right Triangles

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Your Turn- Simplify This Shape

Math Calculations

Right Triangles• Why Right Triangles

–Calculate Length for Rafters

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Right Triangle- Pythagorean Theorem

90°

AC

B

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90°

AC

B

A2 + B2 = C2

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(A2) 3 X 3 = 9, (B2) 4 X 4 = 16, (C2) 9 + 16 = 25 C = √25 = 5

Right Triangle- Pythagorean Theorem

90°

A C

B

A2 + B2 = C2

Solve for: _____________________________

A = √ C2 - B2 _____________________________

B = √ C2 - A2 ______________________________

C = √ A2 + B2 Watch for change in

Sign !!!!

24

Right Triangle- Pythagorean Theorem

B

A2 + B2 = C2

25

(A2) 3 X 3 = 9(B2) 4 X 4 = 16(C2) 9 + 16 = 25

C = √25 = 5

90°

A C

Right Triangle- Pythagorean Theorem

90°

4’ 3”Raft Length ?

15’ 8”

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Right Triangle- Sample Calculation

90°

4’ 3” Raft Length ?

15’ 8”

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Right Triangle- Sample Calculation

3 inches = 3/12 ft = 0.25 ft4’ 3” = 4.25 ft

8 inches = 8/12 ft = 0.67 ft15’ 8’ = 15.67 ft

90°

4’ 3” Raft Length ?

15’ 8”

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Right Triangle- Sample Calculation

A2 = 4.25 x 4.25 = 18.06

B2 = 15.67 x 15.67 = 245.55

C2 = 18.06 + 245.55 = 263.61

C = S263.61 = 16.24 ft

Math Calculations

Ratios• Why Ratios

–Using Roof Pitch in Calculations

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Everyday Use of Ratio’s

• Your going to buy lawn fertilizer– Your lawn is 10,000 ft2

– The fertilizer bag label is:– 1 bag per 2000 ft2

• How many bags do you buy?

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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 ft²X bags = 5

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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

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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

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Units of Ratio’s

They have to be the same on both sides of the =

1 bag = X bags2,000 ft² 10,000 ft²

3 teaspoons = X teaspoons 4 people 2 people

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Roof Pitch

• Roof slope express as a ratio– 4 : 12– 6 : 12– 12 : 12

• Drawn on a Plan as –

• In ratio form = _4_ 12

12

4

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Visualizing Slope

Z12

6

12

6

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Calculating Rise or Run

Slope = 4 : 12 or Rise : Run

On Blueprints, Slope = “X” : 12

”x” = Rise 12 Run

12

4Rise

Z

Run

12

X

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Roof Terms

Z12

6

12

6

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.

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Roof Span is double the Roof Run.

Calculate Run

Z 8Rise16 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 8

12

8

What is the Span ?

Hint: Run is ½ Span

2 x 24 = 48 ft

8

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Calculate RiseExample: 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

Run10ft

12

4

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Calculate Pitch

ZRise15 ft

Run18ft

Example:Pitch “X” : 12

Ratio “X” = 15ft 12 18ft Cross Multiply & Divide

“X” x 18 = 15 x 12

“X” = 12 x 15 = 10 18

Pitch 10 : 12

12

“X”

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Roof Pitch Calculations

Your Turn

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Calculating Perimeter, Area and Volume

Two Most Common Shapes:• Rectangles• Triangles

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P = 2 x length + 2 x width

width

length

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Perimeter = Distance around the outside edge

Calculating Perimeter - Rectangle

P = width + length + slope

length

width

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Calculating Perimeter - Triangle

Slope

width

length

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For a RectangleArea 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

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Volume = length x width x height

height

Calculating Volume - Rectangle

width

length

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V = length x width x height 2

height

Volume - Triangle

width

length

Volume = ½ of Length times Width times Height

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Applying the Calculations

• Floor Area• Wall Area• Conditioned Space Volume

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Area by Component (ft2)

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Area by Component (ft2)

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X Y

Z

Area of a Rectangle Z (ft2)

Area of “Z” = length x width

width

length

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Z

Area of Triangle “X” (ft2)

AX = length x height 2

height

length

X Y

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Area of Triangle Y (ft2)

AY = length x width 2

width

length

X Y

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Total Area (ft2)

AT = AX + AY + AZ

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X Y

Z

Area by Component (ft2)

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Area by Component (ft2)

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Area by Component (ft2)

W

XY Z

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Width W

Area by Component “W”(ft2)

AW = length x width

Length

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X

Area by Component “X”(ft2)

AX= length x width

width

length

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Area by Component “Y”(ft2)

AY = length x width 2

LengthYwidth

length

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Area by Component “Z”(ft2)

AZ = length x width 2

width

length

Z

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Area by Component (ft2)

AT = AW + AX + AY + AZ

W

XY Z

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Calculating Volume (ft3)A Room with a Cathedral Ceiling

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Volume – Cathedral Ceiling

A

B C

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Va = length x width x height

Aheight

Volume by Component “A”(ft3)

width

length

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B

Vb = Rise x Run x length 2

Volume by Component “B” (ft3)

Run(width)

length

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Rise(height)

AB C

C

Vc = Rise x Run x length 2

Run(width)

length

Rise(height)

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Volume by Component “C” (ft3)

AB C

Cathedral Ceiling Volume by Component (ft3)

A

B C

Vt = Va + Vb + Vc

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AB C

Volume - Kneewall

Z

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Volume - Kneewall

Z

A

B CD

Added a Small Cube - DVt = Va + Vb + Vc + Vd

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B

Perimeter (ft)

length

width

P = 2 x length + 2 x width

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Perimeter (ft)

A

B

C

D

E

F

C = ??

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Perimeter (ft)

Y

X

length = e √ X2 + Y2

C

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Perimeter (ft)

A

B

C

D

E

F

P = A + B + C + D + E + F

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-Your Turn-

36

206

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1. What is the Slope ?2. What is Height of Peak ?

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23’-4”

6’-8” 5’-0”

10’-0

”6’

-1 1 /

2”

9’-4

1 /2”

Building is 40’ long

1. Floor Area2. Wall Area3. Roof Area4. Volume5. Perimeter

Calculate:

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-Your Turn-

Working with a Circular Shape

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

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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”

= 3.14D

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Diameter of a Circle

Radius = Distance from the center of a circle to the edge (r)

r

“r” = ½ diameter

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Radius of a Circle

The area of a circle is equal to times the radius (r)

squared.

r a = r²

Remember “” is a constant = 3.14.

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.”

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Area of a Circle

Area of a Circle (ft2)

a = D2 4 = 3.14 * Diameter * Diameter 4 ora = r2 = 3.14 * radius * radius

Diameter

radius.

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Volume of a Cylinder (ft3)

v = D2 * h 4 = 3.14 * Diameter * Diameter * height 4 orv = r2 * L = 3.14 * radius * radius * height

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h = height of the cylinder

Area of a Semi-Circle (ft2)

a = r2 2 = 3.14 x radius x radius 2 Ora = D 8 = 3.14 *Diameter * Diameter 8

Diameter

radius

Area (a)= “pie” times the length of the radius squared divided by 2

2

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Volume of 1/2 a Cylinder (ft3)

h = height of the cylinder

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Volume = r2 x h 2 = 3.14 x radius x radius x height 2or using diameter (D)

Volume = D2 x h 8 = 3.14 x Diameter x Diameter x height 8

C = ??

Perimeter of a Semi-Circle (ft)

A

B D

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

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Area by Component (ft2)

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Area by Component (ft2)

Z

Y

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Area of the Rectangle “Y” (ft2)

AY = length x width

Y

length

width

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Area of the Semi-Circle “Z” (ft2)AZ = r2 2

= 3.14 x radius x radius 2 orAZ = D2 8

= 3.14 x Diameter x Diameter 8

Z

Diameter

radius

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Total Area (ft2)

AT = AY + AZ Z

Y

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Volume (ft3)

Know AY + AZ

VY = AY x L

VZ = AZ x L

VT = VY + VZY

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Z

L = Length

Semi-Circle Calculations

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-Your Turn-

Special Cases

• Ducts• Tray Ceilings

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Duct Surface Area

Rectangular Duct:Surface Area = 2 x (height + width) x length

Round Duct:Surface Area = 3.14 x Duct Diameter x length

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Special Case – Tray Ceiling

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Volume – Tray Ceiling

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Volume – Tray Ceiling

1

2

3

4

5

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Volume – Tray Ceiling

V1 = length x width x height

height

widthlength

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1

Volume – Tray Ceiling

V2 = length x width x height

height

widthlength

105

2

Volume – Tray Ceiling

2 Sloped Sides

V3 = Rise x Run x length

Rise

Runlength

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3

Volume – Tray Ceiling

2 Sloped Sides

V4 = Rise x Run x length

Rise

Runlength

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4

Area – Pyramid

4 Sloped Corners (Pyramid)

a = 2 x length x width x height

lengthwidth

height

108

5

Volume – Tray Ceiling

Sloped Corners = Pyramid

109

5

Volume – Pyramid

Pyramid

V5 = 1/3 x length x width x height

lengthwidth

height

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Volume – Tray Ceiling

1

2

3

4

5

VT = V1 + V2 + V3 + V4 + V5

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Area – Tray Ceiling

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Ceiling Area – Tray Ceiling

1

2

3

4

5

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Ceiling Area – Tray Ceiling

Area 1

12

3

4

5

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Ceiling Area – Tray Ceiling

Area 2

12

3

4

5

115

2

Ceiling Area – Tray Ceiling

Areas 3 & 4

12

3

4

5

width ?

length

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Ceiling Area – Tray Ceiling

Areas 3 & 4

12

3

4

5

width ?

width = e X2 + Y2

X

Y

117

Area – Tray Ceiling

4 Sloped Corners (Pyramid)

A4 = 2 x length x width x height

lengthwidth

height

118

12

3

4

5