9.2 – Curves, Polygons, and Circles
Curves
The basic undefined term curve is used for describing nonlinear figures in a plane.
A simple curve can be drawn without lifting the pencil from
the paper, and without passing through any point twice.
A closed curve has the same starting and ending points, and is
also drawn without lifting the pencil from the paper.
Simple;
closed
Simple; not
closed
Not simple;
closed
Not simple;
not closed
9.2 – Curves, Polygons, and Circles
Polygons
A polygon is a simple, closed curve made up of only straight
line segments.
The line segments are called sides.
The points at which the sides meet are called vertices.
Polygons with all sides equal and all angles equal are regular
polygons.
Polygons
Regular Polygons
9.2 – Curves, Polygons, and Circles
A figure is said to be convex if, for any two points A and B
inside the figure, the line segment AB is always completely
inside the figure.
E
B
A
F
M
C
D
Convex
N
Not convex
9.2 – Curves, Polygons, and Circles
Classification of Polygons According to Number
of Sides
Number of
Sides
Name
Number of
Sides
Names
3
Triangle
13
Tridecagon
4
Quadrilateral
14
Tetradecagon
5
Pentagon
15
Pentadecagon
6
Hexagon
16
Hexadecagon
7
Heptagon
17
Heptadecagon
8
Octagon
18
Octadecagon
9
Nonagon
19
Nonadecagon
10
Decagon
20
Icosagon
11
Hendecagon
30
Triacontagon
12
Dodecagon
40
Tetracontagon
9.2 – Curves, Polygons, and Circles
Types of Triangles - Angles
All Acute Angles
One Right Angle
One Obtuse Angle
Acute Triangle
Right Triangle
Obtuse Triangle
9.2 – Curves, Polygons, and Circles
Types of Triangles - Sides
All Sides Equal
Equilateral Triangle
Two Sides Equal
No Sides Equal
Isosceles
Triangle
Scalene
Triangle
9.2 – Curves, Polygons, and Circles
Quadrilaterals: any simple and closed four-sided figure
A trapezoid is a quadrilateral with one pair of
parallel sides.
A parallelogram is a quadrilateral with two pairs
of parallel sides.
A rectangle is a parallelogram with a right angle.
A square is a rectangle with all sides having equal
length.
A rhombus is a parallelogram with all sides having equal length.
9.2 – Curves, Polygons, and Circles
Triangles
Angle Sum of a Triangle
The sum of the measures of the angles of any triangle is 180°.
Find the measure of each angle in the triangle below.
x°
(x+20)°
(220 – 3x)°
x + x + 20 + 220 – 3x = 180
–x + 240 = 180
– x = – 60
x = 60
x = 60°
60 + 20 = 80°
220 – 3(60) = 40°
9.2 – Curves, Polygons, and Circles
Exterior Angle Measure
The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two opposite interior angles.
2
Exterior angle
4
1
3
The measure of angle 4 is equal to the sum of the measures of
angles 2 and 3.
m4 = m2 + m3
9.2 – Curves, Polygons, and Circles
Find the measure of the exterior indicated below.
(x+20)°
(3x – 50)°
(x – 50)°
3x – 50 = x + x + 20
3x – 50 = 2x + 20
3x = 2x + 70
x = 70
3(70) – 50
160°
x°
9.2 – Curves, Polygons, and Circles
Circles
A circle is a set of points in a plane, each of which is the same
distance from a fixed point (called the center).
A segment with an endpoint at the center and an endpoint on the circle is
called a radius (plural: radii).
A segment with endpoints on the circle is called a chord.
A segment passing through the center, with endpoints on the circle, is
called a diameter.
A diameter divides a circle into two equal semicircles.
A line that touches a circle in only one point is called a tangent to the
circle.
A line that intersects a circle in two points is called a secant line.
A portion of the circumference of a circle between any two points on the circle is called an arc.
9.2 – Curves, Polygons, and Circles
O is the center
P
M
OQ is a radius.
LM is a chord.
Q
O
PR is a diameter.
PQ is an arc.
PQ is a secant line.
(PQ is a chord).
L
R
T
RT is a tangent line.
9.2 – Curves, Polygons, and Circles
Inscribed Angle
An inscribed angle is an angle whose vertex is on the circle and the
sides of the angle extend through the circle and touch or go beyond the
points on the circle.
BAC is an inscribed angle
B
x
C
A
Any angle inscribed in a semicircle must be a right angle.
B
A
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Congruent Triangles
Congruent triangles: Triangles that are both the same size
and same shape.
E
B
A
D
F
The corresponding sides are congruent.
The corresponding angles have equal measures.
Notation:
ABC  DEF.
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Using Congruence Properties
Side-Angle-Side (SAS) If two sides and the included angle of
one triangle are equal, respectively, to two sides and the
included angle of a second triangle, then the triangles are
congruent.
B
E
A
D
F
ABC  DEF.
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Using Congruence Properties
Angle-Side-Angle (ASA) If two angles and the included side
of one triangle are equal, respectively, to two angles and the
included side of a second triangle, then the triangles are
congruent.
E
B
A
D
F
ABC  DEF.
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Congruence Properties - SSS
Side-Side-Side (SSS) If three sides of one triangle are equal,
respectively, to three sides of a second triangle, then the
triangles are congruent.
B
E
A
D
F
ABC  DEF.
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Proving Congruence
Given: CE = ED
C
B
AE = EB
Prove:
ACE  BDE
STATEMENTS
A
E
D
REASONS
1. CE = ED
1. Given
2. AE = EB
2. Given
3. AEC = BED
3. Vertical angles are equal
4. ACE  BDE
4. SAS property
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Proving Congruence
Given: ADB = CBD
B
C
ABD = CDB
Prove:
A
ABD  CDB
STATEMENTS
REASONS
1. ADB = CBD
1. Given
2. ABD = CDB
2. Given
3. BD = BD
3. Reflexive property
4. ABD  CBD
4. ASA property
D
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Proving Congruence
B
Given: AD = CD
AB = CB
Prove:
ABD  CBD
STATEMENTS
A
D
REASONS
1. AD = CD
1. Given
2. AB = CB
2. Given
3. BD = BD
3. Reflexive property
4. ABD  CBD
4. SSS property
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Isosceles Triangles
If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint
of the base AC, then the following properties hold.
B
1. The base angles A and C are equal.
2. Angles ABD and CBD are equal.
3. Angles ADB and CDB are both right angles.
A
D
C
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Similar Triangles: Triangles that are exactly the same shape,
but not necessarily the same size.
For triangles to be similar, the following conditions must hold:
1. Corresponding angles must have the same measure.
2. The ratios of the corresponding sides must be constant.
That is, the corresponding sides are proportional.
Angle-Angle Similarity Property
If the measures of two angles of one triangle are equal to those
of two corresponding angles of a second triangle, then the two
triangles are similar.
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
ABC is similar to DEF.
E
B
8
16
F
24
D
C
32
A
Find the length of side DF.
Set up a proportion with corresponding sides:
EF DF

BC AC
8 DF

16 32
DF = 16.
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Pythagorean Theorem
leg a
c (hypotenuse)
leg b
If the two legs of a right triangle have lengths a and b, and the
hypotenuse has length c, then
a 2  b2  c 2 .
(The sum of the squares of the lengths of the legs is equal to the square
of the hypotenuse.)
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Find the length a in the right triangle below.
39
a
36
a b  c
2
2
2
a2  362  392
a2  1296  1521
2
a  225
a  15
9.3 –Triangles: Congruence, Similarity and the
Pythagorean Theorem
Converse of the Pythagorean Theorem
If the sides of lengths a, b, and c, where c is the length of the
longest side, and if a 2  b 2  c 2 ,
then the triangle is a right triangle.
Is a triangle with sides of
length 4, 7, and 8, a right
triangle?
Is a triangle with sides of
length 8, 15, and 17, a right
triangle?
42  72  82
16  49  64
65  64
Not a right triangle.
right triangle.
9.4 – Perimeter, Area, and Circumference
Perimeter of a Polygon
The perimeter of any polygon is the sum of the measures of
the line segments that form its sides. Perimeter is measured in
linear units.
Perimeter of a Triangle
a
b
c
The perimeter P of a triangle with sides of lengths a, b, and c
is given by the formula:
P = a + b + c.
9.4 – Perimeter, Area, and Circumference
Perimeter of a Rectangle
w
l
The perimeter P of a rectangle with length l and width w is given by the
formula:
P = 2l + 2w or P = 2(l + w).
s
Perimeter of a Square
s
The perimeter P of a square with all sides of length s is given by the
formula:
P = 4s.
9.4 – Perimeter, Area, and Circumference
Area of a Polygon
The amount of plane surface covered by a polygon is called its
area. Area is measured in square units.
Area of a Rectangle
l
w
The area A of a rectangle with length l and width w is given by
the formula:
A = l w.
Area of a Square
s
s
The area A of a square with all sides of length s is given by the
formula:
P = s2.
9.4 – Perimeter, Area, and Circumference
Area of a Parallelogram
h
b
The area A of a parallelogram with height h and base b is
given by the formula:
A = bh.
b1
h
Area of a Trapezoid
b2
The area A of a trapezoid with parallel bases b1 and b2 and
height h is given by the formula:
A = (1/2) h (b1 + b2)
9.4 – Perimeter, Area, and Circumference
h
Area of a Triangle
b
The area A of a triangle with base b and height h is given by
the formula:
A = (1/2) b h
9.4 – Perimeter, Area, and Circumference
Find the perimeter and area of the rectangle.
Perimeter
P = 2l + 2w
15 ft
2(15) + 2(7)
7 ft
P = 44 ft
Area
A = lw
A = (15)(7)
A = 105 ft2
Find the area of the trapezoid.
A = (1/2) h (b1 + b2)
A = (1/2) (5) (7 + 13)
A = (1/2) (5) (20)
A = 50 cm2
7 cm.
6 cm.
5 cm.
13 cm.
6 cm.
9.4 – Perimeter, Area, and Circumference
Find the area of the shaded region.
Area of square – Area of triangle
s2 – (1/2) b h
42 – (1/2) (4)(4)
16 – 8
8 in2
4 in.
4 in.
9.4 – Perimeter, Area, and Circumference
Circumference and Area of a Circle
d
r
The circumference C of a circle of diameter d is given by the
formula: C   d , or C  2 r , where r is a radius.
The area A of a circle with radius r is given by the formula:
2
A r .
9.4 – Perimeter, Area, and Circumference
Find the area and circumference of a circle with a radius that is 6 inches
long (use 3.14 as an approximation for ).
Circumference (  3.14)
Circumference ()
C=2r
C=2r
C = 2 (3.14) 6
C=26
C = 37.68 in
C = 37.699 in
Area (  3.14)
Area ()
A =  r2
A = (3.14) 62
A =  r2
A =  62
A = 113.04 in2
A = 113.097 in2
9.5 – Space Figures, Volume, and Surface Area
Space figures: Figures requiring three dimensions to
represent the figure.
Polyhedra: Three dimensional figures whose faces are made
only of polygons.
Regular Polyhedra: A polyhedra whose faces are made only
of regular polygons (all sides are equal and all angles are
equal.
9-437
9.5 – Space Figures, Volume, and Surface Area
Other Polyhedra
Pyramids are made of triangular sides and a polygonal base.
Prisms have two faces in parallel planes; these faces are
congruent polygons.
9.5 – Space Figures, Volume, and Surface Area
Other Space Figures
Right Circular Cones have a circle as a base and the surface
tapers to a point directly above the center of the base.
Right Circular Cylinders have two circles as bases, parallel
to each other and whose centers are directly above each other.
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume is a measure of capacity of a space figure. It is
always measured in cubic units.
Surface Area is the total region bound by two dimensions. It
is always measured in square units.
Volume and Surface Area of a Rectangular solid (box)
The volume V and surface area S of a box with length l, width
w, and height h is given by the formulas:
h
l
w
V = lwh
and
S = 2lw + 2lh + 2hw
9.5 – Space Figures, Volume, and Surface Area
Find the volume and surface area of the box below.
3 in.
7 in.
2 in.
V = 7(2)(3)
S = 2(7)(2) + 2(7)(3) + 2(3)(2)
V = 42 in.3
S = 28 + 42 + 12
S = 82 in.2
9-441
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume and Surface Area of a Cube
The volume V and surface area S of a cube with side lengths of
s are given by the formulas:
s
s
V = s3
and
S = 6s2
s
Find the volume and surface area of the cube below.
V = 53
V = 125 ft.3
5 ft.
S = 6(5)2
S = 625
S = 150 ft.2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume of Surface Area of a Right Circular Cylinder
The volume V and surface area S of a right circular cylinder
with base radius r and height h are given by the formulas:
V = r2h
h
and
r
S = 2rh + 2r2
Find the volume and surface area of the cylinder below.
10 m
2m
V = (2)2(10)
V = 40
V = 125.6 m3
S = 2(2)(10) + 2(2)2
S = 40 + 8 = 48
S = 150.72 m2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume of Surface Area of a Sphere
The volume V and surface area S of a sphere radius r are given
by the formulas:
V = (4/3) r3
r
and
S = 4  r2
Find the volume and surface area of the sphere below.
9 in.
V = (4/3)(9)3
V = 972
V = 3052.08 in.3
S = 4 (9)2
S = 324
S = 1017.36 in.2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume of Surface Area of a Right Circular Cone
The volume V and surface area S of a right circular cone with
base radius r and height h are given by the formulas:
9.5 – Space Figures, Volume, and Surface Area
Find the volume and surface area of the cone below.
h=4m
r=3m
V = (1/3)(3)2(4)
V = 12
V = 37.68 m3
S = 15 + 9
S = 24
S = 75.36 m2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume of a Pyramid
The volume V of a pyramid with height h and
base of area B is given by the formula:
Note: B represents the area of the base (l w).
Find the volume of the pyramid (rectangular base) below.
7 cm
3 cm
6 cm
cm3