Download SURFACE MEASUREMENTS

SURFACE MEASUREMENTS
Any measurement taken in a two dimensional plane is a surface measurement. There are many objects
evaluated in only two dimensions such as POLYGONS - squares, rectangles, parallelograms, triangles,
trapezoids and CIRCLES. Types of surface measurements taken to evaluate these objects would include
linear measurements to find the length of, or distance around an object, and square measurements to
find the area inside the boundaries of the object.
Taking surface measurements on the job site or off of blueprints is an everyday activity for the contractor.
How many feet of rim joist are needed? How many sheets of plywood sheathing are needed? How many
pieces of bevel siding are needed to cover the wall surface? How many yards of carpeting need to be
ordered?
Luckily calculating surface measurements is pretty easy. You need to memorize some formulas, many of
which you already know, and carry out some very basic math - adding subtracting, multiplying, dividing
and squaring. Lets look at some of the objects you will be evaluating.
POLYGONS
Polygons are plane figures bound by three or more line segments, called the sides, which are joined at their
endpoints. The angle formed where two sides meet is called an interior angle. The point where two sides
meet is called a vertex.(see figure 4-2). A polygon with sides of equal length and equal interior angles is
called a regular polygon. Polygons having four sides are called quadrilaterals. The perimeter of a polygon
is simply the sum of the sides. Figure 4-1 illustrates the names and shapes of polygons you will likely
encountered.
a. Triangle
b. Square
c. Rectangle
e. Parallelogram
f. Trapezoid
i. Hexagon
j. Octagon
d. Rhombus
g. Irregular Polygon
h. Pentagon
k. Irregular Polygon
Figure 4-1
Interior angle
Polygon Terminology
Side
Vertex
Figure 4-2
30
The sum of the interior angles of a polygon can be calculated by using the following formula.
Sum of interior angles = (n – 2) 180 Where n equals the number
of sides or interior angles of a Polygon.
Example3-A: The sum of interior angles of a quadrilateral = (4 – 2) 180 = 360
CIRCLES
Circle – A curved line in a plane that encloses a space. Every point on a circle is the same distance from
the center point.
(C)ircumference – The distance around a circle. C = 2r
(D)iameter – A straight line from one side of a circle to the
r
other side that passes through the center.
D
(r)adius – A straight line from the center of the circle to a
T
point on the circle.
C
Arc – A curved line whose points are equal distance from
a single point.
Arc length – The length if a curved line.
Circle
(T)angent line – A straight line which touches just one point on a circle. A tangent line is also
perpendicular to the radius line or diameter line touching the same point along the circle.
(C)hord – A straight line from one point on a circle to another point on a circle. The longest chord of a
circle is a diameter.
Pi () = 3.1416 The circumference of a circle with a diameter of one.
More about circle later!
LINEAR AND PERIMETER MEASUREMENTS
The simplest surface measurement is a linear measurement, which means to measure length.
Measuring the length of one leg of a triangle or the length of a wall would be a linear measurement. In the
course of a day a carpenter takes dozens of linear measurements.
Another form of linear measurement is a Perimeter (P) measurement. Perimeter means the distance
around an object. If all three legs of a triangle were measured the sum would represent the perimeter of the
triangle. Likewise, if the distance around a circle were measured that distance would represent its
perimeter.
POLYGONS
Try these: You may need to look in the glossary for definitions in order to answer questions 1 and 2
1. List the letters corresponding to the quadrilaterals in figure 4-1
Answer _____________________________________
2. List the letters corresponding to the regular polygons in figure 4-1. Note: triangle a in figure 4-1 is an
equilateral triangle.
Answer _________________
3. Calculate the sum of interior angles for g, h, i, j and k in figure 4-1.
g. ________ h _________ i _________ j __________ k _____________
31
4. A triangle has sides measuring 6-5”, 3-7 and 5-2. What is the perimeter measurement of the
triangle?
Answer _________
5. What is the perimeter measurement of the foundation illustrated below?
10 6
40 9
Answer _________
6. What is the perimeter measurement of the foundation illustrated below? Also write in the missing
measurement
22 11
9 4
7 6
7 3
8 8
63
_______
7 5
38 9
Answer _____________
7. What is the perimeter of the octagon?
13 4
Answer____________
8. A square building has a perimeter measuring 162 feet. What is the length of each side of the building?
Answer ____________
9.
A Pentagon has a perimeter measuring 156 7 1/2. What are the lengths of each side?
Answer __________
10. What is the circumference of a circle with a diameter of 12 6?
Answer ___________
11. What is the circumference of a circle with a radius of 12 3?
Answer ____________
32
CIRCLES
Circles deserve some explanation. To calculate the distance around a circle the circumference formula must
be employed.
Circumference = 2r
 = 3.1416 (rounded)
r = radius
Example 4-B: What is the circumference of a circle with a diameter of 16-0
Step 1: If diameter is given rather than radius divide the diameter by 2 to find the radius.
16/2 = 8 = r
Step 2: Solve for circumference
2r
2r
2  3.1416*  8 = 50.2655 0r 50-3 3/16
*If you use the  on your calculator it will round pi to 11 decimal places and your
answer may be slightly different than if you input 3.1416 into your calculator.
Example 4-C: What is the radius of a circle with a circumference of 46 
Step 1: Setup the formula and plug in what you know.
C =2r
46 = 2  3.1416  r
46 = 6.2832  r
46  6.2832 = r
7.3211 = radius or 3-3 7/8
12. What is the radius of a circle with circumference of 89 11?
Answer ___________
13. What is the arc length of the outer semicircle of the
sidewalk pictured at right?
Answer _____________
What is the arc length of the inner semicircle? Answer _______
3
5
3
14. What is the diameter of a circle with a circumference of 48 11
Answer __________________
33
15. You are asked to dig a trench for direct burial cable, around the outside perimeter of the track below.
How many linear feet must you dig?
75 9
20
Answer ______________
16. What is the perimeter measurement around the inner portion of the track if the track is 5 wide?
Answer _______________
34
SQUARE MEASUREMENTS / AREA MEASUREMENTS
As previously stated square measurements tell us how much space is bound by the sides of a closed
polygon or a circle. As a carpenter you will most often measure area in square feet, square inches or square
yards. Plywood is sold by the square foot, carpeting and vinyl flooring are sold by the square yard, cabinet
doors by the square foot. Because circles and polygons have different area formulas lets look at them
individually.
SQUARE - A square is a four-sided polygon with sides of equal length and equal interior angles. The
square offers an excellent example of how area is calculated. It also provides an example of what happens
when numbers are squared.
Figure 4-3 measures 6 by 6. The same square in figure 4-4 is divided into six equal units horizontally and
six equal units vertically. When you add up all the one unit squares the sum is 36. Therefore 62 = 36
When a number is squared it is simply multiplied times itself.
Area = S2 or S  S
6 units
6 units
Area = S
2
62 = 36
6  6 = 36
figure 4-3
figure 4-4
Try these:
Note: When calculating square footage using feet and inches, first convert feet and inches to decimal feet
and then use in the formula. Leave the answer in decimal feet. Do not convert answer back to feet and
inches.
1. Calculate the area of a square with sides measuring 12 4.
Answer ___________
2. Calculate the area of a square with side measuring 120 13/16.
Answer ___________
3. Calculate the number of square inches in a cabinet door measuring 32 1/4 X 32 1/4.
Answer ___________
4.
Determine the square footage of the inner portion of the track illustrated in the previous
questions 15 and 16.
Answer ___________
35
RECTANGLE - Opposite sides are parallel and equal and all interior angles are 90.
l = length
w = width
Area = l  w
A = lw
w
l
Try these:
5. Calculate the area of a rectangle with length measuring 34 7/8 and width measuring 67 11/16.
Express answer in square feet.
Answer ___________
6. Calculate the area of a rectangle measuring 15 7 3/16 X 67 11/16. Express answer in square feet
Answer ___________
7. Calculate the area of a rectangle measuring 18 9/16 X 42 5/8. Express answer in square feet
Answer ___________
PARALLELOGRAM - Opposite sides are parallel and equal in length. Opposite interior angles are equal
and each diagonal cuts the other diagonal into equal parts (they bisect each other).
Area = bh
b = base
h = height
h
b
RHOMBUS – The rhombus is also a parallelogram except that the diagonals bisect each other at right
angles. Use the parallelogram area formula for the rhombus.
90
36
TRAPEZOID – The trapezoid has two parallel sides, which are called bases, and the altitude or height,
which is the distance measured between and perpendicular to the bases.
Area = b1 +b2
b = base
h = height
h
2
b2
h
b1
Example 4-D: What is the area of a trapezoid with base-1 = 14, base-2 = 8 and height = 17
14 + 8
2
 17
= (22  2)17 = 11  17 = 187 SF
Try this: Express answer in square feet
8. Calculate the area of the trapezoid.
13 5
h =14 1
Answer _______________
8 2
TRIANGLE - Even though triangles come in different shapes they all have two things in common. They
have three sides and the sum of their interior angles is 180.
Types of triangles:
Equilateral – All sides and angles are equal.
h or altitude
b
Isosceles – Two equal sides
h
b
37
Scalene – No equal sides
h
h
b
b
Right Triangle – One interior angle is 90 the other two are acute angles (less than 90).
h
b
Note: The triangles in the illustrations are labeled b and h.
b = base
h = height or altitude
In all of the triangles the height or altitude is the perpendicular distance, measured from a vertex to the
opposite side called the base. As in the scalene example the altitude or height is measured outside the
perimeter of the triangle.
Notice in the rectangle shown in figure 4-5 that a diagonal
has been drawn and half of the rectangle is colored.
We could say that the rectangle has been divided into two
equal triangles.
The area of the rectangle is 30 so what is the area of each triangle?
Because each triangle represents one half of the rectangle the
area of each triangle would be 30  2 = 15. Therefore, the formula
for the area of a triangle is bh
2
10
3
figure 4-5
Area of a triangle = 1/2bh or bh
2
Try these: Express answer in square feet
9. Calculate the area of the triangle
h =25 9
Answer ______________
b = 15 10
10. Calculate the area of the triangle
Answer _______________
h=29 6
32 2
38
11. Calculate the area of the triangle
28 4 5/8
Answer ____________
67 5
HERO’S FOMULA – In the event that the lengths of the sides of a triangle are known but the altitude is
unknown, the area can be calculated by using Hero’s Formula.
Area of a triangle =  s (s – a) (s – b) (s – c)
Where s is half the perimeter, or
S=a+b+c
2
Note: The symbol
means square root. A square root equals a number, when multiplied by
itself equals the given number. For example the square root of 36 is 6 because 6 multiplied by
itself equals 36. Square roots do not always come out even as in this example. The hand held
calculator saves an incredible amount of work solving for square roots.
Example 4-E: Find the area of a triangle with sides measuring 4.6, 7.2, 5.9
Step 1: Solve for s: s =
4.6 + 7.2 +5.9 = 8.85
2
c= 5.9
b =7.2
Step 2: Solve for area using Hero’s formula
Area =  8.85( 8.85 – 4.6) (8.85 – 7.2) ( 8.85 – 5.9)
a = 4.6
Not to scale
Deal with the brackets first, then multiply the final product by 8.85 then find the square root.
Area =  8.85 (4.25) (1.65) (2.95)
Area =  8.85  20.69
Area =  183.08
Area =
13.53 SF
CIRCLES – The area of a circle is solved using the following formula:
Area = r2
The symbol  stands for pi, which is represented numerically as 3.1416….. Pi represents the
circumference of a circle with a diameter of one. To prove this, roll a short piece of one-inch
dowel one revolution on a flat surface and measure the distance traveled.
39
Example 4-F: What is the area of a circle with a diameter of 7 6?
First convert feet and inches to decimal feet 7.5 and set up the formula.
3.1416  7.5 = 176.71 Sq. Ft.
Note: If you encounter a problem where diameter is given instead of radius, remember that radius
is one half the diameter.
SOLVING MORE COMPLEX AREA PROBLEMS
Many times the objects of which you want to know the area are not simple rectangles or triangles, they
are combinations of polygons and sometimes circles. Look at figure 4-6
30
30
20
20
10
10
50
Additive Method
50
Subtractive Method
Figure 4-6
What is the area of the room in figure 4-6? This problem can be approached in several ways.
Additive method - Break the object into two rectangles, solve for the area of each and add together.
30  20 = 600 SF
600 SF
20  10 = 200 SF
+200 SF
800 SF
Subtractive method - Treat the object as if it were one large rectangle and subtract out what is not
included within the boundaries of the object.
50  20 = 1000 SF
20  10  = 200 SF
1000 SF
- 200 SF
800 SF
Try these:
12. You are constructing a dome with a diameter of 24 6. What is the floor area of the dome?
Answer _________
7 6
13 Calculate the area of the object.
Express answer in square feet
8 4
Answer __________
2 0
10 5
40
14. Calculate the area of the gable end. Express
answer in square feet.
12 3
8 0
Answer _____________
25 7
15. Calculate the area of the object pictured. Express answer in
square feet.
16 0
8 0
4 0
Answer ______________
8 0
24 0
16. Calculate the square footage
of the house footprint. Express
answer in square feet.
45 0
10 0
6 0
8 0
22 0 10 0
13 6
11 0
6 0
3
3 4
12 0
4
35 0
Answer ____________
SQUARE YARDS – is a square measure used to quantify building products such as carpeting and vinyl
flooring. We all know that one yard measures 3 feet in length. A square yard would be 3  3 which would
equal 9 SF.
1 Square Yard = 9 SF
Example 4-G: A room measuring 12  10 = 120SF 120  9 = 13.33Yards
41
17. How many yards for carpeting would have to be ordered to cover the floor area of the house in
problem 16? (No waste factor to be added.)
Answer ___________
18. If the wall height of the house in problem 16 is 8 0 what is the total square footage of wall area?
Assume the length of the diagonal walls at the bay are 3 6. Answer __________
19. Using no waste factor, how many sheets of 4 x 8 drywall will it take to cover the walls in problem 16?
Answer _________
20. Using no waste factor how many sheets of 4 x 10 drywall will it take to cover the ceiling in problem
16?
Answer __________
MATERIALS BY THE SQUARE -Roofing and shingle siding are sold by the square. One square is the
area occupied within a 10 X 10  square or 100SF
1 Square = 100 SF
Example 4-H: A shed roof measures 25 0 X 15 0. How many squares of roofing
should be ordered (not including any waist factor)?
25 X 15 = 375 SF
375  100 = 3.75 Squares
21. Calculate the number of squares of shingle siding necessary to cover the walls of the house
illustrated in problem 16. Make no deductions for door and windows and use no waste factor.
Answer ____________
22. Calculate the area of the triangle. Give answer in square feet.
a= 25 4
b= 13 5
c= 17 0
c
b
a
Answer ____________
42
AREA FORMULAS
Square:
Rectangle:
Area = Side  Side
or
A = S2
S
Area = Length  Width
or
A=LW
W
L
Triangle:
Area = 1/2  base  height
or
A = 1/2bh
or
A = bh
2
h
b
Trapezoid:
Area = 1/2 (base1 + base2)  Height
or
A= 1/2(b1 + b2)  h
or
A =(b1 + b2)  h
2
b2
h
b1
Circle:
Area =   radius squared
or
A =   radius radius
or
A = r2
r
Square Yards: 1 Square Yard = 3 feet  3 feet = 9 sq. ft.
Y = Area  9 Example: A room measuring 9 feet by 10 feet would require10 yards of
carpeting. 9  10 = 9 yards
9
43