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179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De piramide, de kegel en de bol 186 M3 Het volume van een piramide, een kegel en een bol 190

M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De ...mathima.be/mediapool/77/770736/data/Frank/Wis2/Matrix/Oplossingen/... · 179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke

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Page 1: M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De ...mathima.be/mediapool/77/770736/data/Frank/Wis2/Matrix/Oplossingen/... · 179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke

179

Meetkunde1 Ruimtemeetkunde

M1 Ruimtelijkesituatiesvoorstellenineenvlak 180

M2 Depiramide,dekegelendebol 186

M3 Hetvolumevaneenpiramide,eenkegeleneenbol 190

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180 M1 Ruimtelijkesituatiesvoorstellenineenvlak

Titel1 RuimtelijkesituatiesvoorstellenineenvlakM1

Het is verstandig om zo nu

en dan een vraagteken te

plaatsen bij de dingen die

je lang voor vanzelfspre-

kend hebt gezien.

Russel564B Tekenhetvluchtpunt(ofvluchtpunten)bijdenatuurlijkeperspectieven.

565V* Onderstaandetekeningstelteenhuisjeinnatuurlijkperspectiefvoor.

Tekenindelangstegeveltweeramenenindeanderegeveleendeureneenraamerboven.

a

c

b

566B Welksoortperspectiefwordtgebruiktindezevlakkevoorstellingen?

a b c

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

567B Welksoortperspectiefofaanzichtherkenjeindezevlakkevoorstellingen?

a b c

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

P

P1 P2

isometrisch perspectief natuurlijk perspectief of natuurperspectief

natuurlijk perspectief of natuurperspectief

natuurperspectief isometrisch perspectief cavalièreperspectief

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181 Ruimtelijkesituatiesvoorstellenineenvlak M1

569B Kunjebijvolgendetekeningenbijdeaangeduidematendewerkelijkelengteaflezen?Indienja,geefdewerkelijkelengte.Indienneen,verklaarwaaromniet.

a

d

h

b

l

h

c

l

h

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

d e f

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

568B Geefdenaamvandevoorstellingswijze.

kenmerken perspectief

aDeafmetingenblijvenbehouden,maarhoekgroottenblijvennietbehouden�

bDeafmetingenblijvenbehoudenenallehoekgroottenblijvenbehouden�

c Deevenwijdigheidblijftsomsbehouden�

d Dehoekgroottenwijzigenallemaal�

e Rechthoekenwordenvoorgestelddoortrapeziums�

f Allevierkantenblijvenvierkanten�

g Rechthoekenwordenparallellogrammen�

570B Vanwelkperspectiefwordtindevolgendegrapgebruikgemaakt?Denkjedatdestudentenhunjobkunnenbehouden?

Twee studenten doen een vakantiejob bij de Belgische Spoorwegen.

De ploegbaas wil ze even testen en neemt ze mee naar een stuk spoorlijn die buiten gebruik is. Samen gaan ze midden tussen de rails staan en de ploegbaas zegt: ‘Mannen, kijk goed in de verte, wat stellen jullie vast?’ Samen turen ze naar de verre horizon tot waar de rails in één punt samen lijken te komen. Zegt de ploegbaas: ‘Mannen, jullie begrijpen dat dit een heel gevaarlijke toestand is. Ga onmiddellijk naar het punt waar de rails samen komen en her-stel de situatie.’ De twee studenten gaan op stap. Vele kilometers en uren later hebben ze ‘het gevaarlijke punt’ nog steeds niet bereikt.Plots draait één van beiden zich om, schrikt en roept: 'Verdorie, dat kan toch niet , we zijn te ver, we zijn dat punt voorbij!'

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

isometrisch perspectief vooraanzicht natuurperspectief

isometrisch perspectief

aanzichten

natuurlijk perspectiefisometrisch perspectiefnatuurlijk perspectief

aanzichtenisometrisch perspectief

h = 2,3 cm b = 1,6 cm d = 3,2 cm l = 2,3 cm h = 1,1 cm b = 1,1 cm Neen in een natuurperspectief blijven

de lengten niet altijd behouden.

natuurlijk perspectief

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G

C

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182 M1 Ruimtelijkesituatiesvoorstellenineenvlak

571V* Tekendefiguurincavalièreperspectief.

572B Deeerstekubusisincavalièreperspectiefgetekend,detweedeinisometrischperspectiefmetalsbovenvlakMNOP.

• Geefdewerkelijkeonderlingeliggingvanonderstaanderechten�• Kiesdemeestpassendeoplossinguit: , , ,kruist

a EH � � � � � � � � � � � � GC

b EA � � � � � � � � � � � � GC

c AD � � � � � � � � � � � � EF

d FC � � � � � � � � � � � � FD

e FH � � � � � � � � � � � � AB

f FB � � � � � � � � � � � � AB

g NM � � � � � � � � � � � � KL

h PL � � � � � � � � � � � � JK

i OM � � � � � � � � � � � � NJ

j OI � � � � � � � � � � � � PL

k MI � � � � � � � � � � � � IL

l NK � � � � � � � � � � � � OK

573B IndevolgendekubuszijndepuntenM,N,K,Lderespectievelijkemiddensvanderibben[EH],[FG],[EF]en[HG].

Zeteenkruisjeindepassendekolom(men)zodatjewareuitsprakenkrijgtoverdewerkelijkheid.

rechthoekig gelijkbenig gelijkzijdig

a ΔABCb ΔCDLc ΔEKAd ΔLDHe ΔBGDf ΔKGHg ΔAEMh ΔNGLi ΔKLEj ΔFGB

kruist

//

kruist

///

kruist

^

//

kruist

kruist

kruist

^

///

x

xx

xx xxx x

x x

x

x

x

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b

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183 Ruimtelijkesituatiesvoorstellenineenvlak M1

574B Devolgenderuimtefigurenzijnopgebouwduitevengrotekubussen. Bijwelk(e)aanzicht(en)ziejedegrijsgekleurdekubusniet?

a � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

d � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

c � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

f � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

575V* Opdevoorstellingvandekubuszijndrierechtengetekend.

a Welkerechtestaatopdezetekeningloodrechtopderechtec?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

b Waaromisditmisleidend?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

576V* Detuinpadenwordenvoorgestelddoorlijnstukken.

a Welklijnstukisopdezetekeninghetlangstelijnstuk:lijnstukABoflijnstukAC?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

b Waaromisditmisleidend?

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rechterzijaanzicht

vooraanzichtrechterzijaanzicht

linkerzijaanzichtachteraanzichtvooraanzichtbovenaanzichtlinkerzijaanzicht

vooraanzicht, bovenaanzichtonderaanzichtrechterzijaanzichtlinkerzijaanzicht

a

De kubus is getekend in isometrisch perspectief.

De rechte c is evenwijdig met de opstaande ribbe en de

rechte b is evenwijdig met de ribbe van het grondvlak.

In werkelijkheid : b ^ c . Als je de kubus wegneemt: a ^ c

De lijnstukken zijn even lang.

We zien de lijnstukken in de tekening in natuur­perspectief. De lijnstukken stellen randen van de wegen voor. De weg AC is langer in het natuurperspectief, want de wegen komen beide in hetzelfde punt toe maar de weg AC vertrekt meer vooraan, dus verder af van A in de context.

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

JanvanEyck(ca�1390-1441)Vlaamskunstschilder

HuwelijkvanArnofini(1434)

Bovenaan zie je een muur en staan de zuilen naast

elkaar, maar als je onderaan kijkt zie je dat 1 zuil voor

de andere staat, de fluitspeler zit tussen de zuilen.

Omdat een tekening maar 2 dimensies heeft, er is geen diepte.

Begin links bovenaan: je ziet het water naar beneden val­

len; daar brengt het een rad in beweging. Daarna stroomt

het weg in een gemetselde afvoergeul. Volg de loop van het

water en het loopt onmiskenbaar telkens lager en het verwij­

dert zich van ons. Plots blijkt het verste en laagste punt iden­

tiek te zijn met het hoogste punt dichtbij.

De vluchtpunten komen niet samen op de horizon

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581V* Hiernaastzieje‘Terrasubterranea’vanRogerShepard.

a Welkperspectiefwordthiergebruikt?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

b Welkeafbeeldingishetgrootst?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

c Meetdemonsters�Waaromisditmisleidend?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Het verdwijnpunt ligt hier voor het schilderij. Het verdwijnpunt is hier de toeschouwer.

De vrouw en het paard zijn deels onzichtbaar door de bomen. Maar

je ziet de verkeerde delen. Er zijn meerdere fouten. Bv. de stam van de

dunnere boom naast de paardrijdster is: achter het paard getekend

op de grond maar is voor het paard en de paardrijdster getekend

ter hoogte van de flank van het paard. Bv. de ‘lege’ ruimte tussen de

bomen kan in de werkelijkheid het paard niet onzichtbaar maken.

Het natuurlijk perspectief.

Ze zijn beide even groot.

Een voorwerp dat zich verderaf bevindt, neem je

kleiner waar en wordt kleiner getekend in een

natuurlijk perspectief. Door de achtergrond en het

perspectief wordt gesuggereerd dat het grote

monster zich achter in de tunnel bevindt en dus ver­

der weg is, waardoor ons brein hem interpreteert als

groter dan het monster op de voorgrond. Zelfs de

uitdrukking van de monsters lijkt anders te zijn door

de situatie waar ze zich in bevinden. Het lijkt wel of

het kleine monster compleet in paniek is…

Page 8: M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De ...mathima.be/mediapool/77/770736/data/Frank/Wis2/Matrix/Oplossingen/... · 179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke

186 M2 Depiramide,dekegelendebol

Depiramide,dekegelendebolM2

582B Vulhetschemain.

a b c

denaamvandepiramide

hetaantalgrensvlakken

hetaantalopstaandegrens-vlakken

hetaantalribben

hetaantalopstaanderibben

devormvandeopstaandegrensvlakken

hetaantalhoekpunten

583B Vulin(geefhetcorrecteaantalofgeefdemeestpassendebenaming).

a Hoeveelgrensvlakkenheefteenvijfzijdigepiramide? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

b Welkevormheefthetgrondvlakvaneenvierzijdigepiramide? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

c Welkevormheefthetgrondvlakvaneenvijfzijdigepiramide? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

d Welkevormhebbendeopstaandegrensvlakkenvaneenachtzijdigepiramide? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

e Hoeveelribbenheefteentwaalfzijdigepiramide? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

f Hoeveelhoekpuntenheefteenpiramidemet10ribben? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

g Hoeveelopstaandegrensvlakkenheefteenpiramidemet12ribben? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

driezijdige piramide

4

3

6

3

driehoek

4

vierzijdige piramide

5

4

8

4

driehoek

5

zeszijdige piramide

7

6

12

6

(gelijkbenige) driehoek

7

6vierhoekvijfhoekdriehoeken2466

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187 Depiramide,dekegelendebol M2

586V** Indevlakkemeetkundeiseenregelmatigeveelhoekeenveelhoekwaarvanallezijdenevenlangzijnenallehoekenevengroot.Inderuimtemeetkundeiseenregelmatigveelvlakeenveelvlakwaar-vanallegrensvlakkenregelmatigeveelhoekenzijnenwaarbijinelkhoekpuntevenveelzijvlakkensamenkomen.Erbestaanslechts5regelmatigeveelvlakken.Vuldetabelaan.

585V** Schrijfinwiskundetaal(niseennatuurlijkgetalgroterdan2).

a Eenpiramidemetngrensvlakkeniseen� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � -zijdigepiramide�

b Eenpiramidemetnhoekpuntenheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ribben�

c Eenpiramidemetnribbenheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � grensvlakken�

Naam Aantalgrens-vlakken

Aantalhoekpun-ten

Aantalribben

tetraëderofregelmatig

� � � � � � � � � vlak

kubusofregelmatig

� � � � � � � � � vlak

octaëderofregelmatig

� � � � � � � � � vlak

584V* Schrijfinwiskundetaal(niseennatuurlijkgetalgroterdan2).

aantalzijdengrondvlak 3 4 5 6 n

aantalgrensvlakken

aantalhoekpunten

aantalribben

aantalopstaandegrensvlakken

a Eenn–zijdigepiramideheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � grensvlakken�

b Eenn-zijdigepiramideheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � hoekpunten�

c Eenn-zijdigepiramideheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ribben�

d Eenn-zijdigepiramideheeft� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � opstaandegrensvlakken�

4463

5584

66

105

77

126

n + 1n + 1

2nn

n + 1n + 1

2nn

(n – 1)2(n – 1)

n _ 2

+ 1

4

6

8

4

6

8

4

8

6

6

12

12

Page 10: M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De ...mathima.be/mediapool/77/770736/data/Frank/Wis2/Matrix/Oplossingen/... · 179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke

188 M2 Depiramide,dekegelendebol

dodecaëderofregelmatig

� � � � � � � � � vlak

icosaëderofregelmatig

� � � � � � � � � vlak

• Inelkregelmatigveelvlakbestaatereenverbandtussenhetaantalgrens-vlakken,hetaantalhoekpuntenenhetaantalribben.Noteerditverband.

Aantalgrensvlakken� � � � � � � � � � � � aantalhoekpunten=� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

WeetjeRegelmatigeveelvlakkenwordenookweleensPlato-nischeveelvlakkengenoemd,naarPlatodegroteGrieksefilosoof(428-348voorChris-tus)�Dezeveelvlakkenston-densymboolvoorhetvuur(regelmatigviervlak);deaarde(kubus);delucht(regelmatigachtvlak);hetuniversum(regelmatigtwaalfvlak)enhetwater(regelmatigtwintigvlak)�

587B Welkeontwikkelingenstelleneenpiramidevoor?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

588B Welkevanonderstaandefigurenzijnvoorstellingenvaneenkegel?

a

a

c

c

d

d

e

e

f

f

g

b

b

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

12

20

12

20

20

12

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SG = ∏ · r² = ∏ . (3cm)² = 28,26 cm²

V = SG · h

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V = 28,26 cm² · 10 cm

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V ≈ 94,2 cm³

Het volume van een hoorntje is 94,2 cm³

Het volume van de bal is 8 181 cm³.

V = 4 _ 3

∏ r³

V = 4 _ 3

· ∏ · (12,5 cm)³

V ≈ 8 181 cm³

180 cm²

339,12 dm²

60 dm

18 cm

V = SG · h

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540 cm³ = SG · 9 cm

_ 3

SG = 540 cm³ · 3 _ 9 cm

SG = 180 cm²

V = SG · h

_ 3

471 cm³ = 78,5 cm² · h

_ 3

h = 471 cm³ · 3 _ 78,5 cm²

h = 18 cm

V = SG · h

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38,4 dm³ = 1,920 dm² · h

__ 3

h = 38,4 dm³ · 3

_ 1,920 dm²

h = 60 dm

V = SG · h

_ 3

226,08 dm³ = SG · 2 dm

_ 3

SG = 226,08 dm³ · 3

__ 2 dm

SG = 339,12 dm²

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Het glas is 9 cm hoog.

15 cl = 150 ml = 150 cm³SG = ∏ · r² SG = ∏ · (4 cm)² SG = ∏ · 16 cm²SG ≈ 50,27 cm²

V = SG · h

_ 3

150 cm³ =

50,27 cm² · h __ 

3

h = 150 cm³ · 3 _ 50,27 cm²

h = 8,952 cm ≈ 9cm

a Volume van de volledige kegel

SG = ∏ · r²

SG = ∏ · (7,5 cm)²

SG = 176,715 cm²

Vvolledige kegel = SG · h

_ 3

Vvolledige kegel = 176,715 cm² · 30 cm __ 

3

Vvolledige kegel = 1 767,15 cm³

b Volume van het onderste deel, het verdwenen deel

SG = ∏ · r²

SG = ∏ · (2,5 cm)²

SG ≈ 19,635 cm²

Vverdwenen gedeelte = SG · h

_ 3

Vverdwenen gedeelte = 19,635 cm² · 10 cm __ 

3

Vverdwenen gedeelte ≈ 65,44 cm³

c Volume van de maatbeker

Vmaatbeker = Vhele kegel – Vverdwenen gedeelte

Vmaatbeker = 1 767,15 cm³ – 65,44 cm³ = 1 701,71 cm³

Antwoord: Het volume van de maatbeker is 1 701,71 cm³

Page 15: M1 Ruimtelijke situaties voorstellen in een vlak 180 M2 De ...mathima.be/mediapool/77/770736/data/Frank/Wis2/Matrix/Oplossingen/... · 179 Meetkunde 1 Ruimtemeetkunde M1 Ruimtelijke

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Je verdeelt het huisje in drie delen en je berekent de drie

volumes.

Deel 1 is een balk:

Vdeel 1 = l · b · h

Vdeel 1 = 1,4 m · 0,8 m · 1,6 m

Vdeel 1 = 1,792 m³

Deel 2 is een balk:

Vdeel 2 = l · b · h

Vdeel 2 = 1 m · 0,8 m · 1,1 m

Vdeel 2 = 0,88 m³

Deel 3 is een piramide met als grondvlak een rechthoek.

Je berekent eerst de oppervlakte van deze rechthoek.

SG = l · b

SG = 1,4 m · 0,8 m

Je verdeelt de kooi in 4 figuren.Figuur 1 is een cilinder. Je berekent eerst de oppervlakte van de cirkel.SG = ∏ · r² SG = ∏ · (1,2m)² SG ≈ 4,5239 m²

1,6 m

1,2 m

0,5 m

0,8 m

2,4 m

1 m

�g 1

�g 2

�g 3

0,5 m

0,6 m

1,8 m

1,6 m

1,2 m

1,2 m

1,4 m

�g 1

�g 2

�g 3

�g 5

�g 4

SG = 1,12 m²

Vdeel3 = SG · h _ 

3

Vdeel3 = 1,12 m² · 1,2 m _ 

3

Vdeel3 = 0,448 m³

Vkippenverblijf = Vdeel 1 + Vdeel 2 + Vdeel 3

Vkippenverblijf = 1,792 m³ + 0,88 m³ + 0,448 m³

Vkippenverblijf = 3,12 m³

Vfiguur 1 = SG · hVfiguur 1 = 4,5239 m² . 1,6 mVfiguur 1 ≈ 7,238 m³

Figuur 2 is een afgeknotte kegel met hetzelfde grondvlak als figuur 1.

Vkegel = SG · h

_ 3

Vkegel = 4,5216 m² · 1,8 m __ 

3

Vkegel ≈ 2,712 m³

Vverdwenen deel = SG · h

_ 3

Vverdwenen deel = ∏ · (0,5 m)² · 0,6 m _ 

3 ≈ 0,157 m3

Vfiguur2 = Vkegel – Vverdwenen deel =

2,712 m³ – 0,157 m³ = 2,555 m³

Figuur 3 is een cilinder.

Je berekent eerst de oppervlakte van de cirkelschijf

SG = ∏ · r²

SG = ∏ · ( 0,5²) m²

SG ≈ 0,785 m²

Vfiguur 3 = SG · h

Vfiguur 3 = 0,785 m² · 1,4 m

Vfiguur 3 ≈ 1,099 m³

Figuur 4 is een kegel met hetzelfde grondvlak als figuur 3

Vfiguur 4 = SG · h

_ 3

Vfiguur 4 = 0,785 m² · 1,2 m __ 

3

Vfiguur 4 ≈ 0,314 m³

Vkooi = Vfiguur 1 + Vfiguur 2 + Vfiguur 3 + Vfiguur 4

Vkooi ≈ 7,238 m³ + 2,555 m³ + 1,099 m³ + 0,314 m³

Vkooi ≈ 11,206 m³

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(15 625 = 25³)

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· ∏ · r³

Vbruisbal Lummen = 4 _ 3

· ∏ · (1 cm)³

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Vbruisbal Beringen = 4 _ 3

· ∏ · r³

Vbruisbal Beringen = 4 _ 3

· ∏ · (3 cm)³

Vbruisbal Beringen ≈ 113,10 cm³

De straal van de bruisbal van Beringen is 3 keer groter.

Het volume van de bruisbal van Beringen is 3³ = 27 keer groter.

Stap 1: Berekening van volume van één sinaasappel

O = 2 · ∏ · r

27,5 cm = 2 · ∏ · r

r = 27,5 cm

_ 2 · ∏

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V = 4 _ 3

· ∏ · r³

V = 4 _ 3

· ∏ · (4,38 cm)³

V = 351,19 cm³

Stap 2: Berekening van de hoeveelheid sap van één sinaasappel

30 % van 351,19 cm³ = 30 _ 100

· 351,80 cm³ = 105,36 cm³

Stap 3: Berekening van het aantal sinaasappels dat nodig is om 1 liter sap te

bekomen

1 l = 1 dm³ = 1000 cm³

1000 : 105,36 = 9,49… ≈ 10

Tom heeft 10 sinaasappels nodig.

V = ( 4 _ 3

∏ · r³ ) : 2V = 2 _ 

3 · ∏ · r³

V = 2 _ 3

· ∏ · (4,5 cm)³ V = 190,852… cm³ ≈ 191 cm²

Het volume van 1 soepbowl is 191cm³

85 % van 191 cm³ = 85 _ 100

· 191 cm³ = 162,35 cm³

8 liter = 8000 cm³

8000 : 162,35 = 49,276 ≈ 49

49 klanten kunnen van deze soep eten.

SG = ∏ · r²

SG = ∏ · (12,5 cm)²

SG = 490,874 cm²

V = SG · h

_ 3

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V = 490,874 cm² · 50 cm

__ 3

V = 8181 cm³

8181 cm³ = 8,181 l ≈ 8,2 l

V1bolletje = 4 _ 3

· ∏ · r³

V1bolletje = 4 _ 3

· ∏ · (2,25 cm)³

V1bolletje = 47,712 9… ≈ 48 cm³

2,5 liter = 2,5 dm³ = 2 500 cm³

2500 cm³ : 48 cm³ = 52,083 3… ≈ 52

Je kunt ongeveer 52 bolletjes scheppen.

Vkegel = SG · h

_ 3

SG = ∏ · r²

De hoogte is het dubbel van de straal: h = 2r

Vkegel = ∏ · r² · 2 · r _ 3

= 2 · ∏ · r³ _ 3

Vbol = 4 _ 3

· ∏ · r³ = 2 · 2 · ∏ r³ _ 3

Vkegel = 2 · ∏ · r³ _ 3

en Vbol = 2 · 2 · ∏ · r³ _ 3

dus het volume van

de bol is het dubbel van het volume van de kegel

Vcilinder = SG · h

Nu is: SG = ∏ · r²

de hoogte is het dubbel van de straal: h = 2r

Dus Vcilinder = ∏ · r² · 2 r = 2 · ∏ · r³

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WeetjeHetlijnstukMVwordtindevlakkemeetkundehetapothemavandekoordeABgenoemd�

Vkegel = 2 · ∏ · r³ _ 3

en Vcilinder = 2 · ∏ · r³

Het volume van de cilinder is het drievoud van het

volume van de kegel met dezelfde hoogte.

Vonderste deel = Vcilinder

Vcilinder = SG · h

Vcilinder = ∏ · r² · h

Vcilinder = ∏ · (21 cm)² · 58 cm

Vcilinder = 80 355,656… cm³ ≈

80 355,66 cm²

Vbovenste deel = Vhalve bol

Vhalve bol = 1 _ 2

· 4 _ 3

· ∏ · r³ =

Vhalve bol = 2 _ 3

· ∏ · (21 cm)³

Vhalve bol ≈ 19 396,19 cm³

Vvogelkooi = 80 355,66 cm³ +

19 396,36 cm³

Vvogelkooi = 99 751,85 cm³

De massa van de tennisbal is 2,819 3 kg.

Berekening massa goud met verhoudingstabelVolume (in cm ³) 1000 146,08Massa (in kg) 19,3 2,8193

Berekening van de straal van de tennisbalO = 2 · ∏ · r21,98 cm = 2· ∏ · r2· ∏ · r = 21,98 cm

r = 21,98 cm _ 

2 · ∏

r ≈ 3,5 cm

Vhele tennisbal = 4 _ 3

· ∏ · r³

Vhele tennisbal = 4 _ 3

· ∏ · (3,5 cm)³Vhele tennisbal ≈ 179,59 cm³

Vlucht = 4 _ 3

· ∏ · r³

Vlucht = 4 _ 3

· ∏ · (2 cm)³Vlucht ≈ 33,51 cm³Vgoud = Vhele tennisbal – VluchtVgoud = 179,59 cm³ – 33,51 cm³ = 146,08 cm³

Verdeel de zeshoek in 6 even grote driehoeken

SG één driehoek = b · h _ 2

SG één driehoek = 3 cm · 2,4 cm

__ 2

SG één driehoek = 3,6 cm²

SG regelmatige zeshoek = 6 · SG één driehoek

2 _ 3

· ∏ · r³

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Antwoord:� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

WeetjeEenregelmatigepiramideiseenpiramidemetalsgrondvlakeenregelmatigeveelhoek�Dehoogtelijnuitdetopgaatdoorhetmid-delpuntvanhetgrondvlak(=middelpuntvandeomge-schrevencirkeldiedoordehoekpuntenvanhetgrond-vlakgaat�

SG regelmatige zeshoek = 6 · 3,6 cm²

SG regelmatige zeshoek = 21,6 cm²

Vpiramide = SG · h

_ 3

Vpiramide = 21,6 cm² · 12 cm

__ 3

Vpiramide = 86,4 cm³

Je berekent eerst de oppervlakte van het grondvlak, dus van de regelmatige achthoek.Hiervoor verdeel je de achthoek in 8 driehoekjes.

SG = ( 2,4 cm · 1,2 cm __ 

2 ) · 8

SG = 11,52 cm²

Véén ijsblokje = SG · h

_ 3

Véén ijsblokje = 11,52 cm² · 2,7 cm

__ 3

Véén ijsblokje = 10,368 cm³

Vzes ijsblokjes = 10,368 cm³ · 6

Vzes ijsblokjes = 62,208 cm³

Het volume stijgt 62 ml

Het volume van de piramide is 86,4 cm3