Chapter 8 (sections 8.1-8.4)
Square units are used to measure area.
AREA formulas for REGULAR polygons:
parallelogram
=
l = length
w = width
P = perimeter
b = base
h = height
d = diagonal
r = radius
m = median
a = apothem (you may need to use trig. to find this)
bh
b
h
rectangle/square =
l
l
w
triangle
lw
w
1
bh
2
=
h
b
Example 1: Find the area of the parallelogram.
80 cm
60 cm
50 cm
Example 2:
Find the height of the triangle. Area = 56
h
14
Example 3:
Find the area of an equilateral triangle. Each side is 16.
You will need to find the height first.
Example 4: Find the area of the shaded region.
11
4
4
5
Perimeter of a Triangle:
a
b
P = a + b +c
P=a+a+b
a
a
c
b
Scalene
Isosceles
a
a
P = a+ a+ a
a
Equilateral
Perimeter is always the sum of the length of the sides.
1
Area of trapezoid= (h)(b1 + b2) OR
2
A = mh
b1
h
h
m
b2
Area Kite or rhombus =
1
(d )(d2)
2 1
( this formula works for any
quadrilateral with perpendicular
diagonals)
d1
d2
Corollary 8.14: The area of a triangle with legs of lengths a and b is given by
Example 5: A right triangle has one leg measuring 12 ft. and hypotenuse of 13 ft. Find the area.
Example 6: These figures are both a rhombus.
a.
A = __________
10
12
12
10
b. A = _____________
10
12
Example 7: Given a rhombus, find the value of x
A = 56
8
x
8
Definition: The center of a regular polygon is the common center for the inscribed and
circumscribed circles of the polygon.
Definition: A radius of a regular polygon is any segment that joins the center of the regular
polygon to one of the vertices.
A
B
Definition: An apothem of a regular polygon is any line segment drawn from the center of that
polygon perpendicular to one of the sides.
A
B
Definition: A central angle of a regular polygon is an angle formed by two consecutive radii of
the regular polygon.
The measure of the central angel of a regular polygon of n sides is given by
360
c
n
Any apothem of a regular polygon bisects the side of the polygon to which it is drawn.
Read 1-6 statements on page 362(beginning of Section 8.3)
Theorem 8.3.1 is summarized by the following
A=
1
Pa
2
where: A = area, P = perimeter, a = apothem*
****an apothem is a segment drawn from the center of the polygon and is perpendicular to
the sides of the polygon.
Example 8: A regular octagon has sides of 12 cm and apothem of 14.5 cm, find the area.
Example 9:
Find the apothem (a), the area (A), and the perimeter (P) of each regular polygon.
A.. a = __________
A = _________
B. . a = __________
P = __________
A = __________
P = __________
8
11
a
a
Definitions:
A circle (symbol ) is the set of all points in a plane that are at the same distance from the
center.
The diameter is a chord through the center of a circle.
The diameter is the distance across the circle.
The circumference of a circle is the distance around the circle.
Definition:
 is a constant equal to 3.14 or 3.1416 or
8.4.1 The circumference of a circle is given by the formula C d or
C 2r .
Definition: The length of an arc is the distance between the endpoints of the arc.
8.4.2 In a circle whose circumference is C , the length l of an arc whose degree measure is
given by:

so for Arc AB

8.4.3 The area A of a circle whose radius has length r is given by A r .
2
Example 10: Find the diameter, circumference, and the area of a circle whose radius is 8
cm.
Example 11: Find the radius and circumference of a circle whose area is 49m2.
Example 12: Find the length of a 720 arc in a circle whose circumference is 45.