Section européenne 3D Geometry 1
Classic Solids
A la fin de ce chapitre, vous devez être capable de :
• manipuler, construire et représenter en perspective des solides ;
• effectuer des calculs, de longueur, d’aire et de volumes.
Representation of 3D solids
6.1 The solid SABCD is a squared-based pyramid with vertex S, EF GH are IJ KL two tetrahedra. Complete their representations.
b b b b
A B
S
D E
F G
H
b b b
b
I
J K
L
b b b
b
6.2 Give the number of faces for each of the the following solids.
1 2 3 4 5
1 2 3 4 5 6
0 1 2 3 4 5
1
1 2 3 4 5
1 2 3 4 5 6 7 8
1 2 3 4 5 6
0 1 2 3 4 5 6
1
1 2 3 4 5 6
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7
1 2 3 4 5 6
0 1 2 3 4 5 6 7 8
1
6.3 Draw in cavalier representation each of the following solids : 1. A cube with edge 2 cm, vanishing angle 45o, report 0,7.
2. A cuboid with length 5 cm, width and depth 2 cm, vanishing angle 45o, report 0,7.
3. A hexagonal-based prismoid (free measures).
4. A pentagonal-based pyramid.
Length, area and volume
6.4 Below is a box made of a cuboid and a semi-cylinder.
1. Compute the volume of the box (rounded to the unit).
2. Draw a net of the box.
3. Compute the area of the box. 9 mm 20 mm
22mm
6.5 Same question as in the previous exercise for the following solids.
5 m 4m
3m2m 5m
4 m 2 m 3 m
2 m
6.6 Let ABCDA′B′C′D′ be a cuboid with dimensions AB = 7, AD = 3 and AA′ = 5, the unit being the centimeter.
b b
b b b
b b
b
A B
D C A′
B′ C′ D′
1. Computing a few lengths.
a. Compute D′B, using twice a famous theorem.
b. LetI be the midpoint of AB and K′ the midpoint of A′D′. Compute the length IK′ with the same method as in question 1.
2. Areas and volumes.
a. Compute the volume of ABCDA′B′C′D′. b. Compute the area of the triangle IBC.
c. Deduce the volume of the prismIBCI′B′C′, where I′ is the midpoint ofA′B′. d. Deduce from the previous questions the volume of the prism AICDA′I′C′D′. e. Check the previous result by computing the volume of the prismAICDA′I′C′D′
directly.
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6.7 Let ABCDEF GH be a cube with side 4m. Let I and K be the midpoints of the line segments BF and AB, and J the point onEF such that EJ = 14EF.
1. What can you say about the lines IK and AF? Explain your answer.
2. Four ants are moving on the surface of the cube, fromAtoG, along different paths : a. AJ+J G;
b. AI+IF +F G;
c. AF +F G; d. AK+KI+IG.
Compute the length of the path followed by each ant, first as an exact value, then an approximate value to 2DP.
3. a. Draw the net of the cube to the scale 1 : 200.
b. Use the net to find the length of the shortest path from A toG.
4. Knowing that a competition ant moves at the incredible speed of 5×10−2km·h−1, find out how long it will take for such an ant to go fromA toG along the shortest path. The answer will be given in minutes and seconds.
6.8 The cuboctahedron
Let ABCDEF GH be a cube with side 4 cm. The pointsI,J andK are respectively the midpoints of the edges [F E], [F G] and [F B].
1. Count the numbers of verticesV, the number of edgesEand the number of faces F for the cube. Then compute
the value of V −E+F. A B
D C
E F
H G I
J
K
2. a. What is the name of the solid F IJ K? b. Compute the volume of the solid F IJ K.
c. Compute, for this solid, the value ofV −E+F.
3. We remove the eight “corners” of the cube ABCDEF GH, which gives the solid shown below. It is called a cuboctahedron.
a. Compute the value of V −E+F for the cuboctahedron.
b. Compute the volume of this cuboctahedron and its area.
Then, try to draw its area.
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Homework #4 – Barycentres of two points
Let [AB] be a segment with length 6 cm. All the points in this exercise will be placed on the same figure.
1. Let Gbe the point defined by the equality 2# »
GA+ 3# » GB= #»0 a. Prove that 5# »
GA+3# »
AB = #»
0 and deduce a simple relation between # »
AGand # » AB. b. Is there more than one position for the point G? Show it on the figure.
2. Let H the point such that B is the midpoint of [AH].
a. Build the pointH on the figure.
b. What can you say about the vectors # »
AH and # »
HB? c. Find two numbersα and β such that α# »
HA+β# » HB = #»0 . The pointGdefined in question 1 by the equality 2# »
GA+3# »
GB = #»0 is called thebarycentre of the points A and B with respective coefficients 2 and 3. More generally, any point K defined by an equality α# »
KA+β# » KB= #»
0 where α and β are two real numbers such that α+β 6= 0 is called the barycentre of the points A and B with respective coefficients α and β.
3. What can you say about the point H?
4. Let P be the barycentre of A and B with respective coefficients 2 and 1.
a. What equality defines the point P?
b. Deduce from this equality a relation between the vectors # »
AP and # » AB. c. Place the point P.
5. Let Qbe the barycentre of A and B with respective coefficients −2 and 1.
a. What equality defines the point Q?
b. Deduce from this equality a relation between the vectors # »
AQ and # » AB. c. Place the point Q.
6. Let R be the barycentre of A and B with respective coefficients −2 and 2.
a. What equality defines the point R?
b. Deduce from this equality a relation between the vectors # »
AR and # » AB.
c. Could you place the point R? What is the problem ? 7. Let S be the point such that # »
AS = 1 4
# » AB. a. Place the point S.
b. Prove that 4# »
AS = # »
AS+# »
SB.
c. Deduce two numbers m and n such thatm# »
SA+n# » SB = #»
0 . d. Give a definition of S using barycentre.
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