HELLO, AND WELCOME TO A QUICK LESSON
ON ARC LENGTH OF A CIRCLE
AND ALSO THE AREA OF A SECTOR OF A CIRCLE.
AND THE GOALS ARE TO DETERMINE THE LENGTH OF AN ARC
AND DETERMINE THE AREA OF A SECTOR.
THE LENGTH OF AN ARC OF A CIRCLE
DEPENDS ON BOTH THE ANGLE OF ROTATION
AND THE RADIUS LENGTH OF THE CIRCLE.
RECALL THAT A RADIAN IS THE MEASURE OF AN ANGLE
THAT CUTS OFF AN ARC LENGTH,
THE SAME LENGTH AS THE RADIUS
SO THAT A HALF ROTATION IS PI RADIANS
OR A LITTLE MORE THAN THREE RADIUS LENGTHS
AROUND THE CIRCLE.
SO IF THE RADIUS IS 4 CENTIMETERS
THE LENGTH OF THE HALF CIRCLE ARC WOULD BE PI/RADIUS LENGTHS
OR 4/PI CENTIMETERS IN LENGTH AS WE SEE HERE BELOW.
SO IT FOLLOWS THAT IF WE TOOK THE ARC LENGTH,
4/PI CENTIMETERS
AND DIVIDED IT BY THE RADIUS OF 4 CENTIMETERS
WE WOULD HAVE THE ANGLE MEASURE IN RADIANS
OR IN THIS CASE WE CAN SEE WE WOULD HAVE PI RADIANS
FOR OUR ANGLE.
SO IN GENERAL WE HAVE THE ARC LENGTH DIVIDED BY THE RADIUS
AS EQUAL TO THE ANGLE IN RADIAN MEASURE.
SO IF WE TOOK THIS RELATIONSHIP
AND PERFORMED CROSS PRODUCTS
WE WOULD HAVE "S" x 1
OR ARC LENGTH IS EQUAL TO RADIUS x THETA,
WHICH WE SEE HERE AS OUR GENERAL FORMULA
FOR ARC LENGTH.
WE NEED TO BE THAT OUR ANGLE THETA
MUST BE IN RADIAN MEASURE.
A CIRCLE HAS RADIUS 9.5 CENTIMETERS.
FIND THE LENGTH OF THE ARC INTERCEPTED BY A CENTRAL ANGLE
WITH MEASURE 120 DEGREES.
SO HERE IS OUR FORMULA FOR ARC LENGTH.
WE KNOW OUR RADIUS IS 9.5 CENTIMETERS
AND OUR CENTRAL ANGLE IS 120 DEGREES.
WELL, FIRST WE NEED TO CONVERT THIS INTO RADIANS.
SO RECALL THAT WE MULTIPLY THAT PI RADIANS
OVER 180 DEGREES TO DO THE CONVERSION.
AND THESE HAVE A COMMON FACTOR OF 60.
THERE'S 2 SIXTIES IN 120 AND 3 SIXTIES IN 180.
SO WE HAVE 2/PI DIVIDED BY 3 RADIANS.
SO NOW WE CAN GO AHEAD AND FIND THE ARC LENGTH.
"S" IS EQUAL TO THE RADIUS
WHICH IS 9.5 CENTIMETERS TIMES THE ANGLE IN RADIANS
WHICH IS 2/PI DIVIDED BY 3.
AND SO OUR ARC LENGTH WILL BE THIS PRODUCT,
9.5 x 2/PI DIVIDED BY 3 RADIANS.
WE'LL GET OUR DECIMAL APPROXIMATION.
IT DOESN'T TELL US WHERE TO ROUND
SO LET'S GO AHEAD AND ROUND THIS TO THE TENTHS.
SO WE HAVE APPROXIMATELY 19.9 CENTIMETERS
FOR THAT ARC LENGTH.
NOW, IF THEY ASK FOR THE EXACT ARC LENGTHS
WE SHOULD LEAVE IT IN TERMS OF PI.
LET'S GO AHEAD AND DO THAT AS WELL.
SO LET'S GO AHEAD AND GO BACK TO OUR CALCULATOR.
BECAUSE WE'RE GOING TO LEAVE PI IN OUR ANSWER,
WE'RE GOING TO MULTIPLY 9.5 x 2/3,
CONVERT THAT TO A FRACTION.
WE HAVE 19/3.
YOU PROBABLY RECOGNIZE THIS AS 6 1/3
BUT SINCE WE'RE LEAVING IT IN TERMS OF PI
WE'LL RIGHT 19PI/3.
SO 19PI/3 CENTIMETERS WOULD BE OUR EXACT ANSWER.
THIS WOULD BE OUR DECIMAL APPROXIMATION.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE.
THE FREE THROW LINE ON A NCAA BASKETBALL COURT IS 12 FEET,
AS WE SEE HERE IN BLUE.
IN INTERNATIONAL COMPETITION IT IS ONLY ABOUT 11.81 FEET,
AS WE SEE HERE INSIDE.
HOW MUCH LONGER IS THE HALF CIRCLE
ABOVE THE FREE THROW LINE IN AN NCAA COURT?
SO WE'LL CALL THIS OUTER ARC LENGTH "A".
AND WE'LL CALL THE INNER ARC LENGTH "B".
SO, AGAIN, OUR ARC LENGTH FORMULA, S= (R) THETA.
SO ARC LENGTH "A" WAS EQUAL TO THE RADIUS
x THE ANGLE IN RADIANS.
WELL, THE RADIUS FOR THE LARGER ARC
WOULD BE HALF OF THE DIAMETER OR 6 FEET.
AND OUR ANGLE THETA IS HALF OF A REVOLUTION OR PI RADIANS.
AND FOR RIGHT NOW WE'LL GO AHEAD
AND LEAVE IT IN EXACT FORM MEANING IN TERMS OF PI.
SO THIS IS 6/PI, AGAIN, OUR UNITS WOULD BE FEET.
AND THE SMALLER ARC LENGTH WE'LL CALL IT S-SUB-B
IS EQUAL TO, AGAIN, THE RADIUS WOULD BE HALF OF 11.81 FEET.
11.81 DIVIDED BY 2.
AND THEN, AGAIN, OUR ANGLE THETA IS PI.
SO, AGAIN, WE'LL LEAVE THIS IN TERMS OF PI FOR NIGHT NOW.
BUT THE QUESTION IS HOW MUCH LONGER
IS THE LARGER HALF CIRCLE THAN THE SMALLER HALF CIRCLE.
SO WE NEED TO FIND THE DIFFERENCE.
SO LET'S GO AHEAD AND GO TO OUR CALCULATOR.
AGAIN, IT DOES NOT SAY WHERE TO ROUND
BUT WE ARE GOING TO ROUND ON THIS PROBLEM.
SO OUR FIRST ARC LENGTH WAS 6/PI - THE SECOND ARC LENGTH.
AND HERE IS OUR DIFFERENCE IN ARC LENGTH.
SO THE DIFFERENCE IN ARC LENGTH
IS APPROXIMATELY 0.298 FEET.
IF WE WANTED TO CONVERT THIS TO INCHES
WE'D HAVE TO MULTIPLY BY 12.
LET'S GO AHEAD AND DO THAT,
AND IT LOOKS LIKE IT'S ABOUT 3.581 INCHES.
SO HERE WE HAVE OUR ANSWER IN FEET
AND HERE'S OUR ANSWER IN INCHES.
LET'S GO AHEAD AND TAKE A LOOK AT THE AREA OF A SECTOR.
A SECTOR OF A CIRCLE IS THE PORTION OF THE INTERIOR
OF A CIRCLE INTERCEPTED BY A CENTRAL ANGLE.
SO HERE WE HAVE A SECTOR THAT IS COLORED GREEN.
YOU CAN THINK OF THIS AS A PIECE OF PIE.
IF THE CENTRAL ANGLE FOR A SECTOR IS MEASURED IN RADIANS
THEN THETA DIVIDED BY 2/PI MAKES UP THE FRACTION
OF A COMPLETE CIRCLE WITH AREA= PI R SQUARED.
THEREFORE, THE AREA OF A SECTOR
IS THETA DIVIDED BY 2/PI x THE AREA OF A CIRCLE.
AND HERE WE CAN SEE THE PI WILL SIMPLIFY OUT
AND WE'RE LEFT WITH 1/2 R SQUARED THETA WHERE THETA,
AGAIN, MUST BE IN RADIANS.
LET'S TAKE A LOOK AT AN EXAMPLE OF THIS.
CROPS ARE OFTEN GROWN USING A TECHNIQUE
CALLED CENTER PIVOT IRRIGATION
THAT RESULTS IN CIRCULAR SHAPED FIELDS.
AS YOU'VE PROBABLY SEEN FROM AN AIRPLANE.
IF THE IRRIGATION PIPE IS 450 METERS IN LENGTH
WHAT IS THE AREA THAT CAN BE IRRIGATED
AFTER A ROTATION OF 240 DEGREES.
SO THE IRRIGATION PIPE OF 450 METERS
WOULD BE THIS LENGTH HERE.
AND SO IF IT ROTATES 240 DEGREES
IT WOULD CARVE OUT THIS AREA IN GREEN
OR WATER THIS MUCH LAND.
SO LET'S GO AHEAD AND FIGURE OUT
WHAT THE AREA OF THIS LAND WOULD BE.
SO OUR AREA WILL BE 1/2 x 450 METERS SQUARE x THETA
WHICH WE'LL HAVE TO CONVERT TO RADIANS.
WE HAVE A COMMON FACTOR OF 60 BETWEEN THESE TWO.
THERE ARE 3 SIXTIES IN 180 AND 4 IN 240.
SO WE HAVE 4/PI DIVIDED BY 3 RADIANS.
LET'S GO AHEAD AND GO TO OUR CALCULATOR.
LET'S FIRST FIND THE EXACT VALUE
AND THEN WE'LL FIND A DECIMAL APPROXIMATION.
SO TO FIND THE EXACT VALUE WE'LL LEAVE OFF THE PI
UNTIL WE GET TO OUR FINAL ANSWER.
SO WE HAVE 1/2 OR .5 x 450 SQUARED x 4/3.
AGAIN, WE'LL ADD THE PI BACK IN OUR FINAL ANSWER.
SO WE HAVE 135,000 PI SQUARE METERS.
AGAIN, WE'RE DEALING WITH AREA NOW SO WE HAVE SQUARE UNITS.
AND LET'S GO AHEAD AND FIND THE DECIMAL APPROXIMATION HERE
BY MULTIPLYING BY PI.
SO WE'LL JUST TAKE THIS PREVIOUS ANSWER
AND MULTIPLY IT BY PI.
AND THEN WE HAVE APPROXIMATELY 424,115 SQUARE METERS.
OKAY. I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.
[ ROCK MUSIC ]