Transcript: Challenge #5 | Mar 26, 1999

An opening clip shows the aerial view from a helicopter circling the Great Pyramid of Giza.

(helicopter engine whirring)

In front of a desk with coloured geometrical building blocks on it, Mister C., curly-haired and bearded in his forties, with glasses, wearing a gray coat over a white T-shirt with the face of a youthful Einstein on it, says
HAVE YOU EVER WONDERED HOW
THEY GOT THE FORMULA FOR THE
VOLUME OF A SQUARE-BASED
PYRAMID IN ANCIENT TIMES?
TO FIND OUT HOW MANY BLOCKS
THEY NEEDED TO MAKE THOSE
MAGNIFICENT PYRAMIDS.
I'M GOING TO TAKE A LOOK
AT THE FORMULA RIGHT HERE.

A formula appears on a blue sheet of paper titled “Volume of a Pyramid.”

Mister C. reads and says THE VOLUME OF A PYRAMID,
A SQUARE-BASED PYRAMID
PARTICULARLY, IS ONE THIRD, L
BEING THE LENGTH SIDE OF THE
PYRAMID, L SQUARED
TIMES EIGHT.
BUT YOU KNOW WHAT'S
REALLY CONFUSED ME?
IS WHERE THEY GOT ONE THIRD.
HOW DID THEY KNOW ONE THIRD?
NOW, YOU MIGHT TRY SOMEDAY
TO TAKE A PYRAMID AND MAYBE
CONSTRUCT SIMILAR LITTLE
PIECES OF PYRAMIDS AND
RECONSTRUCT THEM INTO A SOLID
THAT'S EASILY MEASURABLE AND
EASY TO CALCULATE THE
VOLUME, LIKE A CUBE.
NOW THAT IS IN THE CASE WHERE
THE HEIGHT OF THE PYRAMID IS
THE SAME AS THE
LENGTH OF ONE SIDE.
AND IF YOU WERE TO DO THAT AND
COMPARE THE NUMBER OF PYRAMIDS
THAT YOU CAN PUT INTO THAT
CUBE, YOU'D FIND THAT YOU
SHOULD BE ABLE TO FIT THREE.
BUT I'VE NEVER SUCCESSFULLY
BEEN ABLE TO CUT UP A PYRAMID
AND PUT IT INTO A CUBE.
I DON'T KNOW WHERE TO CUT!
SO I GAVE YOU A LITTLE
EXERCISE THAT WAS SUPPOSED TO
INVESTIGATE THAT VERY IDEA.
SO, WHAT I'M GOING TO DO IS
SORT OF ILLUSTRATE WHAT THE
IDEA OF THE WHOLE
INVESTIGATION WAS
AND HOPEFULLY YOU'VE
GOT THESE ANSWERS.

He lays two cubic building blocks in front of him. One is green, the other red.

Mister C. continues WHAT I WANTED TO DO IS, WE'RE
CALLING THIS A PYRAMID.
DOESN'T LOOK MUCH LIKE A
PYRAMID, BUT AS A STARTING
POINT, WE'VE GOT TO START
SOMEWHERE SO WE'LL CALL THAT
A PYRAMID.
AND WHAT WE'RE LOOKING AT IS
THE CUBE WHICH I CAN BASICALLY
FIT OVER TOP OF THIS PYRAMID.
WELL, IT'S EXACTLY THE SAME.
IT'S 1-1.
WELL, DOESN'T LOOK MUCH LIKE A
PYRAMID SO MAYBE IF WE GO TO
THE SECOND STAGE
WHERE WE DO THIS.

He places a pink cube on top of a square base composed of 4 other cubes. He puts a large cube, composed of 8 smaller cubes - two to a side -, beside it.

Mister C. continues OKAY, THIS IS BEGINNING
TO LOOK LIKE A PYRAMID.
IT HAS FIVE BLOCKS WHICH IS
THE VOLUME OF THAT PARTICULAR
LITTLE PIECE.
THIS IS THE CUBE
THAT THAT FITS INTO.
IT'S TWO-BY-TWO-BY-TWO WHICH
IS EIGHT CUBIC PIECES.
THE RATIO OF THIS IS
5-8 OR FIVE EIGHTHS, HMM.
BUT THAT STILL DOESN'T
LOOK MUCH LIKE A PYRAMID.
SO, PERHAPS IF WE
GO ANOTHER STEP.

He adds another tier of cubes to the two figures. There is now a large cube composed of 27 smaller ones beside a pyramid composed of 14 small cubes.

Mister C. continues YOU WILL HAVE A -- THAT'S
BEGINNING TO LOOK MORE LIKE
A PYRAMID, NO DOUBT.
IT'S GOT ONE, PLUS FOUR, PLUS
NINE, THAT'S FOURTEEN BLOCKS.
THIS IS THREE-BY-THREE-BY-THREE WHICH
IS THE GRAND TOTAL
OF 27 BLOCKS.
THE RATIO IS GETTING
CLOSER TO ONE THIRD,
IT'S NOT QUITE THERE.
WELL, LET'S GO GET
ONE MORE STAGE.

The large cube is now composed of 64 small ones, and the large pyramid consists of 30 blocks.

Mister C. continues CLEARLY THIS FITS INTO THAT.
THIS TIME IT'S 14 WHICH WE
HAD BEFORE PLUS ANOTHER 16,
THERE'S 30 BLOCKS.
THIS IS A FOUR-BY-FOUR-BY-FOUR
WHICH IS 64, AND WHAT YOU'LL
SEE IS THAT THAT FRACTION
IS GETTING SMALLER BUT BY A
SMALLER AND SMALLER
INCREMENT AS WELL.
AND THIS IS BEGINNING TO LOOK
MORE AND MORE LIKE A PYRAMID.
SO WHAT WE'RE INTERESTED IN IS
THE FRACTION OF THIS VOLUME AS
COMPARED TO THE VOLUME OF THE
CUBE OR EVEN THE RECTANGULAR
PRISM THAT COVERS IT.
NOW, IF WE KEEP
MOVING ALONG...

The huge pyramid now consists of eight square storeys, each one made up of small cubes, one at the top and 64 at the bottom.

Mister C. continues THIS IS THE EIGHTH STAGE OF
THAT VERY EXERCISE AND YOU
KNOW, EVERY TIME I ADD A
STAGE, THESE SIDES BEGIN TO
SMOOTH OUT JUST LIKE
THOSE REAL PYRAMIDS.
IT BEGINS TO LOOK MORE AND
MORE LIKE A PYRAMID, AND THE
RATIO THAT WE GET WHEN WE
COMPARE THE VOLUME OF THIS IN
TERMS OF THE NUMBER OF BLOCKS
COMPARED TO THE CUBIC COVERS,
AMAZINGLY, GETS CLOSER
AND CLOSER TO ONE THIRD.
I'D LIKE YOU TO TAKE A LOOK AT
THE SHEET THAT I ASKED YOU TO
WORK WITH WHICH IS
THIS ONE RIGHT HERE.
THIS IS WHAT YOU STARTED
WITH WHEN IT WAS BLANK.
I'VE DONE A LITTLE BIT
OF THE ANSWER FOR YOU.

He produces a blue sheet of paper with a six-column chart with figures written in and totaled at the bottom. He appears in a miniature frame at the top left of the paper.

Mister C. continues THE BASE WIDTH IN ALL CASES
IS EXACTLY THE SAME AS THE
PYRAMID NUMBER.
THE NUMBER OF
BLOCKS ON ONE SIDE.
THE AREA OF THE BASE, WELL,
IT'S SIMPLY THE SQUARE OF
WHATEVER THIS NUMBER IS.
SO ONE, TWO, TWO FOUR, THREE
NINE, FOUR SIXTEEN AND SO ON.
IN THIS CASE WE WERE STICKING
TO A HEIGHT WHICH WAS EQUAL TO
THE LENGTH OF THE BASE.
ONE COULD DO THE SAME
EXPERIMENT INVESTIGATION WHERE
YOU HAD A DIFFERENT HEIGHT BUT
THEN IT WOULD BE A CONSTANT
RATIO TO THIS ANYWAY, AND IT
DOESN'T REALLY MATTER TOO MUCH.
THE CUBE THAT BASICALLY TAKES
IN THAT ENTIRE PYRAMID, IF I
WERE TO DROP A CUBE OVERTOP,
IS THE CUBE OF THE BASE.
ONE CUBED IS ONE, TWO CUBED
IS EIGHT, THREE CUBED IS 27.
IN OUR LITTLE INVESTIGATION,
WHAT WE DID TO SIMULATE THE
VOLUME OF THE PYRAMID, WAS TO
COUNT THE BLOCKS THAT WE USED
TO CONSTRUCT IT.
WE TOOK ONE BLOCK HERE, WE
TOOK FIVE IN THE SECOND STAGE,
14 IN THE THIRD,
30 IN THE FOURTH.
WELL, SIMPLY WHAT WE DID AND
WHAT YOU SHOULD'VE BEEN DOING
UP TO THIS POINT, IS
CALCULATING THE FRACTION.
IN OTHER WORDS, THE NUMBER OF
BLOCKS COMPARED TO THE VOLUME
OR THE NUMBER OF BLOCKS IN
THE CUBE THAT ENCLOSED
THE ENTIRE PYRAMID.
SO THIS IS ONE OVER
ONE WHICH IS ONE.
FIVE EIGHTHS, POINT
SIX, SEVEN, FIVE.
THAT SHOULD BE SIX, TWO, FIVE.
14 OVER 27, POINT
FIVE, ONE, NINE.
30 OVER 64, POINT
FOUR, SIX, NINE.
WHEN WE GO UP TO THE TENTH
STAGE, THESE ARE THE NUMBERS
YOU'LL NOTICE THAT WE GET
POINT THREE, EIGHT, FIVE.
IF YOU HAD A
POWERFUL CALCULATOR,
THESE ARE THE NUMBERS.
AND NOTICE THAT WE'RE GETTING
TO POINT THREE, THREE, FOUR.
NOW, WHAT I CAN SAY AT THIS
POINT IS THAT THESE NUMBERS
ARE GETTING VERY CLOSE TO
A CERTAIN FRACTION, AND THE
FRACTION THAT WE'RE GETTING
CLOSE TO, LET'S SEE.

A blue sheet reads “Approximately, what fraction is this decimal getting close to as “x” increases?” He writes in figures below the text. On the last line, he puts a dot over the last three in the series.

Mister C. continues .334, IF I WENT FURTHER IT WOULD
HAVE MORE AND MORE THREES.
ONE COULD SAY THAT WE
HAVE .33333 REPEATED,
WHICH, OF COURSE, IS
THE FRACTION ONE THIRD.
NOW SOMEBODY HAS DONE SOME
PRETTY SHREWD OBSERVATIONS AND
ANTIQUITY TO BE ABLE TO FIGURE
THAT OUT PERHAPS WITHOUT
THESE TECHNIQUES.
BUT NONETHELESS, WHAT
YOU CAN SEE BY DOING THE
INVESTIGATION, WE HAVE COME
TO THE CONCLUSION OR YOU
SHOULD'VE COME TO THE
CONCLUSION, THAT A SQUARE-BASED
PYRAMID, WITH RESPECT TO THE
CUBE OR THE RECTANGULAR PRISM
IT ENCLOSES, IT'S ABOUT
ONE THIRD ITS SIZE.
IN FACT, EXACTLY ONE THIRD.
SO THAT'S THE ANSWER
TO THAT QUESTION.

He produces another sheet that reads “How does this fraction help you find a formula for the volume of a square-based pyramid?” and writes a formula on it. The formula reads “V equals one third L squared times H.”

Mister C. continues WELL, JUST REPEATING THE
ACTUAL FORMULA, IT IS THIS
CONSTANT AND I WAS USING L SO
I'LL STICK TO L WHICH IS THE
LENGTH OF ONE SIDE.
AND THE HEIGHT.
SO THIS IS THE ACTUAL
FORMULA FOR THE VOLUME OF
A SQUARE-BASED PYRAMID.
HOPEFULLY YOU CAME
TO THAT CONCLUSION.
MY LAST QUESTION WAS TO DO
WITH WAS FINDING THE FORMULA
FOR THE VOLUME OF A
TRIANGLE-BASED PYRAMID.

The new question appears and reads “Can you describe a way to find the formula for the volume of a triangle-based pyramid or tetrahedron?”

Mister C. continues NOW A TRIANGLE-BASED PYRAMID
IS CALLED A TETRAHEDRON.
BASICALLY IT'S A THREE
DIMENSIONAL OBJECT WHICH IS
MADE UP OF FOUR
TRIANGULAR PARTS.
A TRIANGULAR BASE AND
THREE TRIANGULAR SIDES.
NOW TO DO THIS INVESTIGATION,
ONE WAY YOU COULD DO IT IS BY
DOING A SERIES OF TRIANGULAR
BLOCKS IN ESSENCE.
SO THE FIRST ONE WOULD ONE
TRIANGULAR PRISM, THE SECOND
ARRANGEMENT WOULD PROBABLY
LOOK SOMETHING LIKE THIS,
THE THIRD ARRANGEMENT
WOULD BE THE THIRD LAYER.
JUST A LITTLE WEE BIT
LARGER, SOMETHING LIKE... THIS.
WHAT YOU'LL FIND IS THAT THE
SEQUENCE IS EXACTLY THE SAME.
SO THAT ONCE AGAIN YOU'RE
GOING TO GET THAT INTERESTING
FRACTION, ONE THIRD.
WHAT I WANT TO DEAL WITH NOW
IS THAT YOU'RE GOING TO BE
DOING A NEW SET OF
PROBLEMS THIS WEEK.
THESE PROBLEMS INVOLVE A
VARIETY OF DIFFERENT SHAPES,
GOING BACK TO THE SQUARE-BASED
PYRAMID AGAIN, BUT WHAT I'M
INTERESTED IN AT THIS POINT IS
THE WHOLE CONCEPT TO DO WITH
THE SURFACE AREA OF A PYRAMID
WHICH WOULD BE THE AREA OF THE
FOUR TRIANGLES AND MAYBE
THE BASE IF WE WANTED IT.

On a blue sheet of paper in front of him displaying a square-based pyramid with opened-up sides looking like a four-pointed star, he ticks off the opened-up sides and shades in the base.

Mister C. continues BUT I'M NOT WORRIED ABOUT
THAT, THAT'S SIMPLY A SQUARE.
SO GETTING THE AREA OF THE
BASE IS JUST B SQUARED
OR L SQUARED.
BUT WHAT I'M INTERESTED
IN IS THE AREA OF EACH
OF THESE TRIANGLES.
WELL THE AREA OF A TRIANGLE
IS NOT MUCH OF A PROBLEM OR
IT SHOULDN'T BE MUCH
OF A PROBLEM FOR YOU.
BUT YOU KNOW SOMETHING,
USUALLY WE'RE NOT GIVEN
THE RIGHT INFORMATION.
WHEN YOU GET...
A PYRAMID.
IT'S GOING TO LOOK
SOMETHING LIKE THAT.

He takes an opend-up paper pyramid and closes it, then shows the sides to be measured.

Mister C. continues I'M GOING TO CLOSE UP THE
FRONT END FOR A MOMENT.
WHAT YOU'RE NORMALLY GIVEN
WHEN YOU GET A PYRAMID IS THE
LENGTH OF THE BASE AND
THE VERTICAL HEIGHT.
YOU CAN SEE MY HAND JUST
OFF TO THE SIDE THERE.
AND THE VERTICAL HEIGHT.
BUT WHAT WE REALLY NEED IS
THIS PORTION, THIS HEIGHT
RIGHT HERE.

On the drawing, he draws a dotted line showing the base to apex height of one triangular side. He writes “slant” beside it.

Mister C. continues THIS IS SOMETIMES CALLED OR
IS CALLED THE SLANT HEIGHT.
NOW I WON'T PRINT IT TOO BIG,
BUT IT'S SOMETHING CALLED THE
SLANT HEIGHT.
NOW WHEN WE LOOK IN FROM THE
FRONT, HERE'S THE SECRET.
WHAT YOU'LL SEE INSIDE, THERE
WE ARE, WHAT YOU'LL SEE INSIDE
IS ANOTHER TRIANGLE.
THAT'S THE MAGIC TRIANGLE.
THE BASE OF THIS GOES RIGHT
TO THE CENTRE OF THE PYRAMID.
WHAT YOU'LL SEE IS THAT LENGTH
AT THE BOTTOM THERE, WAY DOWN
ON THE BOTTOM, IS HALF
THE WIDTH OF THE PYRAMID.
WHAT WE KNOW ALREADY
IS THE HEIGHT OF THIS.
SO THAT'S THIS
CHUNK RIGHT HERE.
WHAT WE WANT TO
KNOW IS THIS.

He opens up the folded pyramid to show the slant height of one side. the he draws a right-angled triangle and marks in the base and height, and an arrow showing the hypotenuse. Then he writes “L equals 4” and “H equals 6.” Below, he draws another right-angled triangle marking the sides as 6 and 2 and the hypotenuse as “square root of 40.”

Mister C. continues AND THAT'S THE SLANT HEIGHT.
NOW I'M GOING TO HAVE TO DO A
LITTLE BIT OF CALCULATION HERE.
HERE'S OUR TRIANGLE, IT'S
A RIGHT-ANGLE TRIANGLE.
WE KNOW THAT THIS CHUNK DOWN
ALONG THE BOTTOM, IF I CALLED
THE WIDTH OF THE WHOLE THING
L, THIS WOULD BE L DIVIDED BY
TWO AND THIS IS THE HEIGHT.
HOW CAN WE FIND THIS?
THE ANSWER LIES IN THE
PYTHAGOREAN THEOREM.
REMEMBER THAT THE SQUARE ON
THE HYPOTENUSE IS EQUAL TO THE
SUM OF THE SQUARES OF
THE OTHER TWO SIDES.
LET'S SAY FOR ARGUMENTS SAKE,
I'M GOING TO CREATE A SLIGHTLY
SIMPLER SITUATION, LET'S SAY
THAT THE LENGTH IS EQUAL TO
FOUR UNITS AND THE
HEIGHT IS EQUAL TO SIX.
SO THAT IN THIS PARTICULAR
CASE, IN OUR TRIANGLE...
THE LENGTH DIVIDED
BY TWO WOULD BE TWO.
THE HEIGHT WOULD BE SIX.
SO WHAT I WOULD DO TO
CALCULATE THIS LENGTH...
NOW, ONCE YOU KNOW THAT, GO
BACK TO THE ORIGINAL TRIANGLE
THAT WAS ON THE SIDE.
YOU NOW KNOW THIS WHOLE LENGTH
HERE WHICH IS FOUR UNITS.
YOU NOW KNOW THE SLANT HEIGHT
WHICH IS THE ROOT OF 40.
CAN YOU GET THE AREA
OF THAT TRIANGLE?
WHY OF COURSE YOU CAN.
BECAUSE THE AREA OF THE
TRIANGLE IT'S ONE HALF.
THIS LENGTH HERE,
TIMES THE HEIGHT.
SO IN THIS CASE IT'S ONE HALF TIMES 4
TIMES THE ROOT OF 40.
AND YOU GET YOUR ANSWER.
IT'S AS SIMPLE AS THAT.
OKAY, SO, IF YOU'RE FINDING
THE SURFACE AREA OF A PYRAMID,
YOU FIND THE SLANT HEIGHT, YOU
FIND THE AREA OF ONE TRIANGLE,
MULTIPLY IT BY FOUR, AND THAT
GIVES THE AREAS OF ALL FOUR
SIDES AND AWAY YOU GO.
I WANT TO DEAL WITH ONE MORE
WHICH ISN'T DIRECTLY PART OF
YOUR INVESTIGATION, BUT IT KIND
OF FOLLOWS FROM DOING THE
SURFACE AREA OF A PYRAMID.
IN THIS PARTICULAR
CASE WE HAVE A CONE.

On a blue sheet of paper titled “Net Cone nunmber 2” with the drawing of an extended cone surmounted by a circle, he puts a rolled up paper cone.

Mister C. continues THIS CONE IS REPRESENTED
BY THIS NET.
SO THE BASE, WHICH IS THE
BOTTOM, WHICH IS A PERFECT
CIRCLE, WHICH IS THIS PART UP
HERE, AND THIS PART OF THE CONE
WHEN YOU WRAP IT OUT IS
THIS LITTLE CHUNK HERE.
AS IT TURNS OUT, THAT
THIS IS PART OF A CIRCLE. OKAY.
NOW THE QUESTION IS WHAT
PART OF WHAT CIRCLE?
I CAN EXTEND THIS AROUND,
THERE'S A BIG CIRCLE HERE,
IT'S PART OF A CIRCLE.
WHAT DO I KNOW USUALLY WHEN
I'M USUALLY WORKING WITH A CONE?
WHAT I DO KNOW IS THE RADIUS.
AND LET'S SAY FOR INTEREST'S
SAKE THAT THE RADIUS
IN THIS CASE IS TWO.
WHAT CAN I CALCULATE THEN?
I CAN CALCULATE THE
CIRCUMFERENCE OF THIS CIRCLE.
CIRCUMFERENCE OF THIS CIRCLE
IS PI TIMES D WHICH WOULD BE
FOUR, D IS DIAMETER WHICH
IS TWICE RADIUS, TIMES PI.
SO WE GET A FIGURE.
SO THAT'S HOW FAR
IT IS AROUND HERE.
WELL, AS YOU KNOW, WITH A CONE
THIS EDGE HAS TO WRAP RIGHT
AROUND THE CIRCUMFERENCE.
YOU CAN SEE IT WITH THIS.
THIS EDGE AT THE BOTTOM HERE
HAS TO WRAP RIGHT AROUND
THE CIRCLE.
SO WHAT WE DO KNOW, NOW,
IS THE LENGTH OF THIS ARC.
IT'S THE SAME AS THIS.
HMM.
ONE OTHER THING THAT
WE ARE GENERALLY ABLE TO
CALCULATE USING THE SAME
SET-UP THAT WE DID WITH THE
PYRAMID IS THE SLANT
HEIGHT OF THE CONE.
SO LET'S SAY WE KNOW THAT.
LET'S SAY, FOR ARGUMENT'S SAKE,
THAT THE SLANT HEIGHT IS 10.
WELL, REMEMBER THAT THIS IS
PART OF A BIG CIRCLE, AND IF WE
KNOW THIS SLANT HEIGHT, IT IS
THE RADIUS OF THIS BIG CIRCLE,
ISN'T IT?

On the drawing, he writes in “R equals 2” in the small circle, “C equals 4 Pi R” on the arc forming part of the large circle and “Slant S equals 10” on the radius of the large circle.

Mister C. continues SO WE KNOW THE CIRCUMFERENCE
OF THE BIG CIRCLE.
I'M GOING TO CALL IT
CIRCUMFERENCE BIG.
TWO TIMES 10 IS 20 PI.
WELL, THE QUESTION THEN IS I
CAN CALCULATE THIS DISTANCE
AROUND, I CAN CALCULATE THE
AREA OF THE BIG CIRCLE.
LET'S SEE, THAT'S PI
R SQUARED I BELIEVE.
SO THE AREA OF THE BIG CIRCLE
IS PI, R SQUARED WHICH WOULD
BE 10 SQUARED OR 100 PI.
HMM, OKAY.
THIS IS PART OF THAT CIRCLE.
WHAT PART IS IT?
WELL, IF WE COMPARE THE
ARC LENGTH TO THE TOTAL
CIRCUMFERENCE THEN THIS
REPRESENTS THAT PORTION
OF THE CIRCLE.
SO IN OTHER WORDS, OF THE
WHOLE CIRCLE, THIS IS A
FRACTION OF IT, MADE UP OF
FOUR PI DIVIDED BY THE WHOLE
CIRCUMFERENCE WHICH IS 20 PI.
THE NICE THING ABOUT THIS IS
PIs CANCEL, AND THIS REDUCES
TO ONE OVER FIVE.
IN OTHER WORDS, THIS IS ONE
FIFTH OF THE AREA OF THE
ENTIRE CIRCLE.
WELL, IF THE AREA OF THE CIRCLE
IS 100 PI, THEN THE AREA OF
THIS IS EQUAL TO ONE FIFTH
OF THAT CIRCLE TIMES 100 PI.
OR, 20 PI UNITS SQUARED.
SO IF YOU NEED THE SURFACE
AREA OF A CONE, THAT'S
BASICALLY WHAT YOU'RE DOING.
NOW, I WANT TO MENTION A
COUPLE OF THE PROBLEMS THAT
YOU'RE DOING IN
THE UPCOMING WEEK.
ONE OF THEM INVOLVES A MOST
AMAZING STRUCTURE, AND THAT'S
THE “CIRCUS MAXIMUS” IN ROME.
NOW THIS IS AN ANCIENT
BUILDING, AND WE'RE GOING TO
TAKE A LOOK AT A PICTURE OF IT.
AND AS WE DRAW BACK,
WHAT YOU'LL SEE IS AN
ABSOLUTELY MASSIVE BUILDING.
YOU CAN SEE ALL THE OTHER
BUILDINGS AROUND IT.

On an aerial photo of downtown Rome, an enormous roofless oblong building
with rounded ends appears, looking somewhat like a racetrack.

Mister C. continues IT IS HUGE.
IT'S LONG AND, BELIEVE IT
OR NOT, THEY HAD 250,000
SPECTATORS IN THERE AT A
TIME TO WATCH WHATEVER THE
ROMAN GAMES WERE OF
THAT PARTICULAR ERA.
YOUR JOB -- ONE OF THE JOBS --
IS TO CALCULATE THE VOLUME
INSIDE THAT STADIUM BUT ALSO
TO CALCULATE HOW MANY ROWS
OF SEATS YOU WOULD NEED
TO SEAT 250,000 PEOPLE.
IT'S JUST AMAZING.
NOW, WE'VE GOT A FEW SHOTS
HERE COURTESY OF FIDO, OF THE
SKYDOME AND THIS PARTICULAR
PROBLEM ASKS YOU TO CALCULATE
THE VOLUME OF THE SKYDOME.

The Skydome appears from the air, first from above and then from the side. then a Boeing 747 appears in flight.

Mister C. continues AS YOU CAN SEE, IT'S
FUNDAMENTALLY PART OF A
CYLINDER WITH PART
OF A SPHERE ON THE TOP.
[propeller whirring]
NOW, WE'RE LOOKING AT IT
FROM THE SIDE VIEW NOW.
IT IS AN ABSOLUTELY
HUGE BUILDING AS WELL.
SO, PART OF YOUR EXERCISE IS
TO CALCULATE THE VOLUME OF
THAT AND WHAT IT CONTAINS.
WHAT WE'RE REALLY, REALLY
INTERESTED IN IS THEY MAKE A
CLAIM THAT SO MANY 747s
CAN FIT INTO THE SKYDOME.
SO I WANT YOU TO IN JUST A
MOMENT OR TWO, TAKE A LOOK
AT A PICTURE OF 747.
NOW I'M NOT GIVING YOU THE
INFORMATION ON THE ACTUAL
DIMENSIONS OF THE 747.
WHAT YOU HAVE TO DO IS
MAYBE CHECK THE INTERNET,
DO SOME RESEARCH YOURSELF.
JUST AN AMAZING AIRPLANE.
(roaring and whining jet engines)
WHAT YOU'LL HAVE TO DO IS
ACTUALLY DO A BIT OF RESEARCH,
FIND OUT HOW BIG THAT
747 IS AND FIGURE OUT
HOW MANY WOULD FIT INTO IT.
THERE'S ANOTHER PROBLEM
THAT'S ASSOCIATED WITH THAT.
IN LAS VEGAS THERE'S A
CASINO CALLED LUXORS.
AND THE LUXORS CASINO IS AN
EXACT REPLICA, NOT FULL-SIZE
THOUGH, OF THE
PYRAMIDS IN EGYPT.
AND ONCE AGAIN, ONCE YOU KNOW
HOW MANY OF THESE AIRPLANES
CAN FIT INTO THE SKYDOME,
YOU CAN FIGURE OUT HOW MANY
AIRPLANES WOULD FIT
INTO LUXORS AS WELL,
GIVEN ITS VOLUME, TOO.
SO I WISH YOU THE VERY BEST OF
LUCK WITH THE NEXT PROBLEM SET
THAT YOU HAVE TO DO.
I HOPE THAT THIS HAS BEEN
HELPFUL, AND WE'LL SEE YOU NEXT
LIVE PROGRAM.

End of Program.

Watch: Challenge #5