Geometry Applications: Surface Area of Pyramids

Geometry Applications: Surface Area of Pyramids

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[Music]

THE LOUVRE MUSEUM IN PARIS IS ONE OF

THE MOST IMPORTANT MUSEUMS IN THE WORLD.

IN IT YOU'LL FIND THE MONA LISA, THE VENUS DE MILO

AND MANY IMPORTANT WORKS OF ART FROM WORLD HISTORY.

IN 1984 THE MUSEUM WENT THROUGH AN EXTENSIVE

RENOVATION WHICH INCLUDED WHAT IS NOW

THE MOST NOTEWORTHY CHANGE TO THE MUSEUM:

THE GLASS PYRAMID.

IT IS A SQUARE PYRAMID,

MEANING THAT THE BASE OF THE PYRAMID IS A SQUARE.

A SQUARE PYRAMID HAS FOUR TRIANGULAR SIDES.

THE PYRAMID IS MADE UP OF A NUMBER OF GLASS PANELS

SO WHEN DESIGNING THE PYRAMID, THE BUILDERS

NEEDED TO KNOW THE SURFACE AREA OF THIS PYRAMID.

LET'S LOOK AT A NET FOR A SQUARE PYRAMID.

THE SURFACE AREA IS MADE UP OF THE AREA OF THE SQUARE

BASE AND THE AREAS OF THE FOUR TRIANGULAR SECTIONS.

SINCE THE TRIANGLES ARE CONGRUENT,

THEN THE SURFACE AREA OF THE TRIANGULAR SIDES

IS EQUAL TO FOUR TIMES THE AREA OF ONE OF THE TRIANGLES.

IF THE SQUARE SIDE HAS BASE b AND THE TRIANGULAR SIDES

HAVE HEIGHT h THEN THE SURFACE AREA OF

THE PYRAMID IS b SQUARED PLUS 4 TIMES 1/2 bh.

SIMPLIFYING WE GET: b SQUARED PLUS 2bh

FOR THE GLASS PYRAMID AT THE LOUVRE

THE BASE IS PART OF THE ENTRYWAY INTO THE MUSEUM.

SO ONLY THE SURFACE OF THE TRIANGULAR PORTIONS

ARE RELEVANT.

SO THE SURFACE AREA OF THIS PYRAMID IS 2bh.

NOW TURNING TO ONE OF THE TRIANGULAR SIDES

OF THE LOUVRE PYRAMID YOU'LL SEE THAT EACH SIDE

CONSISTS OF A TESSELLATION MADE UP OF RHOMBUSES.

ALONG THE SIDE OF THE TRIANGLE

THERE ARE 18 RHOMBUS SIDES.

ALONG THE BASE THERE ARE 18 RHOMBUS DIAGONALS.

WITH A RHOMBUS ALL FOUR SIDES ARE CONGRUENT.

THIS MEANS THAT A DIAGONAL DIVIDES A RHOMBUS

INTO TWO CONGRUENT ISOSCELES TRIANGLES.

FOCUS ON THE RHOMBUS AT THE APEX OF THE PYRAMID.

SPLIT THIS RHOMBUS INTO TWO CONGRUENT ISOSCELES

TRIANGLES BY DEFINING THIS HORIZONTAL SEGMENT.

NOTICE THAT THE LARGER TRIANGULAR FACE

OF THE PYRAMID AND THE TOP ANGLE OF THE SMALLER

ISOSCELES TRIANGLE SHARE THE SAME ANGLE.

SINCE THE LARGE AND SMALL TRIANGLES

ARE BOTH ISOSCELES TRIANGLES

THEN IT FOLLOWS THAT THE BASE TRIANGLES

ARE CONGRUENT TO EACH OTHER AS SHOWN HERE.

THIS MEANS THAT THE LARGER TRIANGLE

AND THE SMALLER ONE ARE SIMILAR TO EACH OTHER.

THIS IS IMPORTANT FOR FINDING

PROPORTIONAL SIDES OF THE TWO TRIANGLES.

ALSO MAKE A NOTE THAT SINCE THE SMALLER TRIANGLES

ARE SIMILAR TO THE LARGER TRIANGLE,

THE CORRESPONDING SIDES OF THE SMALLER TRIANGLE

ARE PARALLEL TO THE CORRESPONDING SIDES

OF THE LARGER TRIANGLE.

NOW EACH RHOMBUS SHAPED GLASS PANEL

IS A DIAMOND SHAPE

MEANING THAT THE RHOMBUS HAS FOUR 45 DEGREE ANGLES.

WE CAN USE THE PROPERTIES OF SIMILAR TRIANGLES

TO DETERMINE HOW MANY OF THESE GLASS PANELS

ARE NEEDED TO CREATE THE

TRIANGULAR SIDES OF THE PYRAMID.

LET'S USE THE TI-NSPIRE TO SOLVE THIS PROBLEM.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

CREATE A LINE SEGMENT.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER TO THE LOWER LEFT

PART OF THE SCREEN.

PRESS ENTER.

PRESS AND HOLD THE RIGHT ARROW KEY

TO CREATE THE SEGMENT.

WHEN THE SEGMENT COVERS MOST OF THE

LOWER PART OF THE SCREEN PRESS ENTER.

THIS LINE SEGMENT IS THE BASE OF ONE OF THE

TRIANGULAR SIDES OF THE PYRAMID.

IT IS THE BASE OF AN ISOSCELES TRIANGLE.

TO CONSTRUCT AN ISOSCELES TRIANGLE FIRST CONSTRUCT

THE PERPENDICULAR BISECTOR OF THE BASE.

SELECT PERPENDICULAR BISECTOR.

MOVE THE POINTER ABOVE THE SEGMENT AND PRESS ENTER.

YOU'LL SEE THE PERPENDICULAR BISECTOR CONSTRUCTED.

EXTEND THE LENGTH OF THE PERPENDICULAR BISECTOR.

PRESS ESCAPE THEN MOVE THE POINTER

TO THE TOP END OF THE BISECTOR.

PRESS AND HOLD THE CLICK KEY TO SELECT IT.

PRESS AND HOLD THE UP ARROW TO INCREASE THE

LENGTH OF THE PERPENDICULAR BISECTOR.

NOW ADD TWO POINTS TO THE PERPENDICULAR BISECTOR

TO DEFINE THE TRIANGLE'S HEIGHT.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINTS.

MOVE THE POINTER TO THE HORIZONTAL LINE

WHERE IT INTERSECTS THE BISECTOR.

YOU'LL SEE AN ONSCREEN LABEL THAT SAYS

"INTERSECTION POINT".

PRESS ENTER THEN PRESS AND HOLD THE UP ARROW

TO MOVE THE POINTER TO THE END OF THE LINE.

PRESS ENTER AGAIN.

NEXT CONSTRUCT THE TWO REMAINING

SIDES OF THE TRIANGLE.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER TO THE TOP POINT

ON THE PERPENDICULAR BISECTOR.

PRESS ENTER.

THEN MOVE THE POINTER

TO ONE OF THE ENDPOINTS OF THE BASE.

PRESS ENTER AGAIN.

REPEAT THIS TO CONSTRUCT THE OTHER SIDE OF THE TRIANGLE.

WE KNOW THAT THE ANGLE AT THE TOP VERTEX

OF THE TRIANGLE IS 45 DEGREES.

THE PERPENDICULAR BISECTOR

RESULTS IN A 90 DEGREE ANGLE.

THIS MEANS THAT THE BASE ANGLES OF THE TRIANGLE

NEED TO BE 45 DEGREES.

SO MEASURE THE BASE ANGLES.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

MOVE THE POINTER TO THE TOP POINT OF THE TRIANGLE

AND PRESS ENTER.

THEN MOVE THE POINTER TO ONE OF THE OTHER VERTICES

OF THE TRIANGLE AND PRESS ENTER AGAIN.

FINALLY MOVE THE POINTER ABOVE THE THIRD VERTEX

AND PRESS ENTER ONE MORE TIME.

YOU'LL SEE THE ANGLE MEASURE APPEAR.

REPEAT THIS PROCESS

FOR THE OTHER BASE ANGLE OF THE TRIANGLE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

PRESS ESCAPE AND MOVE THE POINTER

TO THE TOP VERTEX OF THE TRIANGLE.

PRESS AND HOLD THE CLICK KEY TO SELECT THE POINT.

USE THE UP OR DOWN ARROW

TO CHANGE THE POSITION OF THE POINT.

NOTICE HOW THE BASE ANGLE MEASURES CHANGE.

YOU WANT TO MOVE THE POINT SO THAT

THE BASE ANGLES ARE 45 DEGREES.

YOU NEED THE MEASURE TO BE PRECISE

SO YOU MAY NEED TO MOVE ALL THE POINTS

ON THE TRIANGLE IN ORDER TO GET TO 45 DEGREES.

THIS TRIANGLE IS A MODEL OF ONE OF THE

TRIANGULAR SIDES OF THE LOUVRE PYRAMID.

MEASURE THE BASE AND HEIGHT OF THE PYRAMID.

THESE ARE THE MEASUREMENTS

WE WILL USE TO FIND THE CORRESPONDING AREA

OF THE RHOMBUS SHAPED PANELS.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE BASE AND PRESS ENTER

TO SEE THE LENGTH MEASUREMENT.

MOVE THE POINTER BELOW THE SEGMENT

AND PRESS ENTER AGAIN

TO PLACE THE MEASUREMENT ON SCREEN.

NEXT MOVE THE POINTER ABOVE ONE OF THE ENDPOINTS

OF THE PERPENDICULAR BISECTOR AND PRESS ENTER.

MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

YOU'LL SEE THE LENGTH MEASUREMENT.

MOVE THE POINTER TO THE SIDE OF THE VERTICAL SEGMENT

AND PRESS ENTER ONE MORE TIME.

ASSIGN THESE LENGTH MEASUREMENTS TO VARIABLES.

PRESS ESCAPE AND MOVE THE POINTER ABOVE

ONE OF THE MEASUREMENTS AT THE BASE OF THE TRIANGLE.

SELECT THE STORE OPTION.

CREATE A VARIABLE CALLED LARGEBASE AND PRESS ENTER.

NEXT, MOVE THE POINTER ABOVE

THE OTHER MEASUREMENT.

AND SELECT THE STORE OPTION ONCE AGAIN.

CREATE A VARIABLE CALLED LARGEHEIGHT

AND PRESS ENTER.

NOW WE CAN CONSTRUCT ONE OF THE RHOMBUS SHAPED REGIONS.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER ALONG THE BASE OF THE TRIANGLE

AND ADD TWO POINTS NEAR EACH OF THE CORNER VERTICES

PRESSING ENTER EACH TIME TO ADD THE POINT.

MEASURE THE DISTANCES ALONG THESE SHORT SEGMENTS.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE ONE OF THE CORNER VERTICES

AND PRESS ENTER.

MOVE THE POINTER ABOVE ONE OF THE POINTS YOU JUST ADDED

AND PRESS ENTER AGAIN.

PLACE THE MEASUREMENT ONSCREEN.

REPEAT THIS FOR THE OTHER SHORT SEGMENT.

EACH OF THESE SHORT SEGMENTS WILL CORRESPOND

TO ONE OF THE RHOMBUS SHAPED SIDES.

AS WE SAW EARLIER THERE ARE 18 SUCH UNITS

ALONG EACH SIDE OF THE TRIANGLE.

PRESS ESCAPE AND MOVE THE POINTER ABOVE ONE OF THE

LENGTH MEASUREMENTS OF ONE OF THE SHORT SEGMENTS.

PRESS ENTER TO GO INTO EDIT MODE.

REPLACE THE MEASUREMENT WITH THE FORMULA

LARGEBASE DIVIDED BY 18 AND PRESS ENTER.

THE LENGTH CHANGES TO THE CORRESPONDING LENGTH

ON THE LOUVRE PYRAMID.

REPEAT THIS PROCESS FOR THE OTHER SHORT SEGMENT.

MOVE THE POINTER ABOVE THE MEASUREMENT

FOR THAT SEGMENT AND PRESS ENTER.

REPLACE THE MEASUREMENT WITH THE FORMULA

LARGEBASE DIVIDED BY 18 AND PRESS ENTER.

WE CAN NOW CREATE LINES

PARALLEL TO THE TRIANGULAR SIDES

THROUGH THE ENDPOINTS OF THE SHORT SEGMENTS.

SELECT PARALLEL.

MOVE THE POINTER ABOVE ONE OF THE SIDES OF THE TRIANGLE

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE END POINT OF

THE SHORT SEGMENT ON THE BASE AND PRESS ENTER AGAIN.

YOU'LL SEE A LINE PARALLEL TO THE SIDE OF THE TRIANGLE.

REPEAT THIS PROCESS FOR THE OTHER SIDE OF THE TRIANGLE.

PRESS ESCAPE AND MOVE THE POINTER

TO THE END OF EACH PARALLEL LINE.

PRESS AND HOLD THE CLICK KEY TO SELECT THE ENDPOINT.

USE THE NAVIGATION ARROWS TO EXTEND THE PARALLEL LINES.

NOTICE THAT YOU NOW HAVE ONE OF THE OUTLINES

OF THE RHOMBUS SHAPED PANEL AT THE APEX OF THE PYRAMID.

CREATE INTERSECTION POINTS AMONG THE VARIOUS LINES.

PRESS MENU AND UNDER POINTS AND LINES

SELECT INTERSECTION POINTS.

CLICK ON PAIRS OF LINES TO CREATE INTERSECTION POINTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW CONSTRUCT THE BASE AND HEIGHT

OF THE SMALLER ISOSCELES TRIANGLE

THAT IS SIMILAR TO THE LARGER TRIANGLE.

PRESS MENU AND UNDER POINTS AND LINES

SELECT SEGMENT.

CONSTRUCT THE TWO DIAGONALS OF THE RHOMBUS

AND CREATE AN INTERSECTION POINT FOR THESE DIAGONALS.

MEASURE THE LENGTHS OF THE BASE AND HEIGHT

OF THE SMALL TRIANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

CLICK ON THE ENDPOINTS THAT THE DEFINE THE BASE

AND HEIGHT OF THIS TRIANGLE TO RECORD THE MEASUREMENTS.

ASSIGN EACH MEASUREMENT TO A NEW SET OF VARIABLES.

PRESS ESCAPE AND MOVE THE POINTER

ABOVE THE BASE MEASUREMENT.

PRESS CONTROL AND MENU AND ASSIGN THIS MEASUREMENT

TO THE VARIABLE SMALL BASE AND PRESS ENTER.

REPEAT THIS PROCESS FOR THE HEIGHT OF THE TRIANGLE

AND ASSIGN IT TO VARIABLE SMALLHEIGHT.

WE CAN NOW CALCULATE THE AREA OF EACH TRIANGLE.

CREATE A NEW CALCULATOR WINDOW.

PRESS THE HOME KEY

AND SELECT THE CALCULATOR OPTION.

USE THE VARIABLES YOU DEFINED EARLIER.

INPUT THE EXPRESSION

ONE HALF TIMES LARGE BASE TIMES LARGEHEIGHT.

ASSIGN IT TO VARIABLE AREA 1

BY PRESSING CONTROL AND THE VARIABLE KEY

AND INPUTTING THE VARIABLE NAME.

PRESS ENTER.

NOW INPUT THE EXPRESSION

ONE HALF TIMES SMALLBASE TIMES SMALLHEIGHT

AND ASSIGN IT TO VARIABLE AREA2.

PRESS ENTER.

INPUT THE EXPRESSION AREA1 DIVIDED BY AREA2

AND PRESS ENTER.

THERE ARE 324 TRIANGULAR PANELS.

TO FIND THE NUMBER OF RHOMBUS SHAPES

DIVIDE THIS VALUE BY TWO.

AND TO FIND THE TOTAL NUMBER OF RHOMBUS PANELS

FOR ALL FOUR SIDES,

MULTIPLY THAT RESULT BY FOUR.

YOUR RESULT SHOULD BE 648 PANELS.

NOTE HOWEVER THAT SOME OF THE RHOMBUS SHAPED PANELS

ARE SPLIT INTO TWO TRIANGLES

ALONG THE BASE OF THE PYRAMID.

IN ADDITION, THE MAIN ENTRANCE TO THE MUSEUM

THROUGH THE GLASS PYRAMID HAS SOME RHOMBUSES AND

TRIANGLES MISSING TO MAKE ROOM FOR AN ENTRYWAY.

SO THE FINAL COUNT IS 603 FULL RHOMBUS SHAPED PANELS

AND 70 TRIANGULAR SHAPED PANELS.

SO THE SIMPLICITY OF THE DESIGN OF THE LOUVRE PYRAMID

REVEALS AN INTRICATE MATHEMATICAL PATTERN

THAT RELIES ON SIMILARITY.