CirclesExpanded--FullProgram.mp4

Geometry Applications: Circles

[Music]

Title: Geometry Applications: Circles

Title: Geometry Applications: Circles: Geometry Basics

IN CHOCO CANYON, NEW MEXICO

THE PUEBLO INDIANS BUILT AN ELABORATE VILLAGE

IN THE MIDST OF THE DESERT.

THE SPANISH REFERRED TO IT AS PUEBLO BONITO,

WHICH MEANS "PRETTY VILLAGE".

BUT BY THE TIME THE SPANIARDS SAW IT

IT WAS ALREADY THE ABANDONED SET OF RUINS

THAT SURVIVE TO THIS DAY.

BUT THE DESERT DOESN'T HIDE EVERYTHING.

SEEN FROM ABOVE, THIS PUEBLO VILLAGE

REVEALS A NUMBER OF CIRCULAR STRUCTURES.

THE PUEBLO INDIANS REFER TO THEM AS KIVAS,

SACRED PLACES WHERE CEREMONIAL RITUALS TOOK PLACE.

WHY A CIRCULAR STRUCTURE?

CIRCLES HAVE CULTURAL MEANING

IN THAT THEY REFER TO THE CYCLES OF NATURE.

IN PARTICULAR, FOUR NATURALLY RECURRING

CELESTIAL EVENTS WERE OF IMPORTANCE

TO ANCIENT CULTURES.

THE WINTER SOLSTICE, WHEN THE EARTH'S AXIS IS TILTED

FARTHEST AWAY FROM THE SUN; THE SUMMER SOLSTICE,

WHEN THE EARTH'S AXIS IS TILTED CLOSEST TO THE SUN;

THE FALL EQUINOX, WHEN THE EARTH'S AXIS IS NOT

TILTED IN ANY DIRECTION AWAY OR TOWARD THE SUN.

DAY AND NIGHT ARE EQUAL IN LENGTH, BUT AFTER THIS POINT

THE DAYS BECOME SHORTER IN LENGTH

WHILE NIGHT TIME IS LONGER.

THE SPRING EQUINOX, LIKE THE FALL EQUINOX,

DAY AND NIGHT ARE EQUAL IN LENGTH,

BUT AFTER THIS POINT THE DAYS BECOME LONGER.

THIS ANNUAL CYCLE IS REPRESENTED BY THE

CIRCULAR SHAPE OF THE KIVA.

AND THE PUEBLO, LIKE MANY ANCIENT CULTURES,

WERE INTERESTED IN IDENTIFYING

THESE IMPORTANT EVENTS.

THE LOCATION OF THE SUN ON THE HORIZON IS AN INDICATOR

OF EACH OF THESE FOUR SEASONAL CHANGES.

PUEBLO BONITO WAS BUILT IN SUCH A WAY TO HIGHLIGHT

WHEN THESE SEASONAL CHANGES OCCURRED.

THE ALIGNMENT OF SHADOWS CREATED BY THE SUN

WAS THE BEST INDICATOR OF THE SOLSTICE OR EQUINOX.

CIRCULAR STRUCTURES ARE IDEAL FOR

TRACKING THESE ASTRONOMICAL CHANGES.

FOR EXAMPLE, THE ARRANGEMENT OF ROCKS AT STONEHENGE

IN ENGLAND IS MEANT TO ALIGN WITH THE SOLSTICES

AND EQUINOXES IN SUCH A WAY THAT

THE SUN APPEARS BETWEEN THE MASSIVE STONES.

WHENEVER YOU'RE ON THE EARTH'S SURFACE

YOUR POSITION MARKS OUT A CIRCULAR LINE OF SIGHT

TO THE HORIZON.

YOUR POSITION IS AT THE CENTER OF THE CIRCLE.

THE POSITION OF THE SUN ON THE HORIZON

FORMS A RADIUS OF THE CIRCLE.

THE VARIOUS POSITIONS ON THE HORIZON

FOR THE VARIOUS SOLSTICES AND EQUINOXES

FORM DIFFERENT RADII WITH THE CENTER OF THE CIRCLE.

FINALLY, THE SEGMENTS CONNECTING THESE

VARIOUS POSITIONS OF THE SUN

FORM CHORDS THAT DEFINE INTERCEPTED ARCS.

WHETHER A KIVA OR AN OBSERVATORY, CIRCULAR

STRUCTURES ARE IDEAL FOR STUDYING THE NIGHT SKY.

TAKING ADVANTAGE OF THE GEOMETRY OF CIRCLES,

THESE STRUCTURES HAVE DEGREES OF FREEDOM

THAT OTHER STRUCTURES DON'T HAVE.

IN THIS PROGRAM YOU'LL SEE HOW CIRCLES

HAVE BEEN USED TO SOLVE REAL WORLD PROBLEMS

AS WELL AS CERTAIN DESIGN CHALLENGES.

IN PARTICULAR, THIS PROGRAM WILL COVER

THE FOLLOWING KEY CONCEPTS

Title: Geometry Applications: Circular Arcs

THE YEAR WAS 80 AD, AND THE FLAVIAN AMPHITHEATER

WHICH WE NOW REFER TO AS THE ROMAN COLOSSEUM,

OPENED ITS GATES.

STARTED UNDER THE REIGN OF EMPEROR VESPASIAN,

THE COLOSSEUM WAS FINISHED WHEN HIS SON,

TITUS, BECAME EMPEROR.

THE SEATING CAPACITY OF THE COLOSSEUM WAS 50,000

- AN ENORMOUS SIZE FOR THE TIME.

SEEN FROM ABOVE, THE COLOSSEUM IS ELLIPTICAL

IN SHAPE.

IT WAS 188 METERS LONG AND 156 METERS WIDE.

IT WAS ALSO 50 METERS HIGH,

OR ROUGHLY THE HEIGHT OF A 12 STORY BUILDING.

PART OF WHAT HELD THE COLOSSEUM TOGETHER

AND ALLOWED FOR SUCH A GRAND STRUCTURE TO BE BUILT

WAS CONCRETE.

THE ROMANS ORIGINATED THE USE OF CONCRETE

AND THE SUBSTANCE CONTRIBUTED TO THE

LONGEVITY OF MANY ROMAN BUILDINGS.

LONG BEFORE THE COLOSSEUM BECAME THE CENTER OF

BLOODY GLADIATORIAL FIGHTS IT WAS USED TO REENACT

FAMOUS NAVAL BATTLES FROM ANCIENT HISTORY.

THE GROUNDS WERE FLOODED WITH WATER

ALLOWING SHIPS TO FLOAT AND SIMULATE

THE CONDITION OF A BATTLE AT SEA.

YET OF THE MANY BATTLES FOUGHT IN THIS STADIUM

THROUGHOUT THE CENTURIES, THE VERY FIRST BATTLE

FOUGHT ON THESE GROUNDS MAY HAVE BEEN MATHEMATICAL.

HOW WERE THE ROMANS ABLE TO BUILD

SUCH A LARGE ELLIPTICAL STRUCTURE?

WHY NOT A CIRCULAR BUILDING?

AS YOU HAVE SEEN,

CONSTRUCTING A CIRCLE IS EASY.

TAKE A COMPASS, DEFINE A CENTER POINT,

AND SWEEP OUT A CIRCLE.

A ROMAN SURVEYOR USING THE SURVEYING TOOL OF THE TIME,

THE GROMA, COULD ACCURATELY CREATE A STRAIGHT LINE

MARKING THE DISTANCE FROM ONE POINT TO ANOTHER.

THIS WAY A LOCUS OF POINTS DEFINING A CIRCLE

OF WHATEVER SIZE COULD EASILY BE MARKED

AND A CIRCULAR STRUCTURE COULD BE BUILT.

THIS IS WHY CIRCULAR BUILDINGS

ARE RELATIVELY EASY TO CREATE.

IF WE LET THE SURVEYOR REPRESENT THE CENTER

OF THE CIRCLE, THEN THE SURVEYOR CAN DETERMINE

A NUMBER OF RADII WHICH CAN BE USED

AS THE FRAMEWORK FOR THE CIRCULAR STRUCTURE.

IN FACT, MOST CIRCULAR STRUCTURES ARE MADE UP OF

AN EMBEDDED REGULAR POLYGON FRAMEWORK.

THE VERTICES OF THE POLYGON

ARE WHAT THE SURVEYOR MAPPED OUT.

THE EASE OF CREATING A CIRCULAR STRUCTURE

IS IN STARK CONTRAST TO THE DIFFICULTY

OF CREATING AN ELLIPTICAL ONE.

THE GEOMETRY OF THE ELLIPSE

PRESENTS SOME STEEP CHALLENGES.

FIRST, THE ELLIPSE HAS TWO POINTS THAT ARE

THE EQUIVALENT TO THE CIRCLE'S CENTER.

THESE TWO POINTS ARE CALLED THE FOCI.

THE ELLIPSE IS THE LOCUS OF POINTS SUCH THAT

THE SUM OF THE DISTANCES FROM THE FOCI TO THE ELLIPSE

IS A CONSTANT.

HERE IS ONE WAY TO CONSTRUCT AN ELLIPSE.

TAKE A LOOP OF STRING, TWO THUMBTACKS,

A PENCIL AND A SHEET OF PAPER.

TACK THE PAPER ON A BULLETIN BOARD OR SIMILAR SURFACE

WITH THE TWO THUMBTACKS.

THESE TACKS REPRESENT THE FOCI OF THE ELLIPSE

YOU ARE ABOUT TO CONSTRUCT.

PLACE THE LOOP OF STRING AROUND THE TACKS.

TAKE THE PENCIL AND EXTEND THE STRING

SO THAT YOU END UP WITH A TRIANGULAR SHAPE AS SHOWN.

THESE TWO SIDES OF THE TRIANGLE REPRESENT THE

DISTANCES TO THE FOCI FROM THE VERTEX OF THE TRIANGLE.

THIS VERTEX IS A POINT ON THE ELLIPSE.

KEEP THE STRING TAUT AND MOVE THE PENCIL TO DIFFERENT

LOCATIONS, IN THE PROCESS CONSTRUCTING THE ELLIPSE.

AS YOU MOVE THE PENCIL AROUND, THE SIDE LENGTHS

OF THE TRIANGLE CHANGE, BUT THE TOTAL AMOUNT OF STRING

THAT REPRESENTS BOTH SIDES IS CONSTANT.

IN OTHER WORDS, THE SUM OF THE LENGTHS

OF THE TWO SIDES OF THE TRIANGLE IS CONSTANT,

FULFILLING THE REQUIREMENT OF AN ELLIPSE.

CONSTRUCTING AN ELLIPSE THIS WAY IS STRAIGHTFORWARD.

BUT TRANSFERRING THAT TECHNIQUE TO THE WORLD

OF SURVEYING AND THE CONSTRUCTION OF BUILDINGS,

ESPECIALLY DURING THE TIME OF THE ROMANS,

IS A CHALLENGE.

THE SIMPLE SITUATION OF ONE SURVEYOR DEFINING

A NUMBER OF RADII BECOMES TWO SURVEYORS

WHOSE MEASUREMENTS MUST ALIGN AT A THIRD POINT.

AND THE SUM OF THEIR SEPARATE MEASUREMENTS

MUST BE A CONSTANT.

FURTHERMORE, THIS THIRD POINT, THE ONE ON THE

ELLIPSE, IS CONSTANTLY CHANGING ITS LOCATION

AND CHANGING THE TWO SIDE LENGTHS OF THE TRIANGLE.

THIS WOULD HAVE BEEN VERY DIFFICULT TO ACHIEVE

WITHOUT RESULTING IN AN INACCURATE CURVE.

FURTHERMORE, CONSTRUCTING ARCHITECTURAL PLANS

OR BLUEPRINTS FOR AN ELLIPTICAL STRUCTURE

WOULD HAVE BEEN IMPOSSIBLE SINCE THERE IS NO WAY TO

CONSTRUCT AN ELLIPSE USING A COMPASS AND A STRAIGHTEDGE.

SO WHY DID THE ROMANS BUILD AN ELLIPTICAL STADIUM?

AND HOW DID THEY DO IT?

THE FIRST QUESTION IS EASY TO ANSWER.

SINCE AN ELLIPSE IS A WIDER VERSION OF A CIRCLE,

THEN THE LARGER AREA WOULD RESULT IN MORE

AVAILABLE STADIUM SEATS.

FOR EXAMPLE, HERE IS A CIRCLE WITH RADIUS R.

AND HERE IS AN ELLIPSE OF WIDTH 2R AND LENGTH 4R.

THE AREA OF THE CIRCLE IS PI R SQUARED

WHILE THE AREA OF THE ELLIPSE IS 2 PI R SQUARED,

TWICE THE AREA OF THE CIRCLE.

THE PERIMETER OF THE ELLIPSE, ON THE OTHER HAND,

IS ONLY 50% LONGER THAN THAT OF THE CIRCLE.

THIS MEANS THAT THE COST OF MATERIALS TO BUILD AN

ELLIPTICAL STRUCTURE OF TWICE THE AREA OF THE CIRCLE

IS NOT TWICE THE COST.

SO THERE'S AN ECONOMICAL REASON

FOR BUILDING AN ELLIPTICAL STRUCTURE.

THE ANSWER TO THE SECOND QUESTION,

HOW DID THEY DO IT? IS MORE DIFFICULT TO ANSWER.

BECAUSE OF THE CONDITION OF THE COLOSSEUM,

IT MAKES IT DIFFICULT TO MAKE A FINAL DETERMINATION

ABOUT HOW IT WAS BUILT.

THERE IS EVIDENCE TO SUGGEST THAT THE ROMANS USED

CIRCULAR ARCS TO APPROXIMATE THE SHAPE OF AN ELLIPSE.

LET'S ANALYZE HOW THIS WOULD WORK.

HERE IS AN ELLIPSE PROPORTIONAL TO THE

ARENA PORTION OF THE COLOSSEUM.

IT IS POSSIBLE TO OVERLAY A SERIES OF CIRCLES

SO THAT PORTIONS OF EACH CIRCLE

OVERLAP THE CURVE OF THE ELLIPSE.

FOR NOW WE WILL LOOK AT JUST THE PORTION OF THE CURVE

IN THE XY COORDINATE SYSTEM'S QUADRANT 1.

PIECING TOGETHER THE THREE ARCS SHOWN REVEALS A CURVE

REMARKABLY CLOSE TO THAT OF THE ELLIPSE.

SINCE THIS ELLIPSE-LIKE SHAPE IS ACTUALLY MADE UP

OF CIRCLES, THEN ONCE AGAIN THE ENGINEERING CHALLENGES

ARE NO DIFFERENT THAN FOR A CIRCULAR STRUCTURE.

THIS WOULD HAVE ELIMINATED THE INSURMOUNTABLE

OBSTACLE TO BUILDING THE COLOSSEUM.

LET'S USE THE TI-NSPIRE TO EXPLORE THIS CONSTRUCTION.

TURN ON THE NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

YOU WILL BE USING THE GEOMETRY TOOLS

WITHIN THE GRAPHING WINDOW.

TO BEGIN WITH, HIDE THE FUNCTION ENTRY LINE.

PRESS CONTROL AND G.

TURN ON THE BACKGROUND GRID BY PRESSING MENU

AND UNDER VIEW SELECTING SHOW GRID.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE ORIGIN OF THE XY COORDINATE GRID.

YOU'LL SEE THE ON SCREEN LABEL "INTERSECTION POINT".

PRESS ENTER.

MOVE THE POINTER TO THE RIGHT

TO CREATE A CIRCLE OF RADIUS 6.

USE THE BACKGROUND GRID AS A GUIDE TO MAKE SURE

THAT THE POINTER IS AT COORDINATE (6,0).

PRESS ENTER.

YOU SHOULD NOW HAVE A CIRCLE OF RADIUS 6 ONSCREEN.

NEXT YOU'LL NEED TO DEFINE THE CENTERS

OF THE THREE CIRCLES YOU'LL BE CONSTRUCTING.

CONSTRUCT A LINE FROM THE LOWEST POINT ON THE CIRCLE

AND CROSS ENDPOINT (2,0).

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO (0,-6) AND PRESS ENTER.

THEN MOVE THE POINTER TO (2,0).

PRESS ENTER.

PRESS ESCAPE AND MOVE THE POINTER

TO THE END OF THE LINE IN ORDER TO EXTEND IT.

PRESS AND HOLD THE CLICK KEY SO THAT THE POINTER

CHANGES FROM AN OPEN HAND TO A GRASPING HAND.

MOVE THE POINTER UP AND TO THE RIGHT

SO THAT THE LINE EXTENDS BEYOND THE CIRCLE.

PRESS ENTER.

NEXT, FIND THE MIDPOINT OF THE LINE SEGMENT

BETWEEN THESE TWO POINTS.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

MOVE THE POINTER TO ONE OF THE ENDPOINTS

OF THE SEGMENT AND PRESS ENTER.

THEN MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

YOU SHOULD NOW SEE THE MIDPOINT OF THE SEGMENT.

MOVE THE POINTER SO THAT IT HOVERS OVER THE MIDPOINT.

YOU WILL NOW CREATE A LINE FROM THIS POINT

TO THE POINT (4,0).

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

PRESS ENTER TO DEFINE THE FIRST ENDPOINT OF THE LINE.

THEN MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

PRESS ESCAPE AND HIGHLIGHT THE ENDPOINT

OF THE LINE YOU JUST CREATED.

PRESS AND HOLD THE CLICK KEY TO HIGHLIGHT THE POINT.

EXTEND THE LENGTH OF THIS LINE

TO BEYOND THE LARGE CIRCLE.

NEXT, FIND THE INTERSECTION POINT

BETWEEN THIS LINE AND THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINT SO THAT IT HOVERS OVER THE LINE.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE CIRCLE

AND PRESS ENTER AGAIN.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU ARE NOW READY TO CONSTRUCT THE

THREE CIRCLES WHOSE ARCS WILL APPROXIMATE

THE SHAPE OF THE ELLIPSE IN THE FIRST QUADRANT.

CREATE THE FIRST CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO COORDINATE (4,0)

WHERE THE SECOND LINE YOU CONSTRUCTED

INTERSECTS THE X AXIS.

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

NEXT, MOVE THE POINTER TO COORDINATE (6,0).

PRESS ENTER.

BEFORE CONSTRUCTING THE SECOND CIRCLE,

CREATE AN INTERSECTION POINT

WHERE THE CIRCLE YOU JUST CONSTRUCTED

AND THE SECOND LINE YOU CONSTRUCTED MEET.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER OVER THE SMALL CIRCLE

AND PRESS ENTER.

THEN MOVE THE POINTER TO THE LINE

AND PRESS ENTER AGAIN.

YOU'RE NOW READY TO CONSTRUCT THE SECOND CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

FOR THE SECOND CIRCLE, MOVE THE POINTER

TO THE MIDPOINT YOU PREVIOUSLY CREATED.

PRESS ENTER.

MOVE THE POINTER TO THE INTERSECTION POINT

YOU CREATED. PRESS ENTER.

BEFORE CONSTRUCTING THE THIRD CIRCLE,

CREATE AN INTERSECTION POINT

WHERE THE SECOND CIRCLE INTERSECTS

THE FIRST LINE YOU CREATED.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER ABOVE THE SECOND CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE LINE

AND PRESS ENTER AGAIN.

NOW YOU ARE READY TO CONSTRUCT THE THIRD CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

FOR THE THIRD CIRCLE,

MOVE THE POINTER TO THE POINT (0,-6).

PRESS ENTER.

THEN MOVE THE POINTER TO THE INTERSECTION POINT

YOU JUST CREATED.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE THE THREE CIRCLES

WHOSE ARCS APPROXIMATE THE SHAPE OF THE ELLIPSE.

IF WE HIGHLIGHT THOSE ARCS YOU WILL SEE THAT THEY ARE

LIKE PUZZLE PIECES THAT SNAP TOGETHER SEAMLESSLY.

HOW GOOD A FIT ARE THESE ARCS?

LOOKING AT THE TANGENTS TO THE CIRCLES

AT THIS POINT WILL TELL US.

IT IS POSSIBLE FOR A LINE TO INTERSECT A CIRCLE

AT TWO POINTS, IN WHICH CASE

SUCH A LINE IS CALLED A SECANT.

A TANGENT, ON THE OTHER HAND,

INTERSECTS THE CIRCLE AT A POINT THAT IS

PERPENDICULAR TO THE RADIUS AT THAT POINT.

THE POINT WE ARE INTERESTED IN

IS WHERE THE SMALL CIRCLE INTERSECTS THE LARGER ONE.

CREATE TANGENT LINES TO THESE CIRCLES AT THIS POINT.

LET'S START BY ZOOMING IN ON THE POINT.

PRESS MENU AND UNDER WINDOW/ZOOM SELECT ZOOM-IN.

MOVE THE POINTER, WHICH SHOULD LOOK LIKE

A MAGNIFYING GLASS, AND PRESS ENTER ONCE OR TWICE

TO ZOOM IN SUFFICIENTLY TO SEE THE POINT CLEARLY

IN THE MIDDLE PART OF THE SCREEN.

NEXT, PRESS MENU AND UNDER "POINTS & LINES"

SELECT TANGENT.

MOVE THE POINTER ABOVE THE SMALLER CIRCLE.

FOR NOW DON'T WORRY ABOUT

CLICKING ON THE INTERSECTION POINT,

BUT JUST ON THE PART OF THE CIRCLE

AWAY FROM THE INTERSECTION.

PRESS ENTER.

YOU'LL SEE THE TANGENT LINE APPEAR.

NEXT, MEASURE THE SLOPE OF EACH TANGENT.

PRESS MENU AND UNDER MEASUREMENT SELECT SLOPE.

MOVE THE POINTER ABOVE THE FIRST TANGENT

AND PRESS ENTER TO MEASURE THE SLOPE.

MOVE THE POINTER TO THE SIDE OF THE TANGENT

AND PRESS ENTER AGAIN

TO RECORD THE SLOPE MEASUREMENT.

REPEAT THIS PROCESS WITH THE OTHER TANGENT LINE.

PRESS ESCAPE AND MOVE THE POINTER

ABOVE THE FIRST TANGENT YOU CREATED.

PRESS AND HOLD THE CLICK KEY

TO HIGHLIGHT THE TANGENT LINE.

MOVE THE TANGENT POINT ALONG THE CIRCLE UNTIL IT

OVERLAPS THE INTERSECTION POINT OF THE TWO CIRCLES.

PRESS ENTER.

REPEAT THIS PROCESS FOR THE OTHER TANGENT.

ZOOM IN IF YOU NEED TO IN ORDER TO MAKE SURE

THAT ALL THREE POINTS OVERLAP.

YOU'LL SEE THAT THE SLOPES ARE NEARLY IDENTICAL

AND MAY ONLY DIFFER BY A SMALL DECIMAL AMOUNT

DUE TO THE FACT THAT THE POINTS

MAY NOT COMPLETELY OVERLAP.

IN FACT, THE TANGENT LINES HAVE THE SAME SLOPE.

WHY IS THIS?

NOTICE THAT ALTHOUGH THE TWO CIRCLES

DO NOT HAVE THE SAME CENTER,

THE TWO CENTER POINTS ARE ON THE SAME LINE.

WHEN RADII ARE COLLINEAR THIS WAY,

THE TANGENT LINES THAT INTERSECT THESE RADII

WILL BE PARALLEL TO EACH OTHER

AND THIS MEANS THAT THE TANGENTS

WILL HAVE THE SAME SLOPE.

HAVING THE SAME SLOPE MEANS THAT

TWO CURVES MEET SMOOTHLY.

THIS HELPS IN PIECING TOGETHER AN ELLIPSE

FROM A SET OF CIRCULAR ARCS.

NOW LOOK AT WHERE THE SECOND

AND THIRD CIRCLES INTERSECT.

WITHOUT EVEN MEASURING THE SLOPES

OF THE TANGENTS ALONG THE INTERSECTION POINT,

YOU KNOW THAT THE TANGENTS ARE PARALLEL

SINCE THE RADII ARE COLLINEAR.

PIECEWISE APPROXIMATIONS OF CURVES

IS A COMMON TECHNIQUE USED TO BUILD THE

FRAMEWORK OF A BUILDING WITH A MORE COMPLEX SHAPE.

AS YOU CAN SEE, THE RESULTS ARE SURPRISINGLY ACCURATE.

HOW ACCURATE?

NOW THAT YOU KNOW HOW TO CONSTRUCT THE COLOSSEUM

USING CIRCULAR ARCS, YOU CAN CONSTRUCT

YOUR OWN SCALE MODEL OF THE COLOSSEUM

USING POPSICLE STICKS OR LEGO BLOCKS.

FOR THIS ACTIVITY USE A COMPASS, STRAIGHTEDGE,

AND ENOUGH BUILDING MATERIALS TO AT LEAST

CONSTRUCT ONE SECTION OF THE COLOSSEUM.

USE THE COMPASS TO CONSTRUCT A LARGE CIRCLE.

USE THE STRAIGHTEDGE TO CONSTRUCT A DIAMETER.

DIVIDE THIS RADIUS INTO THREE PARTS.

USE A RULER TO MEASURE THE RADIUS.

CONSTRUCT A LINE THAT CROSSES THESE TWO POINTS

AND MAKE SURE THAT THE LINE EXTENDS

BEYOND THE LARGE CIRCLE.

FIND THE MIDPOINT OF THIS SEGMENT AND HAVE THAT

BE THE START OF A LINE THAT CROSSES THIS POINT

AND EXTENDS BEYOND THE CIRCLE.

NOW USE THE COMPASS TO CONSTRUCT THESE CIRCLES.

HIGHLIGHT THE THREE ARCS AS SHOWN

AND BEGIN BUILDING A STADIUM SECTION

AROUND THESE ARCS.

ADD AS MANY LEVELS AS NECESSARY

AND YOU WILL SOON SEE THE OUTLINES

OF THE SCALE MODEL COMING INTO SHAPE.

YOU WILL ALSO BEGIN TO APPRECIATE WHY THE ROMANS

HAVE SUCH A REPUTATION FOR CLEVER ENGINEERING

AND CREATING MONUMENTS THAT STAND THE TEST OF TIME.

Title: Geometry Applications: Chords, Secants, and Segments

JUST A SHORT DISTANCE FROM THE ROMAN COLOSSEUM

IS AN EVEN OLDER STRUCTURE

THAT WAS JUST AS WELL KNOWN.

THE PANTHEON WAS ORIGINALLY BUILT

NEARLY A CENTURY BEFORE THE COLOSSEUM.

IT WAS A RELIGIOUS BUILDING MEANT TO HONOR

THE MANY GODS THE ROMANS WORSHIPPED.

ALTHOUGH IN LATER YEARS IT WAS CONVERTED INTO

A CHRISTIAN CHURCH.

FROM THE OUTSIDE YOU CAN SEE THE CIRCULAR DOME

AND THE BODY OF THE STRUCTURE LOOKS CYLINDRICAL.

BUT INSIDE THE CIRCULAR EXPANSE IS DRAMATIC.

THE DOMED STRUCTURE BECOMES A GRAND ARCHED CEILING.

WHAT ADDS TO THE EFFECT IS THAT THE INTERIOR

OF THE PANTHEON IS AS TALL AS IT IS WIDE.

IN OTHER WORDS, IMAGINE A GREAT CIRCLE

DEFINING THE INTERIOR OF THE SPACE.

AT THE TOP OF THE DOME IS AN OPENING

CALLED THE OCULUS, WHICH LETS IN SUNLIGHT.

IT IS THE ONLY SOURCE OF LIGHT FOR THE PANTHEON AND

ADDS A DRAMATIC LIGHTING EFFECT TO THE SPACE.

THE OCULUS LETS IN LIGHT AT DIFFERENT ANGLES

DEPENDING ON THE TIME OF DAY AND THE TIME OF YEAR.

AS LIGHT COMES IN THROUGH THE OCULUS

IT IS DISPERSED AND THE PATTERN OF LIGHT FORMED

CAN BE MODELED ON A CIRCLE WITH TWO CHORDS.

A CHORD IS A SEGMENT WHOSE ENDPOINTS ARE ON THE CIRCLE.

A DIAMETER OF A CIRCLE IS A SPECIAL KIND OF CHORD,

ONE THAT INTERSECTS THE CENTER OF THE CIRCLE.

TO SIMPLIFY, LET ONE POINT BECOME INTO TWO CHORDS

THAT REPRESENT THE SPAN OF LIGHT.

SUCH A PAIR OF CHORDS FORM AN ANGLE

CALLED AN INSCRIBED ANGLE.

THE INSCRIBED ANGLE MARKS OUT A PORTION OF THE CIRCLE

CALLED AN INTERCEPTED ARC.

THROUGHOUT THE YEAR THE OCULUS ALLOWS A PORTION

OF LIGHT THAT DEFINES AN INTERCEPTED ARC.

THE INSCRIBED ANGLE IS DIFFERENT FROM A

CENTRAL ANGLE WHICH IS MADE UP OF TWO RADII.

BUT THERE IS A RELATIONSHIP BETWEEN

AN INSCRIBED ANGLE AND A CENTRAL ANGLE WHICH

REVEALS SOME INTERESTING FEATURES OF THE PANTHEON.

LET'S EXPLORE THIS RELATIONSHIP ON THE NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

CONSTRUCT A HORIZONTAL LINE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO THE LOWER LEFT-HAND

PART OF THE SCREEN.

PRESS ENTER.

PRESS THE RIGHT ARROW

TO CONSTRUCT THE HORIZONTAL LINE.

STOP PRESSING THE RIGHT ARROW

WHEN YOU REACH THE OTHER END OF THE SCREEN.

PRESS ENTER AGAIN.

THIS LINE REPRESENTS THE FLOOR OF THE PANTHEON.

LET'S NOW CONSTRUCT THE GREAT CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

ABOVE THE HORIZONTAL LINE.

PRESS ENTER.

THIS DEFINES THE CENTER OF THE CIRCLE.

MOVE THE POINTER TOWARD THE HORIZONTAL LINE

SO THAT THE CIRCLE INTERSECTS THE LINE.

YOU'LL SEE AN ONSCREEN LABEL THAT SAYS "POINT ON".

PRESS ENTER.

THIS CIRCLE REPRESENTS THE GREAT CIRCLE

THAT MAKES UP THE INTERIOR OF THE PANTHEON.

THE TOP HALF OF THE CIRCLE REPRESENTS THE DOME

WHILE THE BOTTOM HALF OF THE CIRCLE

IS A VIRTUAL SEMI-CIRCLE.

NEXT, CONSTRUCT TWO CHORDS THAT INTERSECT

AT THE TOP OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

MOVE THE POINTER TO THE TOP OF THE CIRCLE

DIRECTLY ABOVE THE CENTER.

PRESS ENTER.

THEN MOVE THE POINTER TO ROUGHLY 5 O'CLOCK

ON THE CIRCLE AND PRESS ENTER AGAIN.

GO BACK TO THE TOP OF THE CIRCLE AND HAVE THE POINTER

HOVER OVER THE FIRST ENDPOINT OF THE SEGMENT.

PRESS ENTER AGAIN TO CREATE THE STARTING POINT

OF THE NEXT SEGMENT.

MOVE THE POINTER TO ROUGHLY 3 O'CLOCK AND PRESS ENTER.

YOU'VE NOW CREATED TWO CHORDS

THAT INTERSECT AT ONE ENDPOINT.

THEY DEFINE AN INTERCEPTED ARC AS SHOWN HERE.

THESE CHORDS REPRESENT THE LIGHT

POURING THROUGH THE OCULUS OF THE PANTHEON.

NOW CONSTRUCT THE CENTRAL ANGLE.

CONTINUING WITH THE SEGMENT TOOL,

MOVE THE POINTER SO THAT IT HOVERS OVER THE CENTER.

PRESS ENTER.

MOVE THE POINTER TO THE ENDPOINT

OF THE SECOND CHORD.

PRESS ENTER.

RETURN THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER TO THE ENDPOINT

OF THE FIRST CHORD.

PRESS ENTER ONCE MORE.

NOW MEASURE THE INSCRIBED ANGLE AND THE CENTRAL ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

TO MEASURE AN ANGLE, SELECT THREE POINTS

THAT DEFINE THE ANGLE.

MOVE THE POINTER ABOVE ONE OF THE ENDPOINTS

OF THE CHORD ON THE CIRCLE.

PRESS ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO THE ENDPOINT

OF THE OTHER CHORD.

PRESS ENTER ONE MORE TIME.

YOU'LL SEE AN ANGLE MEASUREMENT APPEAR

NEXT TO THE VERTEX OF THE ANGLE BEING MEASURED.

REPEAT THIS PROCESS FOR THE INSCRIBED ANGLE.

YOU'LL NOTICE THAT THE CENTRAL ANGLE IS

TWICE THE MEASURE OF THE INSCRIBED ANGLE.

EVEN IF YOU CHANGE THE SIZE OF THE INTERCEPTED ARC,

THE RELATIONSHIP BETWEEN TWO ANGLES STAYS THE SAME.

NOTICE THAT THE POSITION OF THE OCULUS IS AT THE

EXACT OPPOSITE FROM WHERE SOMEONE WOULD BE STANDING

IN THE CENTER OF THE PANTHEON.

FROM THAT POINT OF VIEW, THE INSCRIBED ANGLE

IS IDENTICAL TO THAT OF THE OCULUS.

CREATE AN ANGLE USING THE SEGMENT TOOL

AND MEASURE IT TO VERIFY THIS.

SO THE GEOMETRY OF THE PANTHEON IS SUCH THAT

SOMEONE STANDING IN THE CENTER OF THE FLOOR

WOULD HAVE THE SAME VIEW OF THE LIGHT ON THE WALLS

AS THE OCULUS.

THIS IS IMPORTANT SINCE THE OCULUS WAS MORE THAN

JUST A HOLE IN THE ROOF.

THE OCULUS WAS MEANT TO SYMBOLIZE THE SUN,

THE SOURCE OF ALL LIGHT AND POWER.

SO THE PANTHEON WAS MEANT TO GIVE SOMEONE

THE VIEW FROM THE SUN.

IN A WAY, THE PANTHEON WAS MEANT TO ELEVATE

AND EXALT THE OBSERVER AND GIVE THEM A SENSE

OF THE POWER OF THE UNIVERSE AROUND THEM.

BUT THE OCULUS IS NOT A PIN PRICK OF LIGHT

THAT LETS ONLY A SMALL AMOUNT OF LIGHT IN.

IN FACT IT IS A CIRCLE WITH A DIAMETER OF 27 FEET.

WHILE THIS IS STILL A SMALL PORTION OF THE

OVERALL DIAMETER OF 142 FEET FOR THE GREAT CIRCLE

OF THE PANTHEON, IT IS STILL A SIZABLE HOLE.

SO THE OCULUS LETS IN A COLUMN OF LIGHT

WHICH LEADS TO A CIRCULAR SPOT OF LIGHT

ON THE WALLS OF THE PANTHEON.

LET'S INVESTIGATE THE INTERCEPTED ARC

FORMED BY THIS COLUMN OF LIGHT ON THE NSPIRE.

CREATE A NEW GEOMETRY WINDOW.

PRESS THE HOME KEY AND SELECT GEOMETRY.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE CENTER OF THE SCREEN.

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

MOVE THE POINTER AWAY FROM THE CENTER

TO DEFINE A CIRCLE ABOUT HALF THE SIZE OF THE SCREEN.

PRESS ENTER AGAIN.

USE THE SEGMENT TOOL TO CREATE AND MEASURE

THE RADIUS OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

THEN MOVE THE POINTER TO THE CIRCLE.

PRESS ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER AGAIN.

NOW MEASURE THE RADIUS.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER OVER THE RADIUS AND PRESS ENTER.

MOVE THE POINTER TO THE SIDE OF THE RADIUS

AND PRESS ENTER AGAIN

TO PASTE THE MEASUREMENT ONSCREEN.

WE WANT THIS CIRCLE TO MODEL THE GREAT CIRCLE

OF THE PANTHEON.

THE RADIUS OF THAT CIRCLE IS 71 FEET.

USING A 10:1 SCALE,

CHANGE THE RADIUS OF THE CIRCLE TO 7.1.

PRESS ESCAPE AND HOVER OVER THE CIRCLE.

PRESS AND HOLD THE CLICK KEY TO GRASP THE CIRCLE.

USE THE NAVIGATION ARROWS TO RESIZE THE CIRCLE.

TRY TO GET THE SIZE OF THE RADIUS

AS CLOSE TO 7.1 AS POSSIBLE.

PRESS ENTER.

LOCK THIS VALUE.

MOVE THE POINTER OVER THE MEASUREMENT VALUE.

PRESS CONTROL AND MENU

AND SELECT THE ATTRIBUTES OPTION.

USE THE DOWN ARROW TO HIGHLIGHT THE ICON

THAT LOOKS LIKE AN OPEN LOCK.

USE THE RIGHT ARROW TO CHANGE IT TO A CLOSED LOCK.

PRESS ENTER.

THIS CIRCLE IS NOW A SCALE MODEL OF

THE GREAT CIRCLE OF THE PANTHEON.

SINCE THE MEASUREMENT IS LOCKED,

THE CIRCLE CANNOT BE RESIZED.

NOW HIDE THE RADIUS.

MOVE THE POINTER OVER THE RADIUS,

PRESS CONTROL AND MENU AND SELECT THE HIDE OPTION.

NOW USE THE SEGMENT TOOL TO MODEL THE OCULUS.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

MOVE THE POINTER NEAR THE TOP OF THE CIRCLE

AND PRESS ENTER TO PLACE A POINT ON THE CIRCLE.

NEXT, MOVE THE POINTER TO ANOTHER PART OF THE CIRCLE

MAKING SURE THE SEGMENT REMAINS

AS CLOSE TO HORIZONTAL AS POSSIBLE.

PRESS ENTER.

NOW MEASURE THE SEGMENT.

PRESS MENU AND UNDER MEASUREMENT

SELECT LENGTH.

MOVE THE POINTER OVER THE SEGMENT

AND PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

MOVE THE POINTER AND PRESS ENTER

TO PLACE THE MEASUREMENT ON SCREEN.

PRESS ESCAPE AND MOVE THE POINTER

OVER ONE OF THE SEGMENTS.

PRESS AND HOLD THE CLICK KEY TO HIGHLIGHT THE POINT.

RE-SIZE THE SEGMENT SO THAT IT IS AS CLOSE TO 2.7

AS POSSIBLE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

LOCK THE VALUE OF THE SEGMENT MEASURE.

MOVE THE POINTER OVER THE MEASUREMENT VALUE,

PRESS CONTROL AND MENU.

SELECT ATTRIBUTES AND CHANGE THE OPEN LOCK

TO A CLOSED LOCK.

PRESS ENTER.

NOW CONSTRUCT A SECANT FROM ONE OF THE ENDPOINTS

OF THE SEGMENT TO THE OPPOSITE END OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO ONE OF THE ENDPOINTS

OF THE SEGMENT AND PRESS ENTER.

MOVE THE POINTER TO THE OPPOSITE SIDE OF THE CIRCLE

AND PRESS ENTER AGAIN.

NEXT, CREATE ANOTHER SECANT

PARALLEL TO THE FIRST.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER OVER THE SECANT AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE SECOND ENDPOINT

OF THE SEGMENT AT TOP OF THE CIRCLE.

PRESS ENTER AGAIN.

EXTEND THE SECOND SECANT.

PRESS ESCAPE.

MOVE THE POINTER TO THE END OF THE CIRCLE.

PRESS AND HOLD THE CLICK KEY

AND EXTEND THE LINE BEYOND THE CIRCLE.

THEN ADD AN INTERSECTION POINT

WHERE THE SECOND SECANT INTERSECTS THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER OVER THE CIRCLE AND PRESS ENTER.

THEN MOVE THE POINTER OVER THE SECANT

AND PRESS ENTER AGAIN.

YOU NOW HAVE A MODEL OF THE INSIDE OF THE PANTHEON

AND THE COLUMN OF LIGHT

THAT SHINES THROUGH THE OCULUS.

MEASURE THE INTERCEPTED ARC.

FIRST DEFINE THE INTERCEPTED ARC

USING THE CIRCLE ARC TOOL.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT CIRCLE ARC.

MOVE THE POINTER ABOVE ONE OF THE ENDPOINTS

OF THE ARC AND PRESS ENTER.

NEXT, MOVE THE POINTER TO THE MIDDLE OF THIS ARC,

MAKING SURE THAT THE NEW POINT YOU ADD

IS ON THE CIRCLE.

AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO THE OTHER ENDPOINT

OF THE ARC AND PRESS ENTER ONE MORE TIME.

MEASURE THE ARC.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE ARC AND

MAKE SURE THE "CIRCLE ARC" ONSCREEN TEXT APPEARS.

PRESS ENTER.

THE MEASUREMENT OF THE ARC IS A DISTANCE MEASUREMENT

BUT IT DOES CORRESPOND TO AN ANGLE MEASUREMENT.

THE RELATIONSHIP BETWEEN ARC LENGTH

AND THE ANGLE MEASURE OF THE INTERCEPTED ARC

IS S = R THETA WHERE S IS THE ARC LENGTH,

R IS THE RADIUS OF THE CIRCLE,

AND THE THETA IS THE ANGLE MEASURE

REPRESENTED BY THE INTERCEPTED ARC.

SOLVING FOR THETA WE GET THETA = S OVER R.

THIS GIVES THE ANGLE MEASURE IN A UNIT CALLED RADIANS.

TO CHANGE IT TO A DEGREE MEASURE,

MULTIPLY THIS EXPRESSION BY 360 OVER 2 PI.

SO THE ANGLE FORMULA BECOMES

THETA = 180S OVER PI R.

CREATE A FORMULA TO CONVERT THE ARC LENGTH TO AN ANGLE.

PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN.

PRESS ENTER.

INPUT THE FORMULA 180S OVER PI R.

TO INPUT THE SYMBOL FOR PI, PRESS THE LIBRARY BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS 3 TO BRING UP THE SYMBOL PALLET AND SELECT PI.

POINT THE FORMULA TO THE VALUES ON THE CIRCLE.

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER ABOVE THE FORMULA.

PRESS ENTER.

MOVE THE POINTER ABOVE THE VALUE FOR THE RADIUS.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE VALUE FOR THE ARC LENGTH

AND PRESS ENTER AGAIN.

YOU'LL SEE THE VALUE FOR THE ANGLE MEASURE

OF THE ARC LENGTH.

MOVE THE POINTER NEXT TO THE FORMULA

AND PRESS ENTER TO PLACE THE VALUE ONSCREEN.

THE ANGLE MEASURE SHOULD BE ABOUT 22 DEGREES.

IN FACT, THE ANGLE MEASURE STAYS AT 22 DEGREES

NO MATTER HOW THE COLUMN OF LIGHT IS ORIENTED.

AS YOU MOVE THE PARALLEL LINES

TO DIFFERENT POSITIONS, MAKE SURE TO ALSO

MOVE THE MIDDLE POINT OF THE ARC.

THIS CONSTANT 22 DEGREE PATCH OF LIGHT SHINES

THROUGHOUT THE YEAR, BUT TAKES ON MORE SIGNIFICANCE

DURING FOUR KEY DAYS

THE WINTER AND SUMMER SOLSTICES

AND THE FALL AND SPRING EQUINOXES.

DURING THESE DAYS THE SPOT OF LIGHT

SHINES OVER THE ENTRANCE TO THE PANTHEON.

AS YOU HAVE SEEN, THESE FOUR DATES WERE

VERY SIGNIFICANT TO ANCIENT CULTURES.

SOMEONE ENTERING THE PANTHEON ON THESE DATES

WOULD BE BATHED IN LIGHT, AND ALL FOUR DATES

OCCUR ON OR ABOUT THE 22ND DAY OF MAY, JUNE,

SEPTEMBER AND DECEMBER.

THUS THE ANGLE MEASURE OF THE INTERCEPTED ARC

TAKES ON ADDITIONAL SIGNIFICANCE.

SO THE GEOMETRY OF THE PANTHEON

WAS VERY SIGNIFICANT FOR THE ROMANS.

THEY CREATED A STRUCTURE

THAT HAS LASTED FOR MILLENNIA

AND UTILIZES THE TIMELESS GEOMETRY OF CIRCLES.