Chapter 11:
Areas of Plane Figures
Areas of Rectangles
11-1:
Rectangle
Area is measured in
Postulate 17
(page 422)
(page 423)
Rectangular Region
units.
The area of a square is the square of the length of a
.
A=
s
Postulate 18
If two figures are congruent, then they have the same
Postulate 19
The area of a region is the
overlapping parts.
.
of the areas of its non-
Any side of a rectangle (parallelogram) can be considered to be a
(
).
Any segment perpendicular to the line containing the base from any point on the
opposite side is an
.
The length of an altitude is called the
Theorem 11-1
(
).
The area of a rectangle equals the
of its base and height.
A=
h
b
Examples:
(1)
The area of a square is 9 sq. cm. Find its perimeter.
P=
(2)
The perimeter of a rectangle is 20 cm.
If its height is 4 cm, find its area.
A=
(3)
Consecutive sides of the figure are perpendicular.
Find its area.
A=
9
3
6
4
7
4
1
1
5
9
Assignment: Written Exercises, Pages 426 & 427: 1-19 & 25 -29 odd #’s
11-2:
Areas of Parallelograms, Triangles, and Rhombuses
Any side of a parallelogram may be considered a
(
(page 429)
).
Any segment perpendicular to the line containing a base from any point on the opposite
side is an
.
The length of an altitude is called the
Theorem 11-2
Theorem 11-3
(
), of the parallelogram.
The area of a parallelogram equals the
product of a base and the height to that base.
A=
The area of a triangle equals
the
product of a base and the height to that base.
A=
Hero’s (or Heron’s) Formula: a triangle area formula in terms of the sides of the triangle.
A=
A
s=
B
Theorem 11-4
The area of a rhombus equals
the product of its diagonals.
C
A=
Examples:
(1)
Find the area of a parallelogram with base 12 and height 4.
A=
(2)
Find the area of the parallelogram.
A=
15 m
10 m
60º
(3)
Find the area of the triangle.
13
A=
13
10
(4)
Find the area of the rhombus.
A=
5
4
Assignment: Written Exercises, Pages 431 & 432: 1-17 odd #’s
11-3:
Areas of Trapezoids
In a trapezoid, the bases are the
The altitude of a trapezoid is any segment
from a point on the opposite base.
In a trapezoid, all altitudes have the same
Theorem 11-5
(page 435)
sides.
to a line containing one base
, called the
(
).
The area of a trapezoid equals
the product of the height and the sum of the bases.
A=
The median of a trapezoid is the segment connecting the
The formula for the length of the median is:
of the legs.
.
Therefore, another area formula for a trapezoid is:
A=
=
.
Examples:
(1)
Find the area of the trapezoid.
A=
5
3
|------------------ 15 ---------------------|
(2)
Find the area of the trapezoid.
A=
11
8
h
60º
x
(3)
11
Find the length of the median and the area of the
trapezoid that has bases 18 & 24 and height 16.
m=
A=
(4)
If the area of the trapezoid is 128 u2 and its bases are 12 & 20,
find the height.
h=
Assignment: Written Exercises, Pages 436 & 437: 1-17 odd #’s
11-4:
Student Activity
Areas of Regular Polygons
(page 440)
Using a compass, construct a circle with a given radius.
Mark off congruent arcs with chords equal to the radius of the circle.
Connect the consecutive points on the circle.
The figure formed is a
A regular polygon is both
.
and
Any regular polygon can be inscribed in a
.
.
Many of the terms associated with circles are also used with regular polygons.
The
of a regular polygon is the center of the circumscribed circle.
The
of the regular polygon is the distance from the center to a vertex.
A
consecutive vertices.
is an angle formed by two radii drawn to two
The measure of a central angle of a regular polygon with “n” sides is
The
of a regular polygon is the distance from the center to a side.
.
Theorem 11-6
The area of a regular polygon is equal to
apothem and the perimeter.
the product of the
A=
explanation of proof:
Common Regular Polygons to Know
(1)
Regular Triangle
a.k.a.:
(2)
Regular Quadrilateral
a.k.a.:
(3)
Regular Hexagon
Examples:
(1)
Find the perimeter and area of a regular triangle with its apothem equal to 9 feet.
p=
A=
(2)
Find the apothem and radius of a regular quadrilateral with an area of 100 sq.yd.
a=
r=
(3)
Find the area of a regular hexagon with a side equal to 12.
A=
Assignment: Written Exercises, Page 443: 1-11 odd #’s
11-5:
Note
Circumferences and Areas of Circles
(page 445)
The perimeter of a polygon is defined as the sum of the lengths of the segments
making up its sides. A circle is not made up of line segments and therefore, the
perimeter of a circle must be defined differently.
Student Activity
Look at a sequence of regular polygons.
Imagine more and more regular polygons having more and more sides.
Look at the perimeters of the polygons. The perimeter is approaching the distance
around a circle that is
about the polygon.
This is defined to be the
,(
) of the circle.
The ratio of the circumference to the diameter is
This constant is denoted by the Greek letter
value for this ratio.
Therefore:
C =
in all circles.
, (pi), which represents the exact
=
The area of a circle is defined in a similar way. The areas of the inscribed regular
polygons get closer and closer to a
This limit is defined to be the
number.
of the circle.
As the regular polygon gets closer to looking like a
, the length of the
…a⇒
apothem approaches the length of the
Substitute for “a” and “p” into the
area formula for a regular polygon:
A = 1/2 a p
A=
A=
Therefore, the area formula for a circle is:
A=
and p ⇒
.
Note
Since π is an irrational number, there isn’t any decimal or fraction that expresses
π exactly. Some approximated values for π are:
,
,
, and
.
Examples:
(1)
Find the circumference and area of a circle with its radius equal to 8 3 .
C=
A=
(2)
Find the circumference of a circle if the area is 100π sq.u.
C=
(3)
A circular garden has a radius of 14 feet. What is the area of the garden?
Use π = 22/7.
answer:
(4)
A bicycle wheel has a diameter of 60 cm. How far will it travel if it makes 50 revolutions?
Use π = 3.14.
answer:
(5)
The earth has a diameter of approximately 7,913 miles. What is its circumference?
answer:
Assignment: Written Exercises, Pages 448 to 450: 1-17 odd #’s, * Bonus #26 *
cm
Arc Lengths and Areas of Sectors
11-6:
(page 452)
A
Two different numbers that describe the size of an arc are.
!.
mAB
B
O
(1)
the measure of the arc, ie.
(2)
the arc length, which is the
of a piece of the circumference.
The arc length is a fraction of the whole circumference.
! = x, then the length of AB
! = ____________________________
If mAB
The length of the arc equals the fraction times the circumference of the circle.
A
of a circle is a region bounded by two radii and an arc of the circle.
The area of a sector is a fraction of the area of a whole circle.
! = x, then the area of sector AOB = ____________________________
If mAB
The area of the sector equals the fraction times the area of the circle.
Examples:
(1)
" and ACD
#.
In ! O with radius 6 and m!AOB = 150º, find the lengths of AB
B
10º
D
A
O
C
! = _________
length of AB
! = _________
length of ACD
(2)
Find the area of the shaded sector in the circle with radius equal to 12.
A=
40º
(3)
Find the area of the shaded region in the circle with radius equal to 6.
A=
120º
(4)
Find the area of the shaded region in the circle with radius equal to 2 inches.
A=
300º
Assignment: Written Exercises, Pages 453 & 454: 1-15 odd #’s
4
Ratios of Areas
11-7:
(page 456)
3
Comparing Areas of Triangles - refer to classroom exercises on page
458, #1 and 2.
8
(1)
If two triangles have equal heights, then the ratio of their areas equals
the ratio of their
(2)
9
If two triangles have equal bases, then the ratio of their areas equals
the ratio of their
(3)
.
.
If two triangles are similar, then the ratio of their areas equals
the square of their
Theorem 11-7
.
If the scale factor of two similar figures is a : b , then:
(1) the ratio of the perimeters is
(2) the ratio of the areas is
.
.
Examples:
(1)
The scale factor of two similar figures is 3:5 . Find the ratio of the perimeters and
the ratio of the areas.
ratio of perimeters =
ratio of areas
(2)
=
The ratio of the areas of two similar figures is 1:4 . Find the ratio of their perimeters.
ratio of perimeters =
(3)
(4)
The areas of two circles are 36π and 64π. Find the ratios of the diameters and
circumferences.
ratio of diameters
=
ratio of circumferences
=
The scale factor of two quadrilaterals is 3:5. The area of the smaller quadrilateral is 27 in2.
Find the area of the larger quadrilateral.
A of larger quadrilateral =
Assignment: Written Exercises, Pages 458 & 459: 1-15 odd #’s
Prepare for Test on Chapter 11: Areas of Plane Figures