GEOMETRY
CHAPTER 3
Geometry & Measurement
3.1 Measuring Distance, Area
and Volume
3.2 Applications and Problem
Solving
3.3 Lines, Angles and Triangles
3.1 Rounding Measurements
To round:
1. Underline the place
2. If number to the right of the underlined place is 5 or more, add one
3. Otherwise, do not change
4. Change all digits to the right of
underlined number to zeros
3.1 Rounding Example
Example: Round 38.67 centimeters
to the nearest centimeter
1. 38.67
2. First number to the right of 8 is
“6”, so add one to 8
4. Change all digits to the right to
0’s. The answer is, 39.00 or 39
3.1 Calculating Distances
Linear Measure - a distance which
could be around a polygon (perimeter)
or around a circle (circumference)
Perimeter - sum of the lengths of
the sides
Circumference - distance around
circle C   d (Remember d  2r
)
3.1 Metric Measures
Measure can be in U.S. system
(yd, ft, etc.) or metric (cm,m, etc)
King
km
Henry
hm
Died
dam
kilometer hectometer dekameter
Drinking Chocolate
dm
cm
decimeter
centimeter
Monday
m
meter
Milk
mm
millimeter
3.1 Metric Measures
1 km = 1000 m
1 dm = 0.1 m
1 hm = 100 m
1 cm = 0.01 m
1 dam = 10 m
1 mm = 0.001 m
3.1 Linear Distance
2. What is the distance around the
polygon, in meters?
78 cm
78 + 95 +
80 + 75 = 328 cm
km
hm dam m
A. 328 m
C. 3.28 m
75 cm
95 cm
dm cm
B. 32.8 m
D. 0.328m
80 cm
mm
3.1 Calculating Areas
Rectangle
Parallelogram
Square
Triangle
b2
b1
r
Trapezoid
Circle
3.1 Area - Square Units
4. What is the area of a circular
region whose diameter is 6 cm?
Formula:
A =  r
2
  (3)
2
A. 36 sq. cm
C. 12 sq. cm
If d = 6,
then r = 3
B. 6 sq. cm
D. 9 sq. cm
3.1 Examples of Area
Surface area of a rectangular
solid
H
There are 6
faces of
the solid
L
W
Front/back
Sides (Left/Right)
Top/Bottom
A=2LH +2WH +2LW Square units
3.1 Examples of Area
6. What is the surface area
of a rectangular solid that is
12 in. long, 5 in. wide and 6
in. high?
H=6
L=12
W=5
A=2LH +2WH +2LW
A=2(12)(6)+2(5)(6)+2(12)(5)
A. 360 cubic in.
B. 324 sq. in.
C. 324 cubic in.
D. 360 sq. in.
3.1 Volume - Cubic Units
h
h
w l
r
h
Rectangular Solid
Cylinder
Cone
r
r
Sphere
V=lwh
V  r h
1
2
V  r h
2
3
4
3
V  r
3
3.1 Example of Volume
8. What is the volume of a sphere
with a 12 inch diameter?
4
If d = 12,
3
Formula: V   r
then r = 6
3
4
3
Since (6)(6)(6)= 216, the
V   (6)
3
only reasonable ans. is C
3.1 Identifying the Unit
9. Which of the following would not be
used to measure the amount of water
needed to fill a swimming pool?
A. Cubic feet
B. Liters
C. Gallons
D. Meters
linear
Think of “volume” as capacity or
filling up the inside of a 3D figure.
3.2 Application Example
1. What will be the cost of tiling a room
measuring 12 ft. by 15 ft. if square tiles
cost $2 each & measure 12 in.?
Since 12 inches = 1 ft, one tile is 1 ft on
each side or 1 sq. ft.
Area room: A = bh; (12)(15) = 180 sq ft
And (180)($2) = $360 cost
A. $180
B. $4320
C. $360 D. $3600
3.2 Pythagorean Theorem
For any RIGHT
TRIANGLE
a
c
b
c  a b
2
2
2
Side opposite
the right
angle is the
hypotenuse
“c”
3.2 Pythagorean Theorem
3. A TV antenna 12 ft. high is to be anchored
by 3 wires each attached to the top of
12
antenna and to pts on the roof 5 ft. from
base of the antenna. If wire costs $.75 per
ft, what will be the cost?
c  a b
2
2
2
c 2  (12)2 (5)2  144  25  169
c  13 and 3 wires x 13 ft = 39 ft
Cost is .75 x 39 =$29.25
c
5
A. $27.00
B. $29.25
C. $9.75
D. $38.25
3.2 Infer & Select Formulas
7. The figure shows a regular hexagon
Select the formula for total area
Total area is the area of the
6 identical triangles.
If area of 1 triangle = 1/2xbh,
then 6 x 1/2 x bh = 3 bh
A. 3h+b
B. 6(h+b) C. 6hb
h
b
D. 3hb
3.3 Lines; Angles; Triangles
ANGLES
straight angle 180
right angle
90
obtuse
> 90, < 180
acute angle
< 90
comp. sum to 90
supp. sum to 180
vertical angles-equal
TRIANGLES
Right triangle
Acute triangle
Obtuse triangle
Scalene triangle
Isosceles
Equilateral
3.3 Properties Example
2. What type of triangle is ABC?
Since sum of angles of
55
triangle = 180,
and 55 + 70 = 125,
then angle C = 180 - 125 = 55.
If 2 angles = , then isosceles.
A. Isosceles
C. Equilateral
B. Right
D. Scalene
70
C
3.3 Angle Measures
1.
B S Theorem
All B’s are = ,
All S’s are =
B + S = 180
B S
S B
B S
S B
2. Perpendicular lines intersect
to form right angles.
3.3 Angle Measures
1
2
L1
The parallel lines are
3 4
cut by transversal T
5 6
L2
7 8
Corresponding angles are = T
Terminology
1 and 5, 3 and 7, 2 and 6, 4 and 8
Vertical angles are =
1 and 4, 3 and 2, 6 and 7, 5 and 8
3.3 Angle Measures
1
2
L1
The parallel lines are
3 4
cut by transversal T
5 6
L2
7 8
T
Alternate interior angles are =
Terminology
4 and 5, 3 and 6
Alternate exterior angles are =
1 and 8, 2 and 7
3.3 Angle Measures
3. If 2 angles of a triangle
are = , then sides opposite
are =
4. If 2 sides of a triangle
are =, then angles
opposite are =
3.3
Examples
6. Which statement is
true for the figure shown
at the right given that L1
and L2 are parallel?
After using the BS
theorem, angle T
does = 75 and
angle S=105

60 75
L1 45 R
75 60 45
105 S T7545 135
L2
75 V
105 135
45
A.Since mT  75,mS  60
B.Since mT  75,mS  105
C.mV  mR
D.None
3.3 Similar Triangles
Two triangles are similar if all
angles are = and sides proportional
A
10.Which statements are true?
7.5
x
i. mA = mE
40
D
ii. AC = 6
40
B
C
iii. CE/CA = CB/CD
5
4
E
Since m  D=m  B and  DCE and  ACB are
Vertical angles m  A=m  E
A. i only B. ii only C. i and ii only D. i, ii, iii
3.3 Similar Triangles
Two triangles are similar if all
angles are = and sides proportional
A
10.Which statements are true?
7.5
x
i. mA = mE
40
D
ii. AC = 6
40
B
C
iii. CE/CA = CB/CD
5
4
E
The triangle are similar, thus ratios of corresponding
sides are =. x/4 = 7.5/5 thus x= 4(7.5)/5 = 6
A. i only B. ii only C. i and ii only D. i, ii, iii
3.3 Similar Triangles
Two triangles are similar if all
angles are = and sides proportional
A
10.Which statements are true?
7.5
x
i. mA = mE
40
D
ii. AC = 6
40
B
C
iii. CE/CA = CB/CD
5
4
E
The triangle are similar, thus ratios of corresponding
sides are =. CE/CA = CD/CB thus iii is false!
A. i
only
B. ii only C. i and ii only D. i, ii, iii
REMEMBER
MATH IS FUN
AND …
YOU CAN DO IT