# 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.