Solides - 6e

Pavé droit, cube et prisme droit

Exercice 1 : Nommer et compter sommets/faces/arêtes d'un polyèdre

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 500, "height": 350}}, "data": {"Cylinder_1": {"position": {"x": 0.0, "y": -5.0, "z": 20.0}, "rotation": {"x": -1.57079632679, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "GenericRightCylinder", "path": "Triangle_0", "amount": 40}, "Triangle_0": {"type": "Path2D", "path": [{"moveTo": {"x": 0.0, "y": 0.0}}, {"lineTo": {"x": 40.0, "y": 0.0}}, {"lineTo": {"x": 12.0, "y": 28.0}}]}, "Point3D_E": {"type": "Point", "position": {"x": 0.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": 0.0, "y": -5.0, "z": 20.0}, "text": "E", "position_offset": {"x": 0.0, "y": -4.8, "z": 0.0}}, "Point3D_I": {"type": "Point", "position": {"x": 40.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 40.0, "y": -5.0, "z": 20.0}, "text": "I", "position_offset": {"x": 4.762933808225604, "y": -0.5953667260282005, "z": 0.0}}, "Point3D_R": {"type": "Point", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}}, "Label_for_Point3D_R": {"type": "Label", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}, "text": "R", "position_offset": {"x": 1.8658470193695444, "y": -0.7774362580493255, "z": -4.353643045195604}}, "Point3D_A": {"type": "Point", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}, "text": "A", "position_offset": {"x": 0.0, "y": 4.8, "z": 2.6861438787558914e-11}}, "Point3D_G": {"type": "Point", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_G": {"type": "Label", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}, "text": "G", "position_offset": {"x": 3.612368134593013, "y": 3.160822117768887, "z": 1.7688381215585684e-11}}, "Point3D_M": {"type": "Point", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}}, "Label_for_Point3D_M": {"type": "Label", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}, "text": "M", "position_offset": {"x": 1.2413677786677848, "y": 3.6206560211285552, "z": -2.8965248168712363}}}}

On pourra faire pivoter la figure en cliquant dessus.

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 2 : Reconnaître un polyèdre - Nommer et compter sommets/faces/arêtes

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 300, "height": 300}, "camera": {"starting_position": {"x": 100, "y": 200, "z": 400}}, "materials_options": {"surface": {"color": 16744448}, "solid_line": {"color": 16744448, "linewidth": 2}, "hidden_solid_line": {"color": 16744448, "linewidth": 2}}}, "data": {"Cuboid_(FVIRWPES)": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 25, "ly": 25, "lz": 25, "point_render": {"visible": false}, "label_render": {"visible": true}}, "Label_for_Point3D_F": {"type": "Label", "position": {"x": -13.0, "y": -13.0, "z": -13.0}, "text": "F", "position_offset": {"x": -2.771281292110204, "y": -2.771281292110204, "z": -2.771281292110204}}, "Label_for_Point3D_V": {"type": "Label", "position": {"x": 12.0, "y": -13.0, "z": -13.0}, "text": "V", "position_offset": {"x": 2.623608107563835, "y": -2.842242116527488, "z": -2.842242116527488}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 12.0, "y": 12.0, "z": -13.0}, "text": "I", "position_offset": {"x": 2.6944143536159304, "y": 2.6944143536159304, "z": -2.9189488830839245}}, "Label_for_Point3D_R": {"type": "Label", "position": {"x": -13.0, "y": 12.0, "z": -13.0}, "text": "R", "position_offset": {"x": -2.842242116527488, "y": 2.623608107563835, "z": -2.842242116527488}}, "Label_for_Point3D_W": {"type": "Label", "position": {"x": -13.0, "y": -13.0, "z": 12.0}, "text": "W", "position_offset": {"x": -2.842242116527488, "y": -2.842242116527488, "z": 2.623608107563835}}, "Label_for_Point3D_P": {"type": "Label", "position": {"x": 12.0, "y": -13.0, "z": 12.0}, "text": "P", "position_offset": {"x": 2.6944143536159304, "y": -2.9189488830839245, "z": 2.6944143536159304}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": 12.0, "y": 12.0, "z": 12.0}, "text": "E", "position_offset": {"x": 2.7712812921102032, "y": 2.7712812921102032, "z": 2.7712812921102032}}, "Label_for_Point3D_S": {"type": "Label", "position": {"x": -13.0, "y": 12.0, "z": 12.0}, "text": "S", "position_offset": {"x": -2.9189488830839245, "y": 2.6944143536159304, "z": 2.6944143536159304}}}}

On pourra faire pivoter la figure en cliquant dessus.

Quelle est sa nature ?

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 3 : Reconnaître le patron d'un pavé droit

Soit le pavé suivant :

Parmi les figures suivantes, trouver celle ou celles étant un patron de ce pavé.

A.

{"init": {"range": [[-1.25, 39.0], [-1.0, 27.25]], "scale": [20, 20]}, "label": [[[14.0, 16.0], "10 m", "above", {}], [[33.0, 16.0], "10 m", "above", {}], [[19.0, 21.0], "10 m", "left", {}], [[19.0, 5.0], "10 m", "left", {}], [[23.5, 26.0], "9 m", "above", {}], [[4.5, 16.0], "9 m", "above", {}], [[0.0, 13.0], "6 m", "left", {}]], "line": [[[0.0, 10.0], [0.0, 16.0], {"subtype": "segment"}], [[19.0, 16.0], [19.0, 26.0], {"subtype": "segment"}], [[28.0, 0.0], [28.0, 10.0], {"subtype": "segment"}], [[28.0, 16.0], [28.0, 26.0], {"subtype": "segment"}], [[19.0, 0.0], [28.0, 0.0], {"subtype": "segment"}], [[0.0, 10.0], [9.0, 10.0], {"subtype": "segment"}], [[28.0, 16.0], [38.0, 16.0], {"subtype": "segment"}], [[0.0, 16.0], [9.0, 16.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 10.0], {"subtype": "segment"}], [[19.0, 10.0], [28.0, 10.0], {"subtype": "segment"}], [[38.0, 10.0], [38.0, 16.0], {"subtype": "segment"}], [[19.0, 26.0], [28.0, 26.0], {"subtype": "segment"}], [[19.0, 10.0], [19.0, 16.0], {"subtype": "segment"}], [[28.0, 10.0], [38.0, 10.0], {"subtype": "segment"}], [[9.0, 10.0], [19.0, 10.0], {"subtype": "segment"}], [[19.0, 16.0], [28.0, 16.0], {"subtype": "segment"}], [[9.0, 16.0], [19.0, 16.0], {"subtype": "segment"}], [[9.0, 10.0], [9.0, 16.0], {"subtype": "segment"}], [[28.0, 10.0], [28.0, 16.0], {"subtype": "segment"}]]}
B.

{"init": {"range": [[-1.25, 29.0], [-1.0, 32.25]], "scale": [20, 20]}, "line": [[[19.0, 25.0], [28.0, 25.0], {"subtype": "segment"}], [[0.0, 25.0], [0.0, 31.0], {"subtype": "segment"}], [[9.0, 0.0], [9.0, 10.0], {"subtype": "segment"}], [[0.0, 31.0], [9.0, 31.0], {"subtype": "segment"}], [[19.0, 10.0], [19.0, 16.0], {"subtype": "segment"}], [[9.0, 16.0], [9.0, 25.0], {"subtype": "segment"}], [[19.0, 25.0], [19.0, 31.0], {"subtype": "segment"}], [[9.0, 10.0], [9.0, 16.0], {"subtype": "segment"}], [[0.0, 10.0], [0.0, 16.0], {"subtype": "segment"}], [[19.0, 31.0], [28.0, 31.0], {"subtype": "segment"}], [[0.0, 0.0], [0.0, 10.0], {"subtype": "segment"}], [[28.0, 25.0], [28.0, 31.0], {"subtype": "segment"}], [[19.0, 16.0], [19.0, 25.0], {"subtype": "segment"}], [[9.0, 25.0], [9.0, 31.0], {"subtype": "segment"}], [[0.0, 10.0], [9.0, 10.0], {"subtype": "segment"}], [[0.0, 0.0], [9.0, 0.0], {"subtype": "segment"}], [[0.0, 16.0], [9.0, 16.0], {"subtype": "segment"}], [[9.0, 25.0], [19.0, 25.0], {"subtype": "segment"}], [[0.0, 25.0], [9.0, 25.0], {"subtype": "segment"}], [[9.0, 10.0], [19.0, 10.0], {"subtype": "segment"}], [[9.0, 31.0], [19.0, 31.0], {"subtype": "segment"}], [[9.0, 16.0], [19.0, 16.0], {"subtype": "segment"}]], "label": [[[23.5, 31.0], "9 m", "above", {}], [[9.0, 20.5], "9 m", "left", {}], [[0.0, 13.0], "6 m", "left", {}], [[0.0, 5.0], "10 m", "left", {}], [[4.5, 31.0], "9 m", "above", {}], [[14.0, 31.0], "10 m", "above", {}], [[0.0, 28.0], "6 m", "left", {}]]}
C.

{"init": {"range": [[-1.25, 20.0], [-1.0, 32.25]], "scale": [20, 20]}, "line": [[[0.0, 15.0], [0.0, 21.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 6.0], {"subtype": "segment"}], [[0.0, 21.0], [0.0, 31.0], {"subtype": "segment"}], [[9.0, 15.0], [19.0, 15.0], {"subtype": "segment"}], [[9.0, 15.0], [9.0, 21.0], {"subtype": "segment"}], [[0.0, 31.0], [9.0, 31.0], {"subtype": "segment"}], [[19.0, 15.0], [19.0, 21.0], {"subtype": "segment"}], [[9.0, 6.0], [19.0, 6.0], {"subtype": "segment"}], [[9.0, 0.0], [9.0, 6.0], {"subtype": "segment"}], [[9.0, 21.0], [9.0, 31.0], {"subtype": "segment"}], [[9.0, 21.0], [19.0, 21.0], {"subtype": "segment"}], [[19.0, 6.0], [19.0, 15.0], {"subtype": "segment"}], [[0.0, 15.0], [9.0, 15.0], {"subtype": "segment"}], [[0.0, 21.0], [9.0, 21.0], {"subtype": "segment"}], [[9.0, 0.0], [19.0, 0.0], {"subtype": "segment"}], [[9.0, 6.0], [9.0, 15.0], {"subtype": "segment"}]], "label": [[[0.0, 18.0], "6 m", "left", {}], [[14.0, 21.0], "10 m", "above", {}], [[9.0, 3.0], "6 m", "left", {}], [[0.0, 26.0], "10 m", "left", {}], [[9.0, 10.5], "9 m", "left", {}], [[4.5, 31.0], "9 m", "above", {}]]}
D.

{"init": {"range": [[-1.25, 26.0], [-1.0, 33.25]], "scale": [20, 20]}, "line": [[[0.0, 16.0], [10.0, 16.0], {"subtype": "segment"}], [[10.0, 26.0], [10.0, 32.0], {"subtype": "segment"}], [[10.0, 26.0], [19.0, 26.0], {"subtype": "segment"}], [[19.0, 26.0], [25.0, 26.0], {"subtype": "segment"}], [[10.0, 0.0], [19.0, 0.0], {"subtype": "segment"}], [[10.0, 16.0], [19.0, 16.0], {"subtype": "segment"}], [[19.0, 16.0], [19.0, 26.0], {"subtype": "segment"}], [[10.0, 32.0], [19.0, 32.0], {"subtype": "segment"}], [[10.0, 0.0], [10.0, 10.0], {"subtype": "segment"}], [[10.0, 16.0], [10.0, 26.0], {"subtype": "segment"}], [[0.0, 10.0], [10.0, 10.0], {"subtype": "segment"}], [[10.0, 10.0], [10.0, 16.0], {"subtype": "segment"}], [[25.0, 16.0], [25.0, 26.0], {"subtype": "segment"}], [[19.0, 16.0], [25.0, 16.0], {"subtype": "segment"}], [[10.0, 10.0], [19.0, 10.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 10.0], {"subtype": "segment"}], [[19.0, 10.0], [19.0, 16.0], {"subtype": "segment"}], [[19.0, 26.0], [19.0, 32.0], {"subtype": "segment"}], [[0.0, 10.0], [0.0, 16.0], {"subtype": "segment"}]], "label": [[[10.0, 21.0], "10 m", "left", {}], [[14.5, 32.0], "9 m", "above", {}], [[10.0, 29.0], "6 m", "left", {}], [[22.0, 26.0], "6 m", "above", {}], [[5.0, 16.0], "10 m", "above", {}], [[0.0, 13.0], "6 m", "left", {}], [[10.0, 5.0], "10 m", "left", {}]]}

Exercice 4 : Reconnaître le patron d'un polyèdre (cube, pavé droit, prisme droit)

Voici le patron d'un solide tel que \( YK = 2 cm \).

{"init": {"range": [[-3.0, 2.0], [-4.0, 2.0]], "scale": [100, 100]}, "line": [[[0.04, -2.54], [-0.04, -2.46]], [[0.04, -2.5], [-0.04, -2.42]], [[0.04, -2.58], [-0.04, -2.5]], [[0.0, -3.0], [0.0, -2.0], {"subtype": "segment"}], [[0.04, -1.54], [-0.04, -1.46]], [[0.04, -1.5], [-0.04, -1.42]], [[0.04, -1.58], [-0.04, -1.5]], [[0.0, -2.0], [0.0, -1.0], {"subtype": "segment"}], [[0.04, -0.54], [-0.04, -0.46]], [[0.04, -0.5], [-0.04, -0.42000000000000004]], [[0.04, -0.5800000000000001], [-0.04, -0.5]], [[0.0, -1.0], [0.0, 0.0], {"subtype": "segment"}], [[0.54, 0.04], [0.46, -0.04]], [[0.5, 0.04], [0.42000000000000004, -0.04]], [[0.5800000000000001, 0.04], [0.5, -0.04]], [[0.0, 0.0], [1.0, 0.0], {"subtype": "segment"}], [[1.04, 0.46], [0.96, 0.54]], [[1.04, 0.5], [0.96, 0.5800000000000001]], [[1.04, 0.42000000000000004], [0.96, 0.5]], [[1.0, 1.0], [1.0, 0.0], {"subtype": "segment"}], [[0.54, 1.04], [0.46, 0.96]], [[0.5, 1.04], [0.42000000000000004, 0.96]], [[0.5800000000000001, 1.04], [0.5, 0.96]], [[0.0, 1.0], [1.0, 1.0], {"subtype": "segment"}], [[-0.46, 1.04], [-0.54, 0.96]], [[-0.5, 1.04], [-0.5800000000000001, 0.96]], [[-0.42000000000000004, 1.04], [-0.5, 0.96]], [[0.0, 1.0], [-1.0, 1.0], {"subtype": "segment"}], [[-1.46, 1.04], [-1.54, 0.96]], [[-1.5, 1.04], [-1.58, 0.96]], [[-1.42, 1.04], [-1.5, 0.96]], [[-1.0, 1.0], [-2.0, 1.0], {"subtype": "segment"}], [[-1.96, 0.46], [-2.04, 0.54]], [[-1.96, 0.5], [-2.04, 0.5800000000000001]], [[-1.96, 0.42000000000000004], [-2.04, 0.5]], [[-2.0, 1.0], [-2.0, 0.0], {"subtype": "segment"}], [[-1.46, 0.04], [-1.54, -0.04]], [[-1.5, 0.04], [-1.58, -0.04]], [[-1.42, 0.04], [-1.5, -0.04]], [[-2.0, 0.0], [-1.0, 0.0], {"subtype": "segment"}], [[-0.96, -0.54], [-1.04, -0.46]], [[-0.96, -0.5], [-1.04, -0.42000000000000004]], [[-0.96, -0.5800000000000001], [-1.04, -0.5]], [[-1.0, 0.0], [-1.0, -1.0], {"subtype": "segment"}], [[-0.96, -1.54], [-1.04, -1.46]], [[-0.96, -1.5], [-1.04, -1.42]], [[-0.96, -1.58], [-1.04, -1.5]], [[-1.0, -1.0], [-1.0, -2.0], {"subtype": "segment"}], [[-0.96, -2.54], [-1.04, -2.46]], [[-0.96, -2.5], [-1.04, -2.42]], [[-0.96, -2.58], [-1.04, -2.5]], [[-1.0, -2.0], [-1.0, -3.0], {"subtype": "segment"}], [[-0.46, -2.96], [-0.54, -3.04]], [[-0.5, -2.96], [-0.5800000000000001, -3.04]], [[-0.42000000000000004, -2.96], [-0.5, -3.04]], [[-1.0, -3.0], [0.0, -3.0], {"subtype": "segment"}], [[-0.46, -1.96], [-0.54, -2.04]], [[-0.5, -1.96], [-0.5800000000000001, -2.04]], [[-0.42000000000000004, -1.96], [-0.5, -2.04]], [[-1.0, -2.0], [0.0, -2.0], {"subtype": "segment"}], [[-0.46, -0.96], [-0.54, -1.04]], [[-0.5, -0.96], [-0.5800000000000001, -1.04]], [[-0.42000000000000004, -0.96], [-0.5, -1.04]], [[-1.0, -1.0], [0.0, -1.0], {"subtype": "segment"}], [[-0.46, 0.04], [-0.54, -0.04]], [[-0.5, 0.04], [-0.5800000000000001, -0.04]], [[-0.42000000000000004, 0.04], [-0.5, -0.04]], [[-1.0, 0.0], [0.0, 0.0], {"subtype": "segment"}], [[-0.96, 0.46], [-1.04, 0.54]], [[-0.96, 0.5], [-1.04, 0.5800000000000001]], [[-0.96, 0.42000000000000004], [-1.04, 0.5]], [[-1.0, 0.0], [-1.0, 1.0], {"subtype": "segment"}], [[0.04, 0.46], [-0.04, 0.54]], [[0.04, 0.5], [-0.04, 0.5800000000000001]], [[0.04, 0.42000000000000004], [-0.04, 0.5]], [[0.0, 0.0], [0.0, 1.0], {"subtype": "segment"}]], "label": [[[0.0, -3.0], "Y", "right", {}], [[0.0, -2.0], "K", "right", {}], [[0.0, -1.0], "Z", "right", {}], [[0.0, 0.0], "E", "below right", {}], [[1.0, 0.0], "M", "below", {}], [[1.0, 1.0], "F", "above", {}], [[0.0, 1.0], "Q", "above", {}], [[-1.0, 1.0], "I", "above", {}], [[-2.0, 1.0], "X", "above", {}], [[-2.0, 0.0], "W", "below", {}], [[-1.0, 0.0], "O", "below left", {}], [[-1.0, -1.0], "H", "left", {}], [[-1.0, -2.0], "A", "left", {}], [[-1.0, -3.0], "U", "left", {}]]}

1. Construire ce solide
2. De quel type de solide s'agit-il ?

Exercice 5 : Connaître les propriétés du prisme droit

En utilisant le prisme droit ci-dessous, répondre aux questions posées.

Sélectionner l'arête perpendiculaire à l'arête \([DP]\).

Sélectionner l'arête parallèle à l'arête \([ZP]\).

Sélectionner l'arête perpendiculaire à l'arête \([UK]\).

Sélectionner l'arête ayant la même longueur que l'arête \([KD]\).

Revenir au choix du niveau