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179
Meetkunde1 Ruimtemeetkunde
M1 Ruimtelijkesituatiesvoorstellenineenvlak 180
M2 Depiramide,dekegelendebol 186
M3 Hetvolumevaneenpiramide,eenkegeleneenbol 190
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
181 Ruimtelijkesituatiesvoorstellenineenvlak M1
569B Kunjebijvolgendetekeningenbijdeaangeduidematendewerkelijkelengteaflezen?Indienja,geefdewerkelijkelengte.Indienneen,verklaarwaaromniet.
a
d
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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
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
G
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D
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F
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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
b
a
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a b c
d e f
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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581V* Hiernaastzieje‘Terrasubterranea’vanRogerShepard.
a Welkperspectiefwordthiergebruikt?
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
b Welkeafbeeldingishetgrootst?
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
c Meetdemonsters�Waaromisditmisleidend?
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
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…
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
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
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
30
30
+ aantal ribben + 2
a, e, g
b, d en f
189 Depiramide,dekegelendebol M2
589B Waaromisdeaardegeenbol?
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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
1,6 m
1,2 m
0,5 m
0,8 m
2,4 m
1 m
0,5 m
0,6 m
1,8 m
1,6 m
1,2 m
1,2 m
1,4 m
193 Hetvolumevaneenpiramide,eenkegeleneenbol(uitbreiding) M3
597U(V*) Opdedierenboerderijhebbendedierenleukehuisjes.Berekenhetvolumevanhetkippenhok.
598U(V*) Berekenhetvolumevandekooivandedwergpapegaaien.
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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
194 M3 Hetvolumevaneenpiramide,eenkegeleneenbol(uitbreiding)
599U(V*) Hetatomiumiseenstalenconstructiediebestaatuit9bollenmetelkeendiametervan18meter.
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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
M
2,4 cm1,2 cm
2,7
cm
198 M3 Hetvolumevaneenpiramide,eenkegeleneenbol(uitbreiding)
2,4cm
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AM
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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