ACCELERATED MATHEMATICS
CHAPTER 10
MEASUREMENTS OF POLYGONS AND CIRCLES
TOPICS COVERED:
•
•
•
•
•
•
•
•
•
•
Perimeter of polygons
Area of rectangles and squares
Area of parallelograms
Area of triangles
Area of trapezoids
Circle vocabulary: radius, diameter, chord, tangent, secant
Discovering pi
Circumference of circles
Area of circles
Area of composite shapes
Created by Lance Mangham, 6th grade math, Carroll ISD
Name:
Accelerated Mathematics Formula Chart
Linear Equations
Slope-intercept form
y = mx + b
Constant of proportionality
k=
y
x
y = kx (8th grade)
Slope of a line
m=
y2 − y1 th
(8 grade)
x2 − x1
C = 2π r or C = π d
Circle
Circumference
Direct Variation
Area
1
(b1 + b2 ) h
2
Rectangle
A = bh
Trapezoid
A=
Parallelogram
A = bh
Circle
A = π r2
Triangle
A=
bh
1
or A = bh
2
2
Surface Area (8th grade)
Lateral
Prism
Cylinder
Total
S = Ph
S = Ph + 2 B
S = 2π rh
S = 2π rh + 2π r 2
Volume
Triangular prism
V = Bh
Cylinder
Rectangular prism
V = Bh
Cone
Pyramid
1
V = Bh
3
Sphere
π ≈ 3.14 or π ≈
Pi
V = Bh or V = π r 2 h (8th grade)
1
1
V = Bh or V = π r 2 h (8th)
3
3
4
V = π r 3 (8th grade)
3
22
7
Distance
d = rt
Compound Interest
A = P (1 + r )t
Simple Interest
I = prt
Pythagorean Theorem
a 2 + b 2 = c 2 (8th grade)
Customary – Length
1 mile = 1760 yards
1 yard = 3 feet
1 foot = 12 inches
Metric – Length
1 kilometer = 1000 meters
1 meter = 100 centimeters
1 centimeter = 10 millimeters
Customary – Volume/Capacity
1 pint = 2 cups
1 cup = 8 fluid ounces
1 quart = 2 pints
1 gallon = 4 quarts
Metric – Volume/Capacity
1 liter = 1000 milliliters
Customary – Mass/Weight
1 ton = 2,000 pounds
1 pound = 16 ounces
Metric – Mass/Weight
1 kilogram = 1000 grams
1 gram = 1000 milligrams
Created by Lance Mangham, 6th grade math, Carroll ISD
AREA OF TRIANGLES and QUADRILATERALS
A=s
2
OR
A=l•w
A=l•w
Area of a square
Area of a rectangle
A = b•h
Area of a parallelogram
Area of a trapezoid
Area of trapezoid
1
A=
(b + b ) h
1
2
2
(b + b ) h
A=
1
2
2
Area of triangle
1
A = bh
2
A=
bh
2
Created by Lance Mangham, 6th grade math, Carroll ISD
Area/Volume/Surface Area Computation Page
EXAMPLES
1
A = (b1 + b2 )h
2
1
A = (10 + 20) • 6
2
A = 90 cm 2
V = π r 2h
S = 2 B + Ph
2
V = 3.14 • 10 • 5
V = 1570 m
3
S = 2(8 • 6) + (28) • 10
S = 376 in 2
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-1: Perimeter
Name:
Perimeter: The distance around the outside of a figure. Per means around. Meter means measure.
Thus, the perimeter of a figure is the measure around it.
Find the perimeter of each figure.
2.0 km
1.
2.9 km
2.
3.
1.8 m
2.4 km
Regular
polygon
0.9 m
1.6 km
1.8 m
2.5 km
24 in
Find the missing part of each rectangle.
4.
6.
P = 160 mm
l= 32 mm
w=
P = 21.8 km
w = 4.7 km
5.
l=
Which of the following CANNOT be used to find the perimeter of a square
with side length s?
A. s + s + s + s
B. 2s + 2s
C. 4s
D. s × s
Peter wants to find the perimeter of the isosceles trapezoid shown below. Write
an equation could Peter use to find P, the perimeter of the trapezoid?
7.
8 inches
5 inches
14 inches
Find the perimeter of each regular polygon.
8.
regular hexagon with sides 28.5 millimeters long
9.
regular decagon with sides 2.5 inches long
10.
regular heptagon with sides 10.75 feet long
11.
regular 12-gon with sides 3.25 yards long
12.
regular 25-gon with sides 6 inches long
Mary needs to cut a piece of glass for her table. The table is in the shape of a
1
13. regular hexagon. The glass should measure 1 ft. on each side. What is the
2
perimeter of the piece of glass?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-2: Perimeter/Area of Rectangles
1.
Name:
How would you specifically describe the change in perimeter of a triangle if
all its side lengths are multiplied by 4?
Two rectangles are shown below. The value of x is the same for both rectangles.
II
I
4
7
6+ x
16 − x
What expression represents the perimeter of rectangle I?
2.
A 10 + x
B 10 + 2x
C 20 + 2x
3.
Write an expression that represents the perimeter of rectangle II.
4.
If x is an integer, what are the smallest and largest values x can be?
5.
Suppose the areas of the two rectangles are equal.
What is the value of x?
What is the perimeter of each rectangle?
What is the area of each rectangle?
Use a formula chart to help answer the following problems.
Show all work on separate paper including three steps for each problem: write the correct
formula, fill in the numbers for the variables, and then solve the equation.
Game
Shape
Dimensions
Perimeter
Area
6.
Racquetball
Rectangle
w = ft.
l = 40 ft.
7.
NCAA basketball
Rectangle
w = 50 ft.
l = 94 ft.
8.
Ice hockey
Rectangle
w = 85 ft.
l=
ft.
9.
Volleyball
Rectangle
w = ft.
l = 60 ft.
10.
Lacrosse
Rectangle
w = 180 ft.
l = 330 ft.
A = 800 sq. ft.
P = 570 ft.
A = 1800 sq. ft.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-3: Area of Rectangles
Name:
1.
NCAA soccer
Rectangle
w = 225 ft.
l = 360 ft.
2.
Football
Rectangle
w = 160 ft.
l = 360 ft.
3.
Tennis
Rectangle
w = 36 ft.
l = 78 ft.
4.
Baseball infield
diamond
Square
s=
ft.
A = 8100 sq. ft.
NBA backboard
5. What is the area of the entire backboard?
6. What is the area of the inner rectangle on the backboard?
7.
What is probability that a random shot that hits the backboard will hit the inner
rectangle?
Mrs. Jones wants to paint a wall but not the door on the wall.
15 ft.
8.
10 ft.
Door: 3 ft by 7 ft
How many square feet of wall does Mrs. Jones need to paint?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-4: Area of Parallelograms
Name:
Formula for the area of a parallelogram: A = bh
The height is measured straight up from the base. The height of this parallelogram is 4 m.
A = bh
5m
4m
A = 8• 4
A = 32 m.2
8m
Find the perimeter and the area of each parallelogram. For the area, show all steps.
1.
2.
8 ft
10 ft
15 m
m
16 ft
m
3.
4.
3.6 cm
3.2 cm
6.5 cm
5.1 cm
5.
7.5 cm
9.3 cm
6.
90 ft
100 ft
90 ft
2.0 m
1.8 m
0.7 m
7.
The base of a parallelogram is 10 in. The height is 2 in. more than half the base. Find the area.
8.
The height of a parallelogram is 4.5 cm. The base is twice the height. What is the area?
9.
The area of a parallelogram is 60 ft.2 The height is 5 ft. How long is the base?
10. The area of a parallelogram is 275 cm.2 The base is 25 cm. Find the height.
Mr. Mangham wants to figure out how many bags of fertilizer he needs to cover his yard. You
11. know the following: the area of the yard, area each bag of fertilizer can cover, cost of each bag,
weight of each bag. How would you determine the number of bags needed?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-5: Area of Triangles
Name:
bh
1
(Half of the formula for a parallelogram.)
or bh
2
2
The height is measured straight up from the base. The height of this triangle is 5 in.
Formula for the area of a triangle: A =
bh
2
6•5
A=
2
A = 15 in.2
A=
5 in
6 in
Find the area of each triangle using the formula above. Show all steps on a separate sheet of paper.
1.
2.
5 mm
3.
7.5 cm
in
8 mm
18 cm
in
4.
5.
4.5 km
6.
m
12 yd
m
1.4 km
7.
58 in.
8.
12 yd
2.5 in
72 in.
6 in
9.
10.
16.9 km
11.
15 yd.
19 km
11.2 km.
23.7 km.
12 yd.
Created by Lance Mangham, 6th grade math, Carroll ISD
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-6: Area of Trapezoids
Name:
Formula for the area of a trapezoid: A =
1
(b1 + b2 ) h
2
1
[ (b1 + b2 ) is just the average of the two bases.]
2
1
A = (b1 + b2 )h
2
1
A = (12 + 15)6
2
1
A = • (27) • 6
2
A = 81 in.2
12 in
Example:
6 in
15 in
The two bases are always parallel to each other.
Find the area of each trapezoid using the formula above. Show all steps on a separate sheet of paper.
z
Trapezoid A
Trapezoid B
x
x
z
y
y
Trapezoid A
1.
x=14 cm, y=26.5 cm, z=12 cm
2.
x=4 cm, y=10 cm, z=5 cm
3.
x=40 m, y=50 m, z=20 m
4.
x=7 ft, y=15 ft, z=7 ft
Trapezoid B
5.
x=6 in, y=16 in, z=9 in
6.
x=41 cm, y=78 cm, z=22 cm
7.
x=2.8 m, y=2.5 m, z=1.5 m
8.
x =2
1
3
in, y =12 in, z =9 in
4
4
Cassie draws the following 4 figures.
9.
List the shapes in order
of area from greatest to
least.
8 cm
10
cm
10 cm
12.5
cm
8 cm
6 cm
10 cm
5 cm
10. What happens to the area of a trapezoid if both bases are tripled?
11.
What happens to the area of a trapezoid if both bases and the height are all
divided by 3?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-7: Area of Composite Shapes
Name:
Find the area of each figure. Show all steps.
1.
2.
6m
3.
6m
4m
2m
5 in
4m
2m
2m
4.
3m
10 in
6 in
8 in
5.
12 in
9 cm
7 in
10 cm
12 in
18 cm
5 cm
7 in
Find the area of the shaded region in each figure.
6.
yard with a sandbox
7. wall with windows
Yard: 15 ft by 20 ft
Sandbox: 6 ft by 7 ft
8. sidewalk around pool
Wall: 8 ft by 16 ft
Each window: 5 ft by 4 ft
Sidewalk: 30 ft by 30 ft
Pool: 27 ft by 27 ft
9.
A bedroom is 15 ft long and 12 ft wide. How much will it cost to carpet the room if carpeting
costs $22 per square yard? (1 yd = 3 ft)
10.
A rose garden in the city park is rectangular and is 9 m wide. If the area of the rectangle is 144
m2, what is the length of the garden?
Cindy had a rectangular garden last year with an area of 60 sq. ft. This year the garden is one
11. foot wider and three feet shorter than last year, but it has the same area. What were the
dimensions of the garden last year?
An average gallon of paint will cover 350 sq. feet of wall or ceiling space. All of your ceilings
are 8 feet high. Your living room is 22 feet by 18 feet. Your kitchen is 15 feet by 15 feet and
12.
your dining room is 12 feet by 14 feet. How many gallons of paint would you need to give one
coat of paint to each wall and ceiling? If paint costs $23 a gallon, what would the total cost be?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-8: All Area
Name:
Ms. Wagner painted the outside of the patio door to her house, as shown below. She did not
paint the window or the doorknob.
2 in by
3 in
1.
7 ft.
1 ft by 1 ft
(each square)
4 ft.
Which is the closest to the painted area of the door in square feet?
A. 31 ft.2
B. 28 ft.2
C. 25 ft.2
D. 18 ft.2
A pest-controlled company was hired to spray the lawn represented by the shaded region
shown below. What was the area in square feet that was sprayed?
200 feet
Gar
2.
House
100 feet
24 ft by 30 ft
40 ft by 40 ft
A. 19,280 ft.2
B. 20, 000 ft.2
C. 37, 680 ft.2
D. 17, 680 ft.2
3.
A triangular sail has a base of 5 m and a height of 10 m. If canvas costs $18 a
square meter, find the cost of canvas to make the sail.
4.
A square dinner napkin 8 in. on each side is folded along its diagonal. Find the
area of the folded napkin.
5.
A shuffleboard court is a large isosceles triangle with b = 6 ft and
A = 27 ft2. What is the length of the shuffleboard court?
6.
If you doubled the height of a triangle, what would happen to the area of the
triangle?
7.
If you doubled both the base and the height of a triangle, what would happen to
the area?
8.
The area of ∆ABC is greater than the area of ∆DEF . Which must be true?
A. The height of ∆ABC is greater than the height of ∆DEF .
B. The perimeter of ∆ABC is greater than the perimeter of ∆DEF .
C. The sum of the angles of ∆ABC is greater then the sum of the angles of ∆DEF .
D. Base and height: At least one of them is greater on ∆ABC .
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-9: Area and Perimeter
Name:
Use graph paper for all drawings and all work.
1.
2.
3.
4.
5.
6.
Draw a figure whose perimeter is 24 units.
Draw a different figure whose perimeter is also 24 units.
Draw a figure whose area is 24 square units.
Draw a different figure whose area is also 24 square units.
Can two different figures have the same area but different perimeters? Explain your answer.
Your dog, Benji, needs a new play area. You are in charge of building a fence around the dog’s
play area so that he can’t run away. You are given 80 feet of fencing to build your play area.
Build two different play areas, each using 80 feet of fencing, which you think would be suitable
for a dog using all of the fencing. For each of your SCALE drawings:
• Calculate the perimeter
• Calculate the area
• Explain why/how the shape you chose would be good for a dog’s play area
PART 2
17 in.
7.
The perimeter of the rectangle is 62 in.
Find the length of each side.
8.
What is the area of a rectangle that is formed by connecting the points (1,2), (5,2), (5,4), and
(1,4) on a coordinate grid?
9.
What is the area of a triangle with vertices (1,2), (7,2), and (3,6)?
10.
Amanda bought 40 meters of fencing to make an enclosure for her dog, Sushi. If Amanda
expects a rectangular enclosure, what is the largest area it can have? Explain your answer.
11.
The width of a rectangle is 4.5 inches and its perimeter is 31 inches. What is the length of the
rectangle?
PART 3
12. The club house is a rectangle that is 25 feet by 40 feet in size. The officers voted to put a 6-foot
sidewalk all around the building, leaving a 2-foot space for plants between the building and the
sidewalk. Give the perimeter of the outer edge of the sidewalk and the area of the sidewalk itself.
13. What is the area of each black and white piece if the whole square measures 20 cm on each side?
What percent of the area of the large square is the small shaded square?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-10: Circles
Name:
Symbols
Angle BAC
∠BAC
Ray AB
AB
Arc AB
Line Segment AB
AB
Line AB
AB
AB
Diameter: a segment that passes through the center of the circle and has both endpoints on the circle
Radius: a segment that has one endpoint at the center of the circle and the other endpoint on the circle
Center: the middle point of a circle
Semicircle: half of a circle
Arc: part of a circle
P
Q
P
Point O is the center of the circle.
A
AB is the diameter.
A
B
O
OA is the radius. OP is also a radius.
O
∠AOP and ∠AOQ are central angles.
The diameter is twice the length of the radius.
The radius is half the diameter.
d
r=
2
d = 2r
1. Draw and label the following parts (one per circle).
Diameter KL
Center O
Radius OT
Central angle ∠WOK
Find the unknown length of each circle.
Radius
Diameter
2.
8 cm.
Radius
3.
4.
48 ft.
5.
23 mm.
6.
55 ft.
7.
18 in.
Diameter
110 in.
10.
5
yd.
8
12.3 mi.
11.
3
in.
4
16.4 m.
12.
3.3 cm.
13.
1.25 yd.
8.
14.
6
7
9.
2.6 m.
15.
9.1 in.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-11: Circles
Name:
How can I remember the difference between diameter and radius?
Radius comes from the same root word as “ray”. A ray of sunshine starts at the center of the sun and
goes out from that point. The radius of a circle starts at the center of the circle!
A
B
O
D
C
Solve the following problems using the circle above.
1.
The points on a circle are all the same distance from the...
2.
A line segment from the center to any point on the circle is a…
3.
A diameter of the circle in the drawing above is the segment…
4.
Which of the following is not a radius: OA, OD, or BC ?
5.
Part of a circle, such as between points B and C, is an…
6.
An angle whose vertex is at the center of a circle is a…
7.
Which of the following is not a central angle:
∠AOD, ∠COD, or ∠BCA ?
8.
Points A, B, C, and D are all the same __________ from point O.
9.
If the length of AC is 20 cm, then the length of OC is…
10. If the length of OA is 20 cm, then the length of OD is…
11. If the length of OD is 20 cm, then the length of AC is…
12. The length of a radius is __________ the length of a diameter.
13. The set of points in a plane at a fixed distance from a given point is a…
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-12: Measuring Circles
Name:
Measure at least 6 of the objects provided by your teacher. For each object record the information
below. You may use a calculator as needed.
Item Name
Diameter
(measure in
mm)
Circumference
(measure in
mm)
Radius
(calculate in
mm)
Circumfere nce
Diameter
(round to the nearest
ten-thousandth)
***** MEAN *****
***** MEDIAN *****
***** RANGE *****
***** CLASS MEAN *****
***** CLASS MEDIAN *****
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-13: Extraordinary Pi
Name:
•
Pi is the number of times a circle’s diameter will fit around its circumference.
•
Here is pi to 64 places:
3.1415926535897932384626433832795028841971693993751058209749445923
•
Pi occurs in hundreds of equations in many sciences including describing DNA, a rainbow,
ripples where a raindrop fell into water, distribution of prime numbers, geometry problems,
waves, navigation, etc.
•
Half the circumference of a circle with radius 1 is exactly Pi. The area inside that circle is also
exactly Pi!
•
Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:
0 occurs 599,963,005 times
1 occurs 600,033,260 times
2 occurs 599,999,169 times
3 occurs 600,000,243 times
4 occurs 599,957,439 times
5 occurs 600,017,176 times
6 occurs 600,016,588 times
7 occurs 600,009,044 times
8 occurs 599,987,038 times
9 occurs 600,017,038 times
•
Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b)
where a and b are integers.
•
The Babylonians found the first known value for Pi in around 2000BC; they used (25/8) The
Egyptians used Pi = 3 but improved this to (22 / 7). They also used (256/81).
•
In around 200 BC Archimedes found that Pi was between (223 / 71) and (22 / 7). His error was
no more than 0.008227%. He did this by approximating a circle as a 96 sided polygon.
•
Ludolph Van Ceulen (1540 - 1610) spent most of his life working out Pi to 35 decimal places. Pi
is sometimes known as Ludolph's Constant.
•
The first person to use the Greek letter was Welshman William Jones in 1706. He used it as an
abbreviation for the 'periphery' of a circle with unit diameter.
•
The Pi memory champion is Hiroyoki Gotu (21 years old) who memorized an amazing 42,000
digits.
•
Pi was calculated to 2,260,321,363 decimal places in 1991 by the Chudnovsky brothers in New
York.
•
Most people would say that a circle has no corners - but it is more accurate to say that it has an
infinite number of corners.
•
At position 762 there are six nines in a row. This is known as the Feynman Point.
Created by Lance Mangham, 6th grade math, Carroll ISD
Created by Lance Mangham, 6th grade math, Carroll ISD
CIRCLES
Important information about circles:
• The diameter is exactly twice the radius.
• The radius is exactly half of the diameter.
• π is a number that goes on forever. π represents how many diameters it takes to go around the
circumference of any circle.
RADIUS
DIAMETER
RADIUS to DIAMETER
DIAMETER to RADIUS
***** MULTIPLY BY 2 *****
***** DIVIDE BY 2 *****
If radius = 5, then the diameter = 10.
If diameter = 14, then the radius = 7.
RADIUS TO CIRCUMFERENCE
***** MULTIPLY BY 2π
*****
If the radius of this circle is 9,
C = 2π r
C = 2 • 3.14 • 9
CIRCUMFERENCE TO RADIUS
*****
If the circumference of this circle is 18.84,
C = 2π r
18.84 = 2 • 3.14 • r
18.84 = 6.28r
18.84
=r
6.28
3=r
***** MULTIPLY BY π
*****
If the diameter of this circle is 6,
C =πd
C = 3.14 • 6
C = 18.84
C = 56.52
***** DIVIDE BY 2π
DIAMETER TO CIRCUMFERENCE
CIRCUMFERENCE TO DIAMETER
***** DIVIDE BY π
*****
If the circumference of this circle is 15,
C =πd
15.7 = 3.14d
15.7
=d
3.14
5=d
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-14: Circumference and Area
Name:
Circum means around or bend around. Ferre means to bring or carry. The circumference of a circle is
the measure you need to get around the entire circle. Circumnavigate means to sail around the globe.
22
π ≈ 3.14 ≈
7
Important formulas for circles:
d
Radius to diameter: r =
Diameter to radius: d = 2r
2
Circumference:
C = 2π r or C = π d
Area:
A = π r2
22
for π .
7
Show all work on separate paper including three steps for each problem: write the correct
formula, fill in the numbers for the variables, and then solve the equation.
Radius
Circumference
Area
Diameter
Circumference
Area
Find the circumference and area of each circle given the diameter or radius. Use 3.14 or
1.
3 in
11 2
yd
14
3.
5.
37.68 mi
2.
8 yd
4.
1.2 ft
6.
24 yd
400π ft2
7.
55 m
8.
9.
3
cm
4
10.
20π cm
11.
7.1 cm
12.
157 in
13.
2
1
cm
2
14.
16 in
16.
32 in
18.
y in
60π m
15.
17.
x cm
Write an expression to find the circumference of the outside of the
donut. The radius of the inner circle is 2 cm. Do not solve.
1920.
9 cm
21.
Write an expression to find the area of the donut. The radius of the
inner circle is 2 cm. Do not solve.
If a 14 inch pizza has a 1 inch crust, what is the area of the crust? To
the nearest percent, what percent of the pizza is crust?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-15: Inuit Architecture
Name:
The Inuit are Native Americans who live primarily in the arctic regions of Alaska and Canada. The
Inuit word iglu means “winter house”. Later the term came to mean a domed structure built of snow
blocks, as shown in the figure at the right.
Several families would build a cluster of iglus that were connected by
passageways and shared storage and recreation chambers. Use the
drawing to answer the following questions. Use 3.14 for pi.
List the radius of each of the chambers.
Storage
d = 10
ft.
1.
Determine the circumference of each
chamber.
d=9
ft.
d=9
ft.
Recreation
d = 12 ft.
2.
Determine the area of each chamber. Then
calculate the total area of all chambers.
d=8
ft.
d=8
ft.
Entry
d=
6 ft.
3.
4.
Estimate the distance for the front of the
entry chamber to the back of the storage
chamber.
If Mr. Mangham runs around the storage
iglu 5 times, how far will he run?
5.
6. A person wishes to place a covering on
the floor in the entry, recreation, and
storage rooms. If coverings are sold in 40
square feet pieces, how many pieces will
she need?
7. If the radius of one of the iglus is
tripled, then the area of the iglu is
multiplied by….
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-16: Bull’s Eye & Hat Size
Name:
1. The radius of the inner circle is one inch. Each successive circle has a radius increase of one inch, so
the outside circle has a radius of five inches (note that they are not drawn in actual size).
Which has a greater area – the shaded region or the striped region?
2. Let’s determine your hat size! Have a friend measure around your head at approximately the place a
hat would rest. Measure to the nearest one-eighth of an inch. Divide this by the fraction form of pi,
round up to the nearest eighth, and you have your hat size!
3. Write a procedure that can be used to find the area of the shaded region inside the square. Don’t
solve – just write in words.
8 in.
4. The Square Pizza Company only serves pizza in the shape of squares. Unfortunately, an employee
mistakenly ordered 16-inch diameter circular pizza pans. What is the largest size square pizza that can
be made from this pan? Hint: Make triangles
5. The Square Pizza Company makes pizza dough to place in 16-inch diameter circular pans, then cuts
off the edges of the circle to create the largest square pizza possible. If the area of the cut-off dough can
be reformed into another square pizza, then what side length would this recycled square pizza have?
You can list your answer as between two whole numbers.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-17: Composite Shapes
Name:
An inscribed figure is a figure that is placed inside another figure so that it fits exactly.
1. Mr. Mangham is planning a garden around one of the trees in his backyard. The radius of the large
outer circle is 4 feet while the radius of the inner circle for the tree is 1 foot. Find the area of Mr.
Mangham’s new garden.
GARDEN
2. Find the area of the composite shape below.
15 ft
20 ft
3. The athletic field below is 40 yards wide and 100 yards long. The track is 10 yards wide. What is
the area of the field and what is the area of the track? If you walked along the outside of the entire
track, how far did you walk? If you walked along the inside of the track, how far did you walk?
4. Find 3 composite shapes at DIS. Measure the required lengths to determine the area of each of the
three composite shapes.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-18: Composite Shapes
Name:
1. Find the area of the shaded region in the shape below.
25 yd
50 yd
2. A NBA basketball court is shown below. It is 94 feet long and 50 feet wide. Each circle has a
diameter of 12 feet. The backboard is 4 feet from the baseline. The feet throw line is 15 feet from the
backboard.
Write 4 word problems that can be solved using the basketball court above. The problems should
involve perimeter, circumference, and/or area in some fashion. Make your problems more interesting
than you would find in a math book. Solve your 3 easiest problems.
3. Determine the area of each composite shape below.
0.87 in
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-19: Composite Shapes
Name:
You may use a calculator for this page, but still write out all formulas. Determine the area of the shaded
region on problems 1-6. Determine the area of the combined region of each composite shape on
problems 7-8.
1.
4.
2.
3.
5.
6.
10 in
7.
8.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-20: Area of Composite Figures
Name:
Estimate the area of each figure. Each square represents 1 square foot.
1.
2.
_________________________________
__________________________________
Find the area of each figure. Use 3.14 for π .
3.
4.
____________________
6.
5.
____________________
7.
____________________
____________________
8.
____________________
____________________
9. Marci is going to use tile to cover her terrace. How much tile does
she need?
_____________________________________
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-21: The Margaret Hunt Hill Bridge
Name:
The Margaret Hunt Bridge was completed near downtown Dallas in March 2012, almost five years after
construction began. The bridge cost $182 million to build and it was made out of 11,643,674 pounds of
structural steel. It is a cable stayed bridge, not a suspension bridge.
The bridge cables are two different sizes. There are 44 large cables that have a diameter of 6.3 inches and
are made up of 31 individual strands of wire. The 14 smaller cables have a diameter of 4.9 inches and are
made up of 12 individual strands of wire.
You may use a calculator for all computation, but show all formula steps as usual.
1.
What is the circumference of each of the large cables?
2.
What is the area of a cross-section of one of the large cables?
3.
What is the circumference of each of the smaller cables?
4.
What is the area of a cross-section of one of the smaller cables?
5.
Assuming the strands of wire inside the cables take up 90% of the area, what is your best guess
at the diameter and circumference of a single strand of wire inside the larger cable?
6.
How does the circumference of a circle change if you double the length of its radius?
7.
How does the circumference of a circle change if you double the length of its diameter?
8.
If you triple the length of the radius of a circle, how does its circumference change?
9.
If you make the circumference of a circle half as long, how does its diameter change?
10. If you multiply the radius of a circle by y, how does its diameter change?
11.
A circular swimming pool has a border of 30 curved tiles. There are blue and white tiles in a
pattern of two blue tiles between every white tile. How many white tiles are there?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-22: Do You Want to Build a Snowman?
Name:
You may use a calculator for this activity. Do not use the pi button – use 3.14 for pi.
For all problems involving an equation (circumference, area, etc.) show all steps.
For all problems asking “how many turns” your answer should be a whole number.
Fresh snow has just fallen and Ashley decides to make a snowman.
Ashley makes a snowball 6 inches in diameter. She then rolls the snowball one full turn.
1.
Draw a picture of the original
snowball and draw and label the
diameter.
How far does Ashley roll the snowball?
2.
3.
Show all equations and all steps in the box at
the right.
Ashley finds that 3 inches of
new snow sticks as she rolls the
snowball (3 inches added to
every part on the original
circle). Draw a picture of the
new snowball.
What is the new diameter of the snowball after Ashley rolls it one full turn?
4.
5.
(Hint: It is not 9 inches.)
How many total turns will Ashley have to make so that the diameter of the
snowball becomes 72 inches?
(Hint: The first 6 inches did not require any turns.)
How many total turns will it take for the
circumference to become greater than 300
inches?
6.
(Hint: Start with the circumference formula to
find d.)
Show your original equation and all steps in
the box at the right.
Created by Lance Mangham, 6th grade math, Carroll ISD
How many total turns will it take for the area to
become greater than 1000 in.2?
7.
(Hint: Start with the area formula to find r.)
Show all equations and all steps in the box at
the right.
8.
9.
Ashley makes a second snowball and then rolls it
one full turn. Again 3 inches of snow stick to it.
She measures the new circumference (C) to be
about 44 inches. To the nearest inch, what is the
diameter of Ashley’s rolled snowball?
Show all equations and all steps in the box at
the right.
Using the diameter from #8, what is the area of
the rolled snowball?
Show all equations and all steps in the box at
the right.
10. What was the diameter of the snowball before Ashley rolled it?
What is the area of the snow that stuck to the
original snowball in #8?
11.
Show all equations and all steps in the box at
the right.
Ashley rolls a huge snowball 4 feet in diameter. She decides to make another
2
of its diameter to put on top. In inches, what is the diameter of
12. snowball
3
the snowball Ashley wants to put on the top?
In square inches, what is the area of both
snowballs combined?
13.
Show all equations and all steps in the box at
the right.
Ashley started with a snowball 8 inches in diameter. How many complete
14. turns will Ashley need to roll it before she has a snowball the size she wants
to put on top?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-23: More Circles
Name:
1. Draw a circle similar to the one below with a dot in the middle without lifting your pencil from the
paper. Can it be done?
2. A pizza parlor has 3 size pizzas. Would you get more if you had one small pizza, three-fifths of a
medium pizza, or one-third of a large pizza?
SIZE
Small
Medium
Large
DIAMETER
10 inch diameter
14 inch diameter
18 inch diameter
3. In the picture below, the circle’s area is what percent of the larger squares area? If the larger square
has an area of 10 square inches, what is the area of the smaller square?
4. Queen Dido’s Cutting Trick: Take a 3x5 index card. Fold the card in half hamburger style. Start
your first cut on the fold just barely in from the side. Cut almost all the way up to the top. Next cut
from the top almost all the way back to the bottom. Continue until you end up with an odd number of
cuts. Unfold the card and cut all the folds except for the ones at the end of the card. You will have a
circle big enough to step through!
5. Draw two squares so that each dot is completely
contained in a region by itself.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 10-24: Hot Tubs
Name:
Hot tubs and in-ground swimming pools are sometimes surrounded by borders of tiles. You have a
square hot tub with sides of length b feet. Your tub is surrounded by a border of square tiles. Each
border tile measures 1 foot on each side. How many 1-foot square tiles will be needed for the square
border of the hot tub that has edge length of b feet?
Draw separate pictures in which you group the border tiles in different, logical arrangements and then
express the total number of tiles needed with equivalent expressions (one for each picture you have
drawn).
This problem tells you specifically to draw pictures, so that must be the strategy! Luckily for you, the
pictures have already been drawn for you on the other side. Your job is to look specifically at how the
tiles have been grouped to come up with an appropriate expression using numbers, variables, and
operations. Hint: If you simplified all your expressions, you would get the same answer every time.
THE HOT TUB
1
1
b
b
Created by Lance Mangham, 6th grade math, Carroll ISD
Four segments + Four corners
Four segments – Four corners
Two long segments + Two short segments
All shading – Inside shading
(since they are covered twice)
Can you think of any more logical
configurations? If so, draw them
and determine their formulas.
Four segments
Created by Lance Mangham, 6th grade math, Carroll ISD