#(!04%2
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-EASUREMENT 4WO$IMENSIONAL &IGURES
!2% 9/5 2%!$9 0!'% TRAPEZOID
PARALLELOGRAM
RADIUS DIAMETER
0!'%3 n
4HINK AND $ISCUSS
IN 4HE PAPER
4HE RULER CAN MEASURE PRECISION TO ???
IN
CLIP IS ABOUT ???
-EASURING TO THE NEAREST ?? IN WOULD GIVE THE MORE
ACCURATE MEASUREMENT BECAUSE IT IS MORE PRECISE
THAN MEASURING TO THE NEAREST ?? IN
.EAREST TEN .EAREST HUNDRED .EAREST TEN .EAREST HUNDRED .EAREST TEN .EAREST HUNDRED %XERCISES
3INCE A YARD IS A SMALLER UNIT THAN A MILE YD
IS MORE PRECISE
.EAREST TEN .EAREST HUNDRED G IS MORE PRECISE -EASURING TO THE NEAREST
HUNDREDTH OF A GRAM IS MORE PRECISE THAN
MEASURING TO THE NEAREST TENTH OF A GRAM
.EAREST WHOLE NUMBER .EAREST TENTH LB IS MORE PRECISE -EASURING TO THE NEAREST
TENTH OF A POUND IS MORE PRECISE THAN MEASURING TO
THE NEAREST WHOLE NUMBER OF POUNDS
.EAREST WHOLE NUMBER .EAREST TENTH 3INCE AN INCH IS A SMALLER UNIT THAN A FOOT IN IS
MORE PRECISE
,ENGTH TO THE NEAREST HALF INCH ?? IN ,ENGTH TO THE
NEAREST FOURTH INCH ?? IN ,ENGTH TO THE NEAREST
EIGHTH INCH ?? IN ?? IN IS THE MOST ACCURATE
MEASUREMENT BECAUSE MEASURING TO THE NEAREST
EIGHTH INCH IS MORE PRECISE THAN MEASURING TO THE
NEAREST FOURTH INCH
.EAREST WHOLE NUMBER .EAREST TENTH .EAREST WHOLE NUMBER .EAREST TENTH S D
S D
S D
S D
S D S D QÊ S D R
QÊ R
QÊ R
TRIANGLE
TRAPEZOID
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
PARALLELOGRAM
4HE MASS OF THE BACKPACK IS HALFWAY BETWEEN
KG AND KG
?? KG
4HE MASS OF THE BACKPACK IS ABOUT KG
3INCE MM IS A SMALLER UNIT THAN MM MM
IS MORE PRECISE
3INCE A FLUID OUNCE IS A SMALLER UNIT THAN A CUP
FL OZ IS MORE PRECISE
?? FT IS MORE PRECISE -EASURING TO THE NEAREST ?? FT IS
MORE PRECISE THAN MEASURING TO THE NEAREST ?? FT
3INCE A GRAM IS A SMALLER UNIT THAN A KILOGRAM
G IS MORE PRECISE
,ENGTH TO THE NEAREST HALF INCH IN
,ENGTH TO THE NEAREST FOURTH INCH ?? IN
,ENGTH TO THE NEAREST EIGHTH INCH ?? IN
?
?
IN IS THE MOST ACCURATE MEASUREMENT BECAUSE
MEASURING TO THE NEAREST EIGHTH INCH IS MORE
PRECISE THAN MEASURING TO THE NEAREST FOURTH INCH
(OLT -C$OUGAL -ATHEMATICS
0OSSIBLE ANSWER -ULTIPLY THE RADIUS BY TO FIND
THE DIAMETER AND THEN SUBSTITUTE THE VALUE FOR THE
DIAMETER INTO THE FORMULA
4HE POINTER IS ON THE MARK THAT IS HALFWAY BETWEEN
& AND & %ACH MARK SHOWS SO THE
TEMPERATURE IS ABOUT &
4HE PAPER STRIP IS HALFWAY BETWEEN FT AND FT
?? ??
?
?
??
4HE LENGTH IS ABOUT ?? FT
-ILLILITER IS MORE PRECISE BECAUSE A MILLILITER IS A
SMALLER UNIT THAN A LITER
%XERCISES
0 0 4HE PERIMETER OF THE TRIANGLE IS M
0 0 4HE PERIMETER OF THE PARALLELOGRAM IS IN
0 0 4HE PERIMETER OF THE SQUARE IS FT
/UNCE IS MORE PRECISE BECAUSE AN OUNCE IS A
SMALLER UNIT THAN A POUND
&LUID OUNCE IS MORE PRECISE BECAUSE A FLUID OUNCE
IS A SMALLER UNIT THAN A QUART
0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS IN
4HE LENGTH OF AN OBJECT WOULD BE MORE ACCURATE IF
MEASURED IN DECIMETERS BECAUSE A DECIMETER IS A
SMALLER UNIT THAN A DECAMETER
4HE ANGLE IS ABOUT 0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS M
4HE ANGLE IS ABOUT 0OSSIBLE ANSWER ABOUT M OR CM
TO THE NEAREST MILLILITER
TO THE NEAREST MILLIMETER
0 L W
0 ?? ??
0
0 4HE PERIMETER OF THE RECTANGLE IS FT
S
2ITA SHOULD USE THE MEASURING TAPE OR THE METER
STICK BECAUSE THERE ARE ENOUGH UNITS TO COVER EACH
DISTANCE ON THE WALL
3HIA LOOKED AT THE NUMBER OF DECIMAL PLACES
INSTEAD OF LOOKING AT THE UNITS MM IS MORE
PRECISE THAN M
# Q D
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT FT
# Q R
# y ???
# y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT IN
$ INCHES
)NCHES IS THE SMALLEST UNIT OF THE ONES GIVEN SO
INCHES IS THE MOST PRECISE
-ILLIMETERS ARE MORE PRECISE BECAUSE A MILLIMETER IS
A SMALLER UNIT THAN AN INCH
NEGATIVE CORRELATION 4HE FASTER A CAR TRAVELS THE
LESS TIME IT TAKES TO GO MILES
YES
NO )TS SIDES ARE CURVED
YES
0!'% 4HINK AND $ISCUSS
0OSSIBLE ANSWER !DD THE LENGTHS OF THE FOUR SIDES
/R MULTIPLY TWICE THE LENGTH AND TWICE THE WIDTH
AND THEN FIND THE SUM OF THE PRODUCTS
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
D
# Q D
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT M
0OSSIBLE ANSWER !N ESTIMATE WILL DO IF YOU WANT
TO KNOW ABOUT HOW MUCH TIME HAS PASSED SINCE
BREAKFAST !N ACCURATE MEASUREMENT IS NEEDED IF
YOU WANT TO KNOW WHICH RUNNER HAS THE FASTEST TIME
WHEN RUNNING THE ?? MILE
.O THE TRUCK IS ESTIMATED TO WEIGH TONS WHICH
MEANS IT COULD WEIGH MORE
NO CORRELATION 4HE PRICE OF A CAR IS NOT AFFECTED BY
THE NUMBER OF WINDOWS IT HAS
D S
# QD
y D
y D y D
4HE DIAMETER OF THE &ERRIS WHEEL IS ABOUT FT
0 0 4HE PERIMETER OF THE RHOMBUS IS CM
0 0 4HE PERIMETER OF THE TRIANGLE IS FT
0 0 4HE PERIMETER OF THE TRAPEZOID IS M
(OLT -C$OUGAL -ATHEMATICS
0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS IN
0 L W
0 S D S D
0
0
4HE PERIMETER OF THE RECTANGLE IS FT
0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS CM
# Q D
# y ???
????
#y
# y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT CM
# Q R
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT M
# Q R
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT IN
# QD
y D
y D y D
4HE DIAMETER IS ABOUT IN
# QD
y D
y D y D
RD
4HE DIAMETER IS ABOUT M AND THE RADIUS IS
ABOUT M
# QR
# y # y DR
4HE CIRCUMFERENCE IS ABOUT YD AND THE
DIAMETER IS ABOUT YD
# Q D
# y # y RD
4HE CIRCUMFERENCE IS ABOUT IN AND THE RADIUS
IS ABOUT IN
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
# Q D
Q y Q D
Q Q y Q D Q
y D
RD
??
4HE DIAMETER IS ABOUT UNIT AND THE RADIUS IS
ABOUT A ?? UNIT
# QD
# y Q # y FT
IN ????
X
?????
FT
FT
X X IN
# y IN
IN IN STRANDS STRANDS
STRANDS OF LIGHTS ARE NEEDED TO SURROUND THE
PATIO EDGE
0 (E TRAVELS MI ROUNDTRIP
# QD
y D
y D y D
4HE DIAMETER OF THE ROTUNDA IS ABOUT FT
0OSSIBLE ANSWER ,AY THE STRING ALL THE WAY AROUND
THE OUTSIDE OF THE OBJECT AND CUT IT 4HEN TAKE THE
STRING AND MEASURE IT WITH A RULER
0OSSIBLE ANSWER (OW MANY FEET OF RIBBON WOULD BE
NEEDED TO TRIM THE EDGE OF THE BULLETIN BOARD
3UBTRACT TWICE THE LENGTH FROM THE PERIMETER AND
DIVIDE THE RESULT BY ?? ???
?? ???
??
?? ???
?
?
%ACH SIDE OF THE NONAGON IS IN
$ INCHES
# QD
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT IN
0 L W
0
0 0 4HE PERIMETER IS FT
IS OF WHAT NUMBER
N
N N
IS OF OF IS WHAT NUMBER
X
X
OF IS (OLT -C$OUGAL -ATHEMATICS
FT IS MORE PRECISE BECAUSE A FOOT IS A SMALLER UNIT
THAN A YARD
X X FT WIDE
! LW
! ! 4HE AREA OF THE FIELD IS FT MM IS MORE PRECISE BECAUSE A MILLIMETER IS A
SMALLER UNIT THAN A CENTIMETER
?? IN IS MORE PRECISE )T IS MEASURED TO THE NEAREST
FOURTH INCH RATHER THAN TO THE NEAREST HALF INCH
! LW
! ! 4HE AREA OF THE RECTANGLE IS FT G IS MORE PRECISE )T IS MEASURED TO THE
NEAREST TENTH OF A GRAM RATHER THAN TO THE NEAREST
WHOLE NUMBER OF GRAMS
0!'%3 n
4HINK AND $ISCUSS
! S 0OSSIBLE ANSWER 7HEN A PARALLELOGRAM IS
NONRECTANGULAR YOU HAVE TO MULTIPLY THE HEIGHT AND
THE BASE NOT THE SIDE LENGTHS TO FIND THE AREA
%XERCISES
! LW
! ! 4HE AREA OF THE RECTANGLE IS FT ! LW
!
! 4HE AREA OF THE RECTANGLE IS M ! LW
! ! 4HE AREA OF THE RECTANGLE IS CM ! LW
W
W W
4HE WIDTH OF THE ROOM IS FT
! BH
!
! 4HE AREA OF THE PARALLELOGRAM IS IN ! BH
! ?? ???
!
??
! ???
4HE AREA OF THE PARALLELOGRAM IS ?? CM ! BH
! ! 4HE AREA OF THE PARALLELOGRAM IS M YD
???? FT
X
X
YD
????
FT
! LW
! ?? ??
???
! ???
????
!
??
4HE AREA OF THE RECTANGLE IS ?? IN ! LW
! ! 4HE AREA OF THE RECTANGLE IS IN ! BH
! !
4HE AREA OF THE PARALLELOGRAM IS M ! BH
! ?? ??
???
?
?
!
! ?? ??
???
! ??
4HE AREA OF THE PARALLELOGRAM IS ?? FT ! BH
! ! 4HE AREA OF THE PARALLELOGRAM IS CM YD
??????
X
FT LONG
YD
?????
X
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! LW
L L L
4HE LENGTH OF THE DOG YARD IS M
FT ?????
L
?????
IN
IN
L
L
L L
FT ?????
W
?????
IN
IN
W
W
W W
! LW
! ! 4HE AREA OF ONE BLOCK IS FT (OLT -C$OUGAL -ATHEMATICS
# CM
'UESS AND 4EST OR -AKE A 4ABLE
! LW
!
! 4HE AREA OF THE RECTANGLE IS YD !
"
#
! BH
! ! 4HE AREA OF THE PARALLELOGRAM IS M X
# IN ! BH
! .O DOUBLING THE LENGTH OF EACH SIDE WILL MAKE THE
DECK TIMES AS LARGE BECAUSE AND
S D S D S D 0OSSIBLE ANSWER
DOUBLE THE LENGTH OF ONE SIDE OF THE DECK GIVING
DIMENSIONS OF EITHER FT BY FT OR FT BY FT
"OTH OF THESE ALTERNATIVES WILL GIVE AN AREA OF OR
S D SQUARE FEET
! LW
!
! 4HE AREA OF THE RECTANGLE IS UNITS Y
ACUTE
OBTUSE
STRAIGHT
RIGHT
0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS IN
X
/
0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS M
! LW
!
! 4HE AREA OF THE RECTANGLE IS UNITS 0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS CM
A 0 L W
0 S D S D
0 0 4HE PERIMETER OF THE PAINTING IS CM
! LW
! ! 4HE AREA OF THE PAINTING IS CM B ! LW
! S D S D
! ! 4HE AREA OF THE PAINTING AND THE FRAME IS
CM 0 L W
0 ?? ??
S D S D
0 S ???
S ?? D
D
???
0 ???
???
0 ????
0ETE DID NOT USE THE CORRECT UNIT FOR AREA (E SHOULD
HAVE USED CM #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
!REA
)F THE HEIGHT IS DIVIDED IN HALF THE AREA WILL BE HALF
THE ORIGINAL SIZE )F BOTH THE HEIGHT AND BASE ARE
HALVED THE AREA WILL BE A QUARTER OF THE ORIGINAL
SIZE
/
(EIGHT
.O THE PARALLELOGRAM COULD HAVE A HEIGHT LESS THAN
M BUT THE HEIGHT OF THE RECTANGLE IS M
Y
"ASE
???
???
0 ????
FT
4HE PERIMETER OF THE RECTANGLE IS ???
(OLT -C$OUGAL -ATHEMATICS
! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER -ULTIPLY THE LENGTHS OF THE
PERPENDICULAR SIDES AND THEN DIVIDE BY TWO
0OSSIBLE ANSWER &IND THE SUM OF THE LENGTHS OF
THE TWO PARALLEL SIDES AND THEN FIND ?? OF THE HEIGHT
TIMES THAT SUM
! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS %XERCISES
! ??BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS ! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS ! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS ! ?? H S B B D
?
! ? S D
! ?? S D
! 4HE AREA OF THE TRAPEZOID IS CM ! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS M ! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS FT ! ??H S B B D
! ?? S D
! ?? S D
! 4HE AREA OF 4ENNESSEE IS ABOUT MI #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! ?? BH
?
?
! S D
! 4HE AREA OF THE TRIANGLE IS UNITS ! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS YD ! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS IN ! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS CM ! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS MI ! ?? BH
?
? S H D
H
H
4HE HEIGHT OF THE TRIANGLE IS CM
! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS FT (OLT -C$OUGAL -ATHEMATICS
! ?? BH
?
? S B D
B
B
4HE BASE OF THE TRIANGLE IS IN
Y
X
! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS UNITS Y
X
/
! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS Y
/
! ?? BH
?
! ? S D
! 4HE AREA OF THE TRIANGLE IS UNITS X
/
Y
! ?? H S B B D
?
? H S D
?? H S D
H
H
4HE HEIGHT OF THE TRAPEZOID IS M
0 L W
0 S D S D
0 0 ! LW
! ! 4HE PERIMETER OF #OLORADO IS ABOUT MI AND
THE AREA IS ABOUT MI 0OSSIBLE ANSWER 4HE STUDENT MULTIPLIED THE BASE
AND THE SIDE LENGTH AND THEN MULTIPLIED THE PRODUCT
BY ?? FOR AN AREA OF CM 4HIS IS NOT A RIGHT
TRIANGLE SO THE LENGTH OF ONE SIDE IS NOT THE HEIGHT
OF THE TRIANGLE )N ORDER TO FIND THE CORRECT AREA THE
HEIGHT IS NEEDED
4HINK OF THE STATE AS A RECTANGLE MI BY MI
AND A TRIANGLE WITH HEIGHT MI AND BASE MI
S D AREA OF THE RECTANGLE PART
AREA OF TRIANGLE PART
?? S D AREA OF THE WHOLE STATE
MI X
/
! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS UNITS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! ?? H S B B D
?? S B D
S B D
y B y B 4HE NORTHERN BORDER IS ABOUT MILES LONG
(OLT -C$OUGAL -ATHEMATICS
! CM ! ?? H S B B D
?
! ? S D
?
! ? S D
!
4HE AREA OF THE TRAPEZOID IS CM 0OSSIBLE ANSWER A CLOCK ESTIMATE THE RADIUS OF THE
CLOCK SQUARE THAT NUMBER AND THEN MULTIPLY BY S IN D IN ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT IN 0OSSIBLE ANSWER 4HE FORMULA FOR THE AREA OF A CIRCLE
USES THE RADIUS OF A CIRCLE )F THE DIAMETER IS GIVEN
YOU MUST FIRST DIVIDE THE DIAMETER BY %XERCISES
4HINK AND $ISCUSS
Y
0!'%3 n
X
! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT CM ! ?? BH
?
! ? S D
!
4HE AREA OF THE TRIANGLE IS UNITS ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT YD X
X
X
4HE MEASURE OF THE THIRD ANGLE IS ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT M X
X
X
4HE MEASURE OF THE THIRD ANGLE IS ! QR ! y ???
! y ???
???
!y
! y 4HE AREA OF THE PIZZA IS ABOUT IN X
X
X
4HE MEASURE OF THE THIRD ANGLE IS X
X
X
4HE MEASURE OF THE THIRD ANGLE IS 4HE DIAMETER OF THE CIRCLE IS CM
! QR ! y ! y (ALF OF THE CIRCLE IS SHADED SO 4HE AREA OF THE SHADED PART OF THE CIRCLE IS ABOUT
CM &IRST CONVERT THE YARDS TO FEET
YD
YD
????
????
L
FT
L
L FEET
YD
YD
????
????
W
FT
W
W FEET
! LW
! *USTIN NEEDS FT OF TILE
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT IN ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT FT (OLT -C$OUGAL -ATHEMATICS
! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT YD ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT CM ! QR
! y ???
???
!y
! y ???
! y 4HE AREA OF THE PIZZA IS ABOUT CM 4HE DIAMETER OF THE CIRCLE MEASURES CM
! QR ! y ! y ! QUARTER OF THE CIRCLE IS SHADED
SO 4HE AREA OF THE SHADED PART OF THE CIRCLE
IS ABOUT CM ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS ABOUT MI ! QR ! y ! y ! y 4HE AREA OF +AYS FLOWER BED IS ABOUT FT ! QR ! y ! y ! y 4HE AREA OF ONE LID IS ABOUT CM # QR
# y # y 4HE CIRCUMFERENCE IS ABOUT M
! QR ! y ! y ! y 4HE AREA IS ABOUT M # QD
# y # y 4HE CIRCUMFERENCE IS ABOUT IN
! QR ! y ! y ! y 4HE AREA IS ABOUT IN #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
# QD
# y # y 4HE CIRCUMFERENCE IS ABOUT FT
! QR ! y ! y ! y 4HE AREA IS ABOUT FT # QR
# y # y 4HE CIRCUMFERENCE IS ABOUT CM
! QR ! y ! y ! y 4HE AREA IS ABOUT CM ! QR y R y R yR
4HE RADIUS IS CM
! QR y R y R
yR
4HE RADIUS IS IN
! QR y R y R
yR
4HE RADIUS IS FT
A 4HE HIKER WALKED MILES IN ANY DIRECTION 4HIS IS
EQUIVALENT TO A CIRCLE WITH A RADIUS OF ! QR ! y ! y ! y 4HEY WOULD HAVE TO SEARCH ABOUT MI B 4HE HIKER WOULD HAVE WALKED MILES IN ANY
DIRECTION WHICH IS A CIRCLE WITH A RADIUS OF ! QR ! y ! y ! y 4HEY WOULD HAVE TO SEARCH ABOUT MI 4HIS IS AN ADDITIONAL MI 4HE DIAMETER OF THE TURBINE IS FT
! QR ! y ! y ! y 4HE AREA COVERED BY THE TURBINE WHEN IT IS ROTATING
IS ABOUT FT .O THE COMBINED AREA OF THE TWO CIRCLES OF THE
SAME RADIUS IS QR WHILE THE AREA OF A CIRCLE WITH
TWICE THE RADIUS IS QS R D QR 0OSSIBLE ANSWER 7HAT IS THE AREA THAT #HANG
PAINTED USING FOR Q 0OSSIBLE ANSWER &IRST USE THE FORMULA FOR THE
CIRCUMFERENCE # QR TO SOLVE FOR THE RADIUS R
4HEN PLUG THE RADIUS INTO THE FORMULA FOR THE AREA OF
A CIRCLE ! QR (OLT -C$OUGAL -ATHEMATICS
4HE AREA IS MULTIPLIED BY A FACTOR OF N &OR
INSTANCE A CIRCLE WITH A RADIUS OF HAS AN AREA OF
ABOUT UNITS )F THE RADIUS IS MULTIPLIED BY THEN THE AREA IS ABOUT UNITS WHICH IS
TIMES OR TIMES AS LARGE
X
! ?? H S B B D
?
! ? S D
?
! ? S D
! 4HE AREA OF THE TRAPEZOID IS UNITS 0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER $IVIDE THE FIGURE HORIZONTALLY INTO
TWO RECTANGLES WITH AREAS IN AND IN OR
DIVIDE THE FIGURE VERTICALLY INTO TWO RECTANGLES
WITH AREAS IN AND IN 0OSSIBLE ANSWER 4HE AREA WILL STAY THE SAME IN BUT THE PERIMETER WILL CHANGE 4HE PERIMETER OF THE
IRREGULAR FIGURE IS IN WHEREAS THE PERIMETER OF
THE TWO RECTANGLES IS IN IN IN
%XERCISES
FULL SQUARES PARTIALLYFILLED SQUARES IS ABOUT
FT FULL SQUARES HALFFILLED SQUARES PARTIALLYFILLED SQUARES IS ABOUT FT 4OP RECTANGLE
! LW
! ! "OTTOM RECTANGLE
! LW
! ! 4HE AREA OF THE FIGURE IS FT X
! ?? BH
! ?? S D
! 4HE AREA OF THE TRIANGLE IS UNITS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
Y
AND ARE VERTICAL ANGLES SO THEY ARE
EQUAL IN MEASURE
SINCE LINES A AND B ARE PARALLEL THEIR
CORRESPONDING ANGLES ARE EQUAL IN MEASURE
LINES A AND B ARE PARALLEL SO THEIR
CORRESPONDING ANGLES ARE EQUAL IN MEASURE
4HEREFORE IS 3INCE AND ARE
SUPPLEMENTARY 4HE M IS Y
! SQUARE FEET
! QR y R y R y R
! QR ! y ! y 4HE AREA OF THE SECOND CIRCLE IS SQUARE FEET
.O THE IN PIZZA IS ACTUALLY TIMES THE SIZE OF
THE IN PIZZA 7HEN THE DIAMETER IS DOUBLED THE
AREA BECOMES OR TIMES AS LARGE
IN DIAMETER PIZZA
! QR ! y ! y IN IN DIAMETER PIZZA
! QR ! y ! y IN 2ECTANGLE
! LW
! ! (ALF CIRCLE
! ?? QR ! y ?? ?
! y ? ! y 4HE AREA OF THE FIGURE IS M (OLT -C$OUGAL -ATHEMATICS
2ECTANGLE
! LW
!
!
4RIANGLE
! ?? BH
?
! ? ! ?? ! 4HE AREA OF THE FIGURE IS FT 2ECTANGLE
! LW
! !
4WO HALF CIRCLES
! ??QR ! y ?? ! y S D
! y 4HE AREA OF THE FIGURE IS FT S
S
D
D
FULL SQUARES HALFFILLED SQUARES PARTIALLYFILLED SQUARES IS ABOUT FT FULL SQUARES PARTIALLYFILLED SQUARES IS ABOUT
FT 2ECTANGLE
! LW
! ! 4WO HALF CIRCLES
! ??QR ?
! y ? ! y S D
! y 4HE AREA OF THE FIGURE IS M S
S
D
D
2ECTANGLE ! LW
! ! 2ECTANGLE ! LW
!
! 2ECTANGLE ! LW
!
!
4HE AREA OF THE FIGURE IS FT #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
2ECTANGLE
! LW
!
! 4RIANGLE ! ?? BH
?
! ? ! ?? !
4RIANGLE ! ?? BH
?
! ? ?
! ? !
4HE AREA OF THE FIGURE IS CM 4RAPEZOID
! BH
! ! (ALF CIRCLE
! ?? QR ?
! y ? ! y 4HE AREA OF THE FIGURE IS M 4OP RECTANGLE
! LW
!
! "OTTOM RECTANGLE
! LW
!
! 4HE AREA OF THE FIGURE IS FT 0 4HE PERIMETER OF THE FIGURE IS FT
3QUARE
! LW
!
! 4RIANGLE
! ?? BH
?
! ? ! ?? !
4HE AREA OF THE FIGURE IS M 0 4HE PERIMETER OF THE FIGURE IS M
(OLT -C$OUGAL -ATHEMATICS
2ECTANGLE
! LW
! ! (ALF CIRCLE
! ?? QR ?
! y ? ! y ?? ! y 4HE AREA OF THE FIGURE IS M # QD
# y # y (ALF CIRCLE
# y 0 4HE PERIMETER OF THE FIGURE IS M
.O REARRANGING THE COMPOSITE FIGURE DOES NOT
CHANGE ITS PERIMETER OR AREA
0SSSSS
X X X X X X
FT X
Y
!
"
#
( $ X
% &
' 2ECTANGLE ! LW
!
! 2ECTANGLE ! LW
!
! 2ECTANGLE ! LW
!
!
4HE AREA OF THE FIGURE IS UNITS 0 4HE PERIMETER OF THE FIGURE IS UNITS
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
! M
$RAW A 0ICTURE AND 'UESS AND #HECK
!REA OF TOTAL FIGURE AREA OF TRIANGLE AREA OF SQUARE
! S
S MS
$RAW A VERTICAL LINE TO DIVIDE THE FIGURE INTO A
RECTANGLE AREA IN AND TRIANGLE AREA IN 4HE TOTAL AREA IS IN &IND THE AREA OF THE ENTIRE RECTANGLE 4HEN SUBTRACT
THE AREA OF THE TWO HALF CIRCLES
2ECTANGLE
! LW
! ! 4WO HALF CIRCLES
! ?? QR ?
! y ? ! y S D
! y 4HE AREA OF THE FIGURE IS CM # QD
# y # y (ALF CIRCLE
# y 0 4HE PERIMETER OF THE FIGURE IS CM
S
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! ?? BH
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COMBINATION 0 IN
COMBINATION 0 IN
COMBINATION 0 IN
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PERIMETER OF THIS RECTANGLE
FULL SQUARES HALF SQUARES FULL SQUARES
OR PATCHES
PATCHES CARROTS PER PATCH CARROTS
3HE CAN EXPECT TO GROW CARROTS IN HER GARDEN
(OLT -C$OUGAL -ATHEMATICS
R D ! QR ! ! M R D ! QR ! ! IN ! QR ! y ! y y /2
! y ???
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y 4HE AREA OF THE CIRCLE IS FT OR FT R D ! QR ! ! CM R D ! QR ! ! FT S D
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IS MORE PRECISE
0 4HE PERIMETER IS M
# QD
y D
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4HE DIAMETER IS ABOUT CM
! LW
W
W W
4HE WIDTH OF THE COURTYARD IS M
YD
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! LW
! 4HE AREA OF THE KITCHEN IS FT ! ?? H S B B D
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! ! 2ECTANGLE ! LW
! ! 4HE AREA OF THE FIGURE IS CM 2ECTANGLE
! LW
! ! 4RIANGLE ! ?? BH
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! ? !
4RIANGLE ! ?? BH
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! ? ! 4HE AREA OF THE FIGURE IS FT ! ?? BH
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? H
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4HE HEIGHT OF THE TRIANGLE IS CM
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
2ECTANGLE
! LW
! ! 4WO HALF CIRCLES
! ?? QR ?
! y ? ! y S D
! y 4HE AREA OF THE FIGURE IS YD ! LW
!
! 4HE SQUARE OF IS 0!'%3 n
S
S
D
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4HINK AND $ISCUSS
0OSSIBLE ANSWER LIES BETWEEN THE PERFECT
SQUARES AND 3INCE IS CLOSER TO THAN TO
^
^
^
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q IS CLOSER TO q THAN TO q q SO
^
q y 0OSSIBLE ANSWER 3INCE SQUARING AND FINDING A
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! LW
!
! 4HE SQUARE OF IS ! LW
! ! 4HE SQUARE OF IS ^
q ^
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q y ^
! LW
! ! 4HE SQUARE OF IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
q ^
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^
q q q ^
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q y ^
q ^
^
^
q q q ^
q 3INCE IS NEARER IN VALUE TO THAN TO ^
q y (OLT -C$OUGAL -ATHEMATICS
4HE LENGTH OF EACH SIDE OF THE PATROL SQUARE
^
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^
^
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q q q ^
q ^
3INCE IS NEARER TO THAN TO q y %ACH SIDE OF THE PATROL SQUARE IS ABOUT MILES LONG
! LW
!
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q ^
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! LW
! ! 4HE SQUARE OF IS ! LW
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^
! LW
! ! 4HE SQUARE OF IS #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
q ^
^
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q q q ^
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q y (OLT -C$OUGAL -ATHEMATICS
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^
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4HE WINDOW WITH THE PERIMETER OF IN IS LARGER
BECAUSE ITS SIDES ARE IN LONG AND ITS AREA IS
IN 4HE SIDES OF THE WINDOW WITH THE AREA OF
IN ARE ABOUT ?? IN LONG AND THE PERIMETER
IS ABOUT IN
q 4HE LENGTH OF EACH SIDE IS IN
0 4HE PERIMETER OF THE SQUARE IS IN
^
$ q! ^
$ q $ y $ y 4HE HORIZON IS ABOUT KM AWAY
^
q %ACH SQUARE IN THE QUILT IS IN BY IN 4HERE ARE
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%ACH SIDE OF THE QUILT IS IN 4HE DIMENSIONS ARE
IN IN
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! QR y R y R M y R
! QR y R y R YD y R
# QD
# y # YD
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^
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#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
^
q & ^
q ROUNDED TO THE NEAREST WHOLE
NUMBER IS SCALENE
(OLT -C$OUGAL -ATHEMATICS
EQUILATERAL
ISOSCELES
# QD
# y # y IN
! QR ! y ! y IN # QD
# y # y CM
! QR ! y ! y CM A y A y 4HE LADDER TOUCHES THE WALL ABOUT FT ABOVE
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A B C
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q q C C
4HE LENGTH OF THE HYPOTENUSE IS YD
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# QD
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! QR ! y ! y FT B qB q 0OSSIBLE ANSWER 4HE 0YTHAGOREAN 4HEOREM CAN BE
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B 4HE LENGTH OF THE LEG B IS FT
A B A A ?
0OSSIBLE ANSWER ! LEG CAN NEVER BE GREATER THAN
THE HYPOTENUSE 4HE HYPOTENUSE OF A RIGHT TRIANGLE
IS ALWAYS THE LONGEST SIDE SINCE IT IS ALWAYS OPPOSITE
THE GREATEST ANGLE IN THE TRIANGLE THE RIGHT ANGLE
3INCE THE SQUARE OF THE HYPOTENUSE IS GREATER THAN
THE SQUARE OF EITHER LEG THE HYPOTENUSE MUST BE
LONGER THAN EITHER LEG
A A B C
C C C ^
^
q q C C
4HE LENGTH OF THE HYPOTENUSE IS M
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A B C A A ?
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A B C
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qA q ^
#OPYRIGHT © BY (OLT -C$OUGAL
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B B y MM
q A 4HE LENGTH OF THE LEG A IS CM
C
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A B C C
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A B C
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A B C A A ?
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CUBIT y CUBITS
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FT
4HE HEIGHT OF THE PYRAMID IS ABOUT CUBITS
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ALWAYS THE HYPOTENUSE
A A y CM
A 9ES 4HE LENGTHS ARE UNITS AND UNITS AND
THE HYPOTENUSE IS UNITS
3INCE THE 0YTHAGOREAN 4HEOREM IS SATISFIED IT IS
A RIGHT TRIANGLE
B 9ES 4HE LENGTHS ARE UNITS AND UNITS AND THE
HYPOTENUSE IS UNITS
3INCE THE 0YTHAGOREAN 4HEOREM IS SATISFIED IT IS
A RIGHT TRIANGLE
C .O 4HE LENGTHS ARE UNITS AND UNITS AND THE
LONGEST SIDE IS UNITS
p BECAUSE AND DOES NOT EQUAL 3INCE THE 0YTHAGOREAN 4HEOREM IS NOT SATISFIED
IT IS NOT A RIGHT TRIANGLE
D 9ES 4HE LENGTHS ARE UNITS AND UNITS AND THE
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3INCE THE 0YTHAGOREAN 4HEOREM IS SATISFIED IT IS
A RIGHT TRIANGLE
^
A q 4HE LENGTH OF EACH SIDE OF THE BASE IS M
B
A B C
C C C y C
4HE LENGTH OF A DIAGONAL OF THE BASE IS
ABOUT M
! ?? BH
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! 4HE AREA OF ONE SIDE OF THE PYRAMID IS M A B C
C C C y C
4HE DISTANCE FROM ONE CORNER TO THE PEAK IS
ABOUT M
A B C
B B ??
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B B y 4HE HEIGHT OF THE PYRAMID IS M
" M
A B C
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A B C B B B B y 4HE LADDER IS ABOUT FEET TALL
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q #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
^
q (OLT -C$OUGAL -ATHEMATICS
^
KG IS MORE PRECISE BECAUSE IT IS MEASURED
TO THE NEAREST TENTH OF A GRAM RATHER THAN TO THE
NEAREST GRAM
^
q q ^
q ^
q ^
q ^
q 3INCE A FOOT IS A SMALLER UNIT THAN A MILE FT IS
MORE PRECISE
?? LB IS MORE PRECISE BECAUSE IT IS MEASURED TO
THE NEAREST ?? LB RATHER THAN TO THE NEAREST POUND
G IS MORE PRECISE BECAUSE IT IS MEASURED TO
THE NEAREST TENTH OF A GRAM RATHER THAN TO THE
NEAREST GRAM
^
^
^
q q q ^
q 3INCE IS NEARER IN VALUE TO THAN TO ^
q y ^
q ^
^
^
q q q ^
q 3INCE IS NEARER IN VALUE TO THAN TO ^
q y 3INCE A CENTIMETER IS A SMALLER UNIT THAN A METER
CM IS MORE PRECISE
3INCE A QUART IS A SMALLER UNIT THAN A GALLON QT
IS MORE PRECISE
0 0 4HE PERIMETER OF THE TRAPEZOID IS M
^
q ^
^
^
q q q ^
q 3INCE IS NEARER IN VALUE TO THAN TO ^
q y 0 L W
0 S D S D
0 0 4HE PERIMETER OF THE RECTANGLE IS CM
^
q ^
^
^
q q q ^
q 3INCE IS NEARER IN VALUE TO THAN TO ^
q y A B C C
C C CM C
A B C B B B B M
A B C
C C C FT C
A B C B B B B FT
A B C
C C C YD C
# QD
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT FT
A B C
C C C MI y C
A B C
A A A A y M
A B C
C C C IN y C
A B C
A A A A y CM
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CIRCUMFERENCE
PRECISION
SQUARE ROOT
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
# QR
# y # y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT IN
! LW
! ! 4HE AREA OF THE RECTANGLE IS CM ! BH
! ! 4HE AREA OF THE PARALLELOGRAM IS YD ! LW
! ! 4HE AREA OF THE ART PAPER IS IN ! ?? BH
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! ? S D
! 4HE AREA OF THE TRIANGLE IS IN ! ?? H S B B D
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! ? S D S D
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! ? S D S D
! 4HE AREA OF THE TRAPEZOID IS CM (OLT -C$OUGAL -ATHEMATICS
! ?? BH
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! ?? H S B B D
! ?? S D S D
! ?? S D S D
! 4HE AREA OF THE TRAPEZOID IS IN ! ??BH
! ?? S D
! 4HE AREA OF BOTH TRIANGLES IS M ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS M ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS FT ! QR ! y ! y ! y 4HE AREA OF THE CIRCLE IS IN !REA OF RECTANGLE
! LW
! ! !REA OF HALF CIRCLE
! ??QR ?
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!
! !REA OF RECTANGLE ! LW
!
!
!REA OF RECTANGLE ! LW
!
!
4HE AREA OF THE FIGURE IS M #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
^
q ^
^
^
q q q ^
q ^
q y ^
q ^
^
^
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q ^
^
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q y ! S
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4HE LENGTH OF EACH SIDE OF THE GARDEN IS ABOUT
FEET
A B C
C C C C
4HE LENGTH OF THE HYPOTENUSE IS CM
A B C
B B ?
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B B 4HE LENGTH OF THE LEG B IS FT
A B C
A A ?
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A A
4HE LENGTH OF THE LEG A IS M
A B C
C C C C
4HE LENGTH OF THE HYPOTENUSE IS MM
(OLT -C$OUGAL -ATHEMATICS
A B C A A ??
A A y 4HE POLE TOUCHES THE WALL ABOUT FT ABOVE THE
GROUND
#(!04%2 4%34 0!'% 3INCE A CENTIMETER IS A SMALLER UNIT THAN A METER
IS MORE PRECISE
3INCE A YARD IS A SMALLER UNIT THAN A METER YD IS
MORE PRECISE
3INCE AN OUNCE IS A SMALLER UNIT THAN A POUND
OZ IS MORE PRECISE
0 0 4HE PERIMETER OF THE TRAPEZOID IS IN
# QD
D
x D
RD
R R 4HE RADIUS OF THE TUNNEL OPENING IS ABOUT FEET
! BH
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! ! !REA OF RECTANGLE ! LW
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4HE HEIGHT OF THE TRIANGLE IS M
# QR
# y ??
# y 4HE CIRCUMFERENCE OF THE CIRCLE IS ABOUT FT
! QR ! LW
L L
4HE LENGTH OF THE LAB IS FEET
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
^
q ^
q A B C B B B B y x
4HE MISSING LENGTH B IS ABOUT CM
A B C
C
C
C x y C
4HE LENGTH OF THE THIRD SIDE C IS ABOUT MI
A B C
A A A A 4HE MISSING LEG IS CM
S
D
S D
S D
! y ??
! y ??
! y 4HE AREA OF THE CIRCLE IS ABOUT FT ! ?? BH
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! ? S D
! 4HE AREA OF THE TRIANGLE IS CM (OLT -C$OUGAL -ATHEMATICS