Math Calculations For
HERS Raters
1
Why Worry
2
Why Worry
3
Why Worry
4
Calculating Areas
5
Calculating Areas
6
Calculating Areas
7
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 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 / 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 = 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
8 inches = 1/12 = 0.67
Answer
4.67 feet
12
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 =
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
16
Simplify The Shape
Hint: Look for Rectangles and Right Triangles
17
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
20
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
A2 + B2 = C2
C
A
90
°
B
(A2) 3 X 3 = 9, (B2) 4 X 4 = 16, (C2) 9 + 16 = 25 C = √25 = 5
23
Right Triangle- Pythagorean Theorem
A2 + B2 = C2
Solve for:
_____________________________
A = √ C2 - B2
B = √ C2 - A2
C = √ A2 + B2
_____________________________
Watch for change in
Sign !!!!
______________________________
C
A
90
°
B
24
Right Triangle- Pythagorean Theorem
C
A
A2 + B2 = C2
90°
B
(A2) 3 X 3 = 9
(B2) 4 X 4 = 16
(C2) 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”
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”
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
28
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 ft2
– The fertilizer bag label is:
– 1 bag per 2000 ft2
• 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
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
– 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
4 people
=
X teaspoons
2 people
34
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
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
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 _ = 16ft
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
Run
2 x 24 = 48 ft
39
Calculate Rise
Example:
Pitch 4:12
(Ratio) _4_ = Rise
12 10ft
Cross multiply & Divide
12
4 x 10 = Rise x 12
Rise = 10 * 4 = 3.33 ft
12
4
Rise
Convert to feet – inches
3 ft – 4”
Run
10ft
40
Calculate Pitch
Example:
Pitch “X” : 12
Ratio “X” = 15ft
12
18ft
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
18ft
41
Roof Pitch Calculations
Your Turn
42
Calculating Perimeter, Area and Volume
Two Most Common Shapes:
• Rectangles
• Triangles
43
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
length
width
45
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 width
2
length
width
47
Calculating Volume - Rectangle
Volume = length x width x height
height
width
length
48
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
Area by Component
2
(ft )
51
Area by Component (ft2)
X
Y
Z
52
Area of a Rectangle Z (ft2)
Area of “Z” = length x width
length
width
Z
53
Area of Triangle “X” (ft2)
AX = length x height
2
height
X
Y
length
54
Area of Triangle Y (ft2)
AY = length x width
2
X
Y
width
length
55
Total Area
2
(ft )
AT = AX + AY + AZ
X
Y
Z
56
Area by Component (ft2)
57
Area by Component (ft2)
58
Area by Component (ft2)
Y
X
Z
W
59
Area by Component “W”(ft2)
AW = length x width
Width
W
Length
60
Area by Component “X”(ft2)
AX= length x width
width
length
X
61
Area by Component “Y”(ft2)
AY = length x width
2
width
length
Y
Length
62
Area by Component “Z”(ft2)
AZ = length x width
2
length
Z
width
63
Area by Component (ft2)
AT = AW + AX + AY + AZ
Y
X
Z
W
64
3
(ft )
Calculating Volume
A Room with a Cathedral Ceiling
65
Volume – Cathedral Ceiling
B
C
A
66
Volume by Component “A”(ft3)
Va = length x width x height
height
A
width
length
67
Volume by Component “B” (ft3)
B C
A
Vb = Rise x Run x length
2
Rise
(height)
B
length
Run
(width)
69
Volume by Component “C” (ft3)
B C
A
Vc = Rise x Run x length
2
Rise
(height)
C
Run
(width)
length
71
Cathedral Ceiling Volume by Component (ft3)
B C
A
Vt = Va + Vb + Vc
B
C
A
72
Volume - Kneewall
Z
73
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)
C = ??
D
C
E
B
F
A
76
Perimeter (ft)
X
Y
C
length = e √ X2 + Y2
77
Perimeter (ft)
P=A+B+C+D+E+F
D
C
E
B
F
A
78
-Your Turn1. 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-
Calculate:
1.
2.
3.
4.
5.
Floor Area
Wall Area
Roof Area
Volume
Perimeter
80
Working with a Circular Shape
81
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 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 the edge (r)
“r” = ½ diameter
r
84
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 (ft2)
radius
.
a =  D2
4
= 3.14 * Diameter * Diameter
4
or
a =  r2
= 3.14 * radius * radius
Diameter
86
Volume of a Cylinder (ft3)
v =  D2 * h
h = height of the cylinder
4
= 3.14 * Diameter * Diameter * height
4
or
v =  r2 * L
= 3.14 * radius * radius * height
87
Area of a Semi-Circle (ft2)
Area (a)= “pie” times the length of the radius squared divided by 2
a =  r2
2
= 3.14 x radius 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 =  r2 x h
h = height of the cylinder
2
= 3.14 x radius x radius x height
2
or using diameter (D)
Volume =  D2 x h
8
= 3.14 x Diameter x Diameter x height
8
89
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
2
or
Diameter
radius
C =  x radius
C = 3.14 x radius
91
Area by Component (ft2)
92
Area by Component (ft2)
Z
Y
93
Area of the Rectangle “Y” (ft2)
AY = length x width
width
Y
length
94
Area of the Semi-Circle “Z” (ft2)
AZ =  r2
2
= 3.14 x radius x radius
2
or
AZ =  D2
Diameter
Z
radius
8
= 3.14 x Diameter x Diameter
8
95
Total Area (ft2)
AT = AY + AZ
Z
Y
96
Volume (ft3)
Know AY + AZ
L = Length
VY = AY x L
VZ = AZ x L
VT = VY + VZ
Z
Y
97
Semi-Circle Calculations
-Your Turn-
98
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
2
4
3
1
5
103
Volume – Tray Ceiling
length
width
height
1
V1 = length x width x height
104
Volume – Tray Ceiling
width
length
2
height
V2 = length x width x height
105
Volume – Tray Ceiling
2 Sloped Sides
V3 = Rise x Run x length
3
length
Rise
Run
106
Volume – Tray Ceiling
4
Rise
length
Run
2 Sloped Sides
V4 = Rise x Run x length
107
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 – Pyramid
height
width
length
Pyramid
V5 = 1/3 x length x width x height
110
Volume – Tray Ceiling
VT = V1 + V2 + V3 + V4 + V5
2
4
3
1
5
111
Area – Tray Ceiling
112
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 2
2
1
4
2
3
5
115
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
2
Y
3
width = e X2 + Y2
5
117
Area – Tray Ceiling
height
width
1
4
2
3
length
4 Sloped Corners (Pyramid)
A4 = 2 x length x width x height
5
118