Name:
Period
Pre - AP
UNIT 11: PERIMETER AND AREA
I can define, identify and illustrate the following terms:
Perimeter
Area
Base
Height
Diameter
Radius
Circumference
Pi
Regular polygon
Radius of polygon Altitude
Apothem
Geometric probability
Composite figure
Dates, assignments, and quizzes subject to change without advance notice.
Monday
Tuesday
Block Day
Friday
15
Basic Shapes
18
19
HOLIDAY
25
Geometric Probability
Parallelograms,
Rhombus, Trapezoid,
and Regular Polygons
26
Dimensional Changes
4
5
Review
Review
20/21
Applications and
Composite Figures
22
27/28
March 1
Dimensional Changes
BA #3
Composite Figures
6/7
Test: Perimeter & Area
Friday, 2/15
Perimeter and Area of Circles, Rectangles, Squares, and Triangles
I can find the perimeter of triangles, squares, rectangles, and circles.
I can find the area of triangles, squares, rectangles, and circles.
I can find missing measurements given the area or perimeter.
PRACTICE: Basic Area and Perimeter Assignment
Tuesday, 2/19
Perimeter and Area of Parallelograms, Rhombi, Trapezoids, and Regular Polygons.
I can find the perimeter of Parallelograms, Rhombi, Trapezoids, and Regular Polygons.
I can find the area of Parallelograms, Rhombi, Trapezoids, and Regular Polygons
I can find missing measurements given the area or perimeter.
PRACTICE: Perimeter and Area of Parallelograms, Rhombi, Trapezoids, and Regular Polygons.
Block day, 2/20-21
Perimeter and Area of Applications and Composites
I can find the perimeter and area of a regular polygon.
I can find missing measurements when given an area or perimeter.
I can solve application problems using perimeter and area
PRACTICE: Perimeter and Area of Applications and Composites
Friday, 2/22
9-3: Perimeter and Area of Composite Figures
I can find the perimeter and area of a composite figure.
I can find the area of a shaded OR un-shaded region.
PRACTICE: Composite Figures Assignment
1
Monday, 2/25
9-6 Geometric Probability
I can calculate geometric probabilities.
PRACTICE: Geometric Probability Assignment
Tuesday, 2/26
9-5: Dimensional Changes
I can describe the effect on perimeter and area when one or more dimensions of a figure are
changed.
I can solve problems using dimensional changes.
PRACTICE: Dimensional Changes Assignment
Block Day, 2/27-28
CBA #3
Friday, 3/1
9-5: Dimensional Changes
I can describe the effect on perimeter and area when one or more dimensions of a figure are
changed.
I can solve problems using dimensional changes.
PRACTICE: Dimensional Changes Assignment
Monday, 3/4 AND Tuesday, 3/5
Review
PRACTICE: Review Assignment: TBA
Block Day, 3/6-7
Test 11: Perimeter and Area
I can find area and perimeter of a variety of figures using a variety of methods.
2
Basic Area and Perimeter Notes
Vocabulary: Area, perimeter, diameter, radius, height, base, Pi, in terms of Pi, altitude, diagonal, circumference
What is the perimeter of an object? _____________________________________________________________
How do you find it for ANY shape that is formed by straight lines?____________________________________
How do you find the perimeter of ANY figure? ___________________________________________________
What is the perimeter of a circle called? ________________________ Write the formula: _________________
How do you find the perimeter of a figure with straight lines and part of a circle? ________________________
What is the area of an object?_________________________________________________________________
Write the formulas for a rectangle/parallelogram: __________ What other shape can be found this way?______
Write the formula for the area of a triangle: ________ Write the formula for the area of a circle: _________
The base and height can always be found by looking for the
____________
_______________!
“In terms of Pi or π” means to NOT multiply by _________ but to leave the π in your answer.
Examples:
1. Find the perimeter and area of a
rectagle with length (s+3) and (s - 7).
3. Find the circumference and area.
Leave answers in terms of
π.
5. Find the perimeter and area.
2. Find the perimeter of a square with
an area of 64 square centimeters.
4. The area of a circle is 144 π ft 2 .
Find the circumference.
6. Find the perimeter and area.
13 in 5 in
3
13 in
Basic Area and Perimeter Assignment
1.
2.
4.
3.
6.
5.
7.
8.
9.
10.
11.
15.
16.
17.
18.
19.
20.
4
12.
21.
22.
23. Which of the following could be used to find the perimeter of the given figure?
A. 5+3+4+13+x
B. 5(3)(x)(13)
C. 5+3+13
D. 5+3+x+13
24.
26.
5
25.
27.
28. Which of the following could not be used to find the perimeter of the given figure?
18.2 cm
11.6 cm
A. 4(11.6) + 2(18.2) + 2(13x)
B. 2(11.6 +18.2+13x)
13x cm
C. 46.4 + 36.4 + 26x
13x cm
D. 11.6 + 11.6 + 11.6 + 11.6 + 18.2 + 18.2 + 13x + 13x
18.2 cm
29. Which of the following could be used to find the area of the given figure?
A. 4(3+x)
B. 4(3)(x)(13)
C. 1/2[4(3+x)]
D. 1/2[5(3+x)]
30. Which of the following could not be used to find the area of the given figure?
A. 0.5( 6 2 )( 6 2 )
12 inches
B. 0.5( 6 2 )2
C.
(6 2) 2
2
6 2
D. 0.5( 6 2 )(12)
31.
32.
6
Notes: Area and Perimeter of Parallelograms, Trapezoids, and Rhombi
Consider the following diagram with given areas:
4
10
4
10
4
4
What can you conclude about the area of a parallelogram compared to the area of a rectangle?
What is the formula for the area of a parallelogram? ___________________________
Examples:
1. Find the height of a parallelogram with base length 5 inches and area 12 inches.
What is the formula for the area of a trapezoid? ___________________________
Given: The area of trapezoid CGBE is 900 cm2..
10 cm
Prove: The perimeter of the trapezoid is___________.
(Not drawn to scale!)
50 cm
What is the formula for the area of a rhombus? ___________________________
Given that SV = 30 and RT = 40 find the area of Rhombus RSTV.
7
Notes: Regular Polygons
EXAMPLES:
Ex 1: Find the area of the regular polygon below.
A=
13 in
1
aP
2
a = _____
5
in
P = _____
For some figures you can use special right triangles to find the apothem of side length if it is not given.
Squares are made up of _____________ triangles.
Equilateral triangles and Hexagons are made up of ______________ triangles.
Label the sides of the triangle with the correct ratios.
30°
45°
60°
Ex 2:
A=
1
aP
2
Ex 3:
a = _____
4(rm+
P = _____
r
Ex 4:
1
A = aP
2
a = _____
6 cm
P = _____
8
apothem
2m
A=
1
aP
2
a = _____
P = _____
6)
Assignment: Area and Perimeter of Parallelograms, Trapezoids, and Rhombi
2.
3. Find the perimeter of the isosceles trapezoid.
4. Find the height of a trapezoid if the bases have lengths of 6 and 17 and the area of the trapezoid is 46 square units.
5.
7.
8.
9.
9
6.
10.
11.
13.
12.
14.
10
Regular Polygon and Application Worksheet
Find the area of each regular polygon.
1.
2.
3.
8 cm
x+3
Area:______________
4.
Area:______________
y
x
Area:______________
5.
Area:______________
Area:______________
6.
Area:______________
7. A regular heptagon has a perimeter of 35 feet, and an apothem of 3 3 feet. What is the area?
8. A regular octagon has side lengths of 12 inches and an apothem of 4 2 inches. What is the area?
9. The perimeter of a regular hexagon is 48 ft. What is the area of this polygon?
11
3
y–1
4
12
13
Perimeter and Area of Composite Figures
“I can …
find the perimeter and area of composite figures.”
I. Composite Figures
A. Composite figures are made up of __ or _________ geometric shapes.
B. To find perimeter:
1. Determine all _____ ____________
2. ____ like terms.
C. To find area:
1. You may find it easier to find more than you need then __________ out the extra piece(s)
2. You may find it easier to break the shape into __________ shapes and ______ all of the areas.
II. Formulas
A. Perimeter - ______ all side lengths. In a circle the perimeter is called ______
B. Area
1. Parallelogram: A = ____
2. Triangle: A = _____
3. Trapezoid: A = _______
4. Circle: A = _______
5. Regular Polygon: A = ______
6. Rhombus/Square: A = ______
Application and Composite Examples
1) Look at the rectangular prism below. Write an equation to represent the area of the shaded rectangle located
diagonally in the prism.
w
w
3w
2) A parallelogram has a base (x + 6) units and a height of (x + 2) units. If the area of the parallelogram is 60
units2, what are its dimensions?
3) The perimeter of a rectangle is 72 in. The base is 3 times the height. Find the area of the rectangle.
14
4) A trundle wheel is used to measure distances by rolling it on the ground and counting its number of turns. If
the circumference of a trundle wheel is 1 meter, what is its diameter?
5) The perimeter of each triangle is 40 units. The perimeter of the inside pentagon is 50 units. The perimeter of the
concave “star” is ______.
6)Find
the perimeter and area of the shaded region.
7. Find the shaded area
2m
3m
2m
8) A regular hexagon is to be cut out of a circular piece of paper that has a radius of 6 inches. Approximately
how many square centimeters of paper will be left over as scraps?
9) Find the area.
All regular
polygons
A = ________
15
10) Find the area and perimeter.
16 in.
D
(–h, k)
A
(–h, 0)
C (h, k)
B
(h, 0)
11)
2006 Exit
12) Look at the figure shown below.
Which expression does not represent the area of the figure?
A
B
C
D
bc − ef
af + ad − de
de + af + ad
af + cd
13) Which of the following could be used to find the area of the shaded region?
 16π 
A 
 (16 )( 8 )
 2 
8m
B(16π ) + (16 )( 8 ) − [.5(6 + 2)(8)]
6m
16 m
C  1 16π  + (16 )( 8 ) 6 + [ 1 (6 + 2)(8)] D  16π  + (16 )( 8 ) − [.5(6 + 2)(8)]
2
2

 2 
8m
9m
3m
14) Mr. Ike wants to put brown tile in his living room except in the center where he wants ivory tile in a square
shape. The diagram below shown the layout of the room. If each tile is a 6 inch square, how many brown tiles
will he need? How many ivory tiles?
42 in
10 ft
24 ft
16
15) Delta’s backyard is rectangular. Its dimensions are 15 m by 10 m. Delta’s family is making a garden from
the patio doors to the corners at the back of the yard. The patio doors are 2 m wide. If 1 m is approximately 3.28
feet, what is the area of the garden in square feet?
Wrap-up:
1. How did you figure out the perimeter of a composite figure? Write a few sentences explaining your process
to someone having trouble with this concept.
2. How did you figure out the area of a composite figure? Write a few sentences explaining your process to
someone having trouble with this concept.
Applications and Composite Assignment
1) The area of a triangle is 50 cm2. The base of the triangle is 4 times the height. Find the height of the triangle.
2) The length of a rectangle is four less than three times its width. Write the expression to find the perimeter. If
the rectangle has a perimeter of 22 inches, what is its width?
3) The area of a circle is six times the radius. Write and equation and use it to solve for the radius. Leave your
answer in terms of π.
4) A rectangular sheet of paper has dimensions of (x + 2) and (x + 3). The area of the paper is 61 square feet.
What are the dimensions of the paper?
5) The perimeter of an isosceles trapezoid is 40 ft. The bases of the trapezoid are 11 ft and 19 ft. Find the area of
the trapezoid.
17
6) Two circles have the same center. The radius of the larger circle is 5 units longer than the radius of the
smaller circle. Find the difference in the circumferences of the two circles.
7) A stop sign is a regular octagon. The signs are availiable in two sizes: 30 in. or 36 in.
a) Find the area of a 30 in. sign.
b) Find the area of a 36 in. sign
c) Find the percent increase in metal needed to make a 36 in. sign instead of a 30 in. sign.
8) Alisa has a circular tabletop with a 2-foot diameter. She wants to paint a pattern on the table top that includes
a 2-foot-by-1-foot rectangle and 4 squares with sides 0.5 foot long. Which information makes this scenario
impossible?
A There will be no room left on the tabletop after the rectangle has been painted.
B A 2-foot-long rectangle will not fit on the circular tabletop.
C Squares cannot be painted on the circle.
D. There will not be enough room on the table to fit all the 0.5-foot squares.
9) The area of a parallelogram is thirty-four square inches. Write the factors that can be used to solve for x.
A = (x2 – 5x – 50)
10) The dimensions of the rectangular prism below are 8k′ by 10k′ by 12k′. What is the area of the shaded
rectangle located diagonally inside the prism?
8′
10′
12′
11 – 12: Find the area and outside perimeter for each figure. Assume all angles are right angles
11. A = _____________
P = _____________
12. A = ________
C
(0,k)
P = _________
A
(0,0)
B
(k,0)
13-16. Find the area of the shaded regions and the outside perimeter. Leave answer as simplified radicals and in
terms of Pi.
13. A = ________
14.A = ________
2m
P = _________
18
P = _________
2m
3m
15.
16.
All regular
polygons
10 in.
12 in.
All regular
polygons
A = ________
A = ________
17. Mr. Ike wants to put brown tile in his living room except in the center where he wants ivory tile in a square
shape. The diagram below shown the layout of the room. If each tile is a 4 inch square, how many brown tiles
will he need? How many ivory tiles?
42 in
12 ft
24 ft
2003 Exit
18. Find the equation that can be used to determine the total area of the composite figure shown below.
1
A A = lw + w2
2
B A = lw + w2
C A = w + 2l + w 2
1
D A = w + 2l + w 2
2
19. A store sells circular rugs in two different sizes. The rugs come in diameters of 9 ft and 12 ft. Find the difference in
the areas of the two rugs. Round to the nearest tenth.
20. A circle with a diameter of 12 centimeters is to be cut out from a square piece of paper that measures 12
centimeters on each side, as shown below. Which is closest to the amount of paper that will be left over after the
circle is cut out?
12cm
F 31 cm2
H 21 cm2
G 25 cm2
J 9 cm2
12cm
19
21.
22.
1 in = 2.54 cm
23.
1 ft = 0.30 m
25.
24.
26. Four square pieces are cut from the corners of a square sheet of metal. As the size of the small squares
increases, the remaining area decreases, as shown below.
If this pattern continues, what will be the difference between the first square’s shaded area and the fifth square’s
shaded area?
A 4 square units
20
B 24 square units
C 49 square units
D 96 square units
27.
28.
29. The area of a rectangle is (x2 – 9x + 20) square units. If the area is six square units, what factors can be used
to solve for x?
30.
31.
32.
33. A diagram of a ticket stub is shown below. The shaded region is composed of a rectangle and a cut-out of a
semicircle with a diameter 8 cm. If 1 inch is approximately equal to 2.54 cm, what is the area of the shaded
region in square inches?
8 in
21
5 in
14 in
34. The dimensions of a puzzle block are shown below.
Cut it up differently!
22 cm
5 cm
8 cm
6 cm
10 cm
4 cm
Write three different number equations that can be used to find the composite area.
A. __________________________________________________________________
B. __________________________________________________________________
C. __________________________________________________________________
35. Rectangle ABCD is graphed on the coordinate
grid below. Which of the following equations
F
G
H
J
best represents a line perpendicular to
4
y = x + 11
5
4
y = – x + 11
5
5
y = x + 11
4
5
y = – x + 11
4
CD ?
36. If AD is the perpendicular bisector of
∆ABC with vertices A(7, -7), B(2, 4),
and C(-4, -2), what is the length of
A
3 2
6
B
C
D
BD ?
6
18
37. On Triangle TOY, Point T (–6, 9) and Point O (9, 1) are graphed on the coordinate grid below.
If Segment YK is a median, what is the length of segment TK? ________________________
*recall that a median bisects a side of a triangle!
Method A: find midpoint K, use dist formula
T
K
Method B: use dist formula for TO, divide by 2
Y
22
O
9.6: Geometric Probability
“I can …find the geometric probability involving lengths and area.”
I. Probability
A. Recall: Probability is a number between __ and __.
B. Probability =
C. Geometric Probability – A probability involving a geometric measure, such as __________ or
_______.
III. Model Problems
1. A circle is drawn inside a rectangle. Point W in the rectangle is chosen at random. What is the probability
that W lies in the shaded region?
5 in.
12 in.
2. A dartboard has 3 scoring sections formed by concentric circles with radii 5 in, 10 in, and 12 in. If a dart hits
the board at a random point, what is the probability that it hits the middle section?
5 in
12 in 10 in
3. A map of a town is shown. If a random point on the map is chosen, what is the probability that it is on the
east side of town?
2 mi
West
7 mi
East
5 mi
4. A swimming pool is surrounded by a fenced-in deck as shown below. If a ball is randomly tossed into the
fenced area, what is the probability that it lands in the pool?
10 yd
15 yd
25 yd
40 yd
5. Two concentric circles have known radii of 4 cm and 6cm. Point M is chosen at random. What is the
probability that M is inside the larger circle but outside the smaller circle?
23
Use the picture below for questions #1-3
13 cm
B
8 cm A
C
8 cm
5 cm
4 cm
3 cm
Use the picture for questions# 7-9
1. What is the probability of throwing a dart on
section A if thrown at random?
Area of A
7. What is the probability of landing on a shaded
region?
=
Total area
2. What is the probability of randomly throwing a
dart and landing on A or C ?
8. What is the probability of landing on the square?
Area of A+C
=
Total area
3. What is the probability of randomly throwing a
dart and NOT landing on C ?
Area of A+B
9. What is the probability of landing on the triangle?
Use the picture for questions #10-12
=
Total area
2m
Use the picture for questions #4-6
2m
3m
4. Find the probability of choosing
pasta at random.
10. What is the probability that a dart thrown at
random will hit a shaded region?
Degrees of pasta section
=
Total degrees in a circle
5. Find the probability of NOT choosing peach
cobbler at random.
11. What is the probability that a dart thrown at
random will hit the bull’s eye? (center)
6. What is the probability of choosing peas at
random?
12. What is the probability that a dart thrown at
random will hit the white section of the board?
24
13.
14.
17) A square is drawn inside a rectangle. Point A
in the rectangle is chosen at random. What is the
probability that A lies in the shaded region?
9m
15 m
16
15.
25
Dimensional Changes
Use the figures below to answer questions #1 - 6.
A
10 cm
E
15 cm
F
B
4 cm
6 cm
1. Find the scale factor of the sides
D
2. Find the perimeter of ABCD
C
H
G
3. Find the perimeter of EFGH
4. Find the scale factor of the perimeters (EFGH / ABCD)
6. Find the area of ABCD
7. Find the area of EFGH
8. Find the scale factor of the areas (EFGH/ABCD)
9. How does the scale factor of the sides compare to the scale factor of the area?
10 Tony and Edwin each built a rectangular garden. Tony’s garden is twice as long and twice as wide as
Edwin’s garden. If the area of Edwin’s garden is 600 square feet, what is the area of Tony’s garden?
11 The similarity ratio of two similar polygons is 3:5. The perimeter of the larger polygon is 150 centimeters.
What is the perimeter of the smaller polygon?
2003 9th grade
13. Describe the effect on the area of a circle when the radius is doubled.
F
G
H
J
1
.
2
The area remains constant.
The area is doubled.
The area is increased four times.
The area is reduced by
2004 9th grade
14. The similarity ratio of two similar polygons is 2:3. The perimeter of the larger polygon is 150 centimeters.
What is the perimeter of the smaller polygon?
A
B
C
D
26
100 cm
75 cm
50 cm
150 cm
27
28
Effects of Changing Dimensions Proportionally – Recall from Tuesday
Changing both dimensions:
1. Describe the effect on the perimeter and area of the rectangle if the base and height are multiplied
by 4.
6
2
Perimeter: ______
Conclusion:
The perimeter changes by a SF of
__________________________.
Perimeter: ______
Area: ______
The area changes by a SF of
_________________________.
Area: ______
Changing a circle:
2. Describe the effect on the area and circumference of a circle if the radius is doubled.
Conclusion:
The circumference changes by a SF of
_____________________________.
7
The area changes by a SF of
____________________________.
Circumference: ______
Area: ______
Circumference: ______
Area: ______
Changing one dimension:
3. Describe the effect on the area of the rectangle if the height is tripled.
6
2
Conclusion:
The area changes by a SF of
____________________________.
Area: ______
29
Area: ______
Effects of Changing Dimensions Proportionally
Change in Dimensions
Perimeter or Circumference
Area
All dimensions
multiplied by the
_____________ by the
_____________ by the
_________________
_________________
_________________
Describe the effect of each change on the perimeter (circumference) and the area of
the given figure:
1. The base and height of the triangle are both doubled.
2. The radius of E is multiplied by ¼.
Describe the effect of each change on the area of the given figure:
3. The diagonals of a rhombus are both multiplied by 8.
4. The base of a rectangle is multiplied by 4, and the height is multiplied by 7.
5. The diagonal of a square is divided by 4.
6. One diagonal of a kite is multiplied by 1 7 .
Practice: Pg 625-626 #9 – 21 odd, 22, 30 – 35 all
30