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Chapter 11 - Measuring Length and Area

Areas of triangle/parellolgrams, areas of trapezoids/kites/rhombuses, perimeter/area of similar figures, circumference/arc length, areas of circles/sectors, areas of regular polygons and geometric probability
by

Christopher Cole

on 20 September 2014

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Transcript of Chapter 11 - Measuring Length and Area

Examples
.
Chapter 11
Measuring Length and Area
Prequisite Skills
Vocabulary Check
Skills and Algebra check
4. Use a formula to find the width of the rectangle that has a perimeter of 24 cm and a length of 9 cm.

In ABC, angle C is a right angle. Use the given information to find AC.

5. AB = 14, BC = 6________ 6. m<A= 35 , AB = 25________ 7. m<B = 60 , BC = 5 __________

8. Which special quadrilaterals have diagonals that bisect each other?

9. Use the proportion to find XY if UVW XYZ.
Give the indicated measure for P.
.
.
C
A
B
P
D
.
70
o
3
o
o
U
V
W
Z
Y
X
12
8
5
Postulates
Postulate 24
Postulate 25
Postulate 26
The area of a square is the square of the length of its side.
A = s
2
s
If two polygons are congruent, then they have the same area.
The area of a region is the sum of the areas of its nonoverlapping parts.
Area of a Rectangle Theorem
The area of a rectangle is the product of its base and height.
b
h
A = bh
Area of a Parallelogram Theorem
Area of a Triangle Theorem
The area of a parallelogram is the product of a base and its corresponding height.
A = bh
The area of a triangle is one half the product of its base and its correspomding height.
A = 1/2bh
Areas of Triangles and Parallelograms
Chapter 11.1
Chapter 11.2
Area of Trapezoids, Rhombuses,
and Kites
Vocabulary
The height of a trapezoid is the perpendicular distance between bases.
height
Base
Base
diagonals
diagonals
Area of a Trapezoid Theorem
The area of a trapezoid is one half the product of
the height and the sum of the lengths of the bases.
h
b2
b1
A = 1/2h(b1+b2)
Area of a Rhombus Theorem
Area of a Kite Theorem
The area of a rhombus is one half the product
of the lengths of its diagonals.
The area of a kite is one half the product of
the lengths of its diagonals.
A = 1/2d d
1
2
A = 1/2d d
d
d
1
2
1
2
d
d
1
2
Anyone know where this is?
(hint...Mr. Hill grew up near here.)
Chapter 11.3
Perimeter and Area
of Similar Polygons
Vocabulary
Do you remember...
regular polygon
corresponding sides
similar polygons
Can you label them here?
Areas of Similar Polygons Theorem
2
2
=
=
a
a
b
b
_
_
2
2
If two polygons are similar with the lengths of corresponding sides in the ratio of a : b, then the ratio of their areas a : b.

side length of Polygon I
side length of Polygon II

Area of Polygon I
Area of Polygon II
I
II
a
b
S.W.B.A.T.
Use ratios to find areas of similar figures.
Are all Twins Similar?
Are these pyramids similar polygons?
Chapter 11.4
Circumference and Arc Length
S.W.B.A.T.
Find arc lengths and other measures.
r
Vocabulary
The circumference of a circle is the distance around the circle. For all circles, the ratio of the circumference to the diameter is always the same. The ratio is known as pi or .

An arc length is a portion of the circumference of a circle. You can use the measure of the arc (in degrees) to find its length (in linear units.)
Some twins are identical...
EXTRA CREDIT!!! If you can name Pi to 10 decimals - right now???
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164
Circumference of a Circle Theorem
The circumference C of a circle C is C = d or C = 2 r, where d is the diameter and r is the radius of the circle.
π
π
π
π
π
Arc Length Corollary
In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360.
Arc length of AB
π
π
π
2 r
=
360
mAB
,
or Arc length of AB
=
360
o
o
mAB
.
2 r
.
P
A
B
r
Chapter 11.5
Areas of Circles and Sectors
S.W.B.A.T.
Find the areas of circles and sectors.
S.W.B.A.T.
Find the areas of triangles and parallelograms.
S.W.B.A.T.
Find the areas of other types of quadrilaterals.
QUIZ
Area of Circle Theorem
The area of a circle is times the square of the radius.
π
.
r
A = r
π
π
2
Area of a Sector Theorem
The ratio of the area of a sector of a circle to the ratio of the whole circle ( r ) is equal to the ratio of the measure of the intercepted arc to 360 .

area of the sector APB = mAB, or Area of the sector APB = mAB r
π
π
2
2
.
o
o
π
o
r
2
360
360
.
A
B
P
r
Vocabulary
A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.
Quiz?
Examples
Areas of Regular Polygons
Chapter 11.6
S.W.B.A.T
Find the areas of regular polygons inscribed in circles.
Vocabulary
The center of the polygon and the radius of the polygon are the center and the radius of the circumscribed circle.

The distance from the center to any side of the polygon is called the apothem of the polygon. The apothem is the height to the base of an isosceles triangle that has two radii as legs.

The central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the polygon. Find the measure of each central angle, divide 360 by the number of sides.
Area of a Regular Polygon Theorem
The area of a regular n-gon with side length s is one half the product of the apothem a and the perimeter P,

so A = 1/2 aP, or A = 1/2 a ns
.
a
s
.
P
M
N
Q
<MPN is a central angle
Finding Lengths in a Regular n-gon
To find the area of a regular n-gon with a radius (r), you may need to first find the apothem (a) or the side length (s).
You can use...
...when you know n and...
...as in...
Pythagorean Theorem: (1/2 s) + a =r

Special Right Triangles

Trigonometry
Two measures: r and a, or r and s

Any one measure: r or a or s
And the value of n is 3, 4, or 6

Any one measure: r or a or s
Example 2 and Guided Practice Ex. 3

Guided Practice Ex. 5

Example 3 and Guided Practice Exs. 4 and 5
Concept Summary
2
2
2
Who can tell me what's going on here?
Where is this?
http://www.timepassguru.com/2010/06/infinity-pool-on-skypark.html
Chapter 11.7
Use Geometric Probability
Vocabulary
The probability of an event is a measure of the likelihood that the event will occur. It is a number between 0 and 1, inclusive, and can be expressed as a fraction, decimal, or percent. The probability of event A is written P(A).

A geometric probability is a ratio that involves geometric measure such as area or length.
Key Concept
Probability and Length
Let AB be a segment that contains the segment CD. If a point K on the line AB is chosen at random, then the probability that it is on CD is the ratio of the length of CD to the length of AB.
.
.
.
.
A
B
C
D
P(k is on CD) = Length of CD
Length of AB
Key Concept
Probability and Area
Let J be a region that contains M. If a point K in J is chosen at random, then the probability that it is in region M is a ratio of the area of M to the area of J.
J
M
P(K is in region M) = Area of M
Area of J
Examples
S.W.B.A.T.
Use lengths and areas to find geometric probability.
'
Theorem from Ch6:
Perimeters of Similar Polygons
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
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