Aire et volume - 6e

Volume d'un solide

Exercice 1 : Calcul d'un volume d'un parallélépipède rectangle

Soit un parallélépipède rectangle de dimensions 9 x 8 x 4. Calculer son volume.

Exercice 2 : Calcul de dimensions à partir du volume d'un parallélépipède

Soit un parallélépipède de dimensions 5 x 12 et de volume 180. Calculer la hauteur du parallélépipède.

Exercice 3 : Calculer le volume d'un solide à partir d'une unité de volume

Déterminer le volume de la figure suivante à l'aide de l'unité de volume choisie.

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240, "point_size": 0}, "size": {"width": 500, "height": 300}, "materials_options": {"surface": {"opacity": 0}, "hidden_solid_line": {"dashSize": 0}, "solid_line": {"color": 255}}}, "data": {"Cuboid_0": {"position": {"x": -20.0, "y": -10.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_9": {"position": {"x": -20.0, "y": -10.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_18": {"position": {"x": -20.0, "y": 0.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_27": {"position": {"x": -20.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_36": {"position": {"x": -10.0, "y": -10.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_45": {"position": {"x": -10.0, "y": -10.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_54": {"position": {"x": -10.0, "y": 0.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_63": {"position": {"x": -10.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_72": {"position": {"x": 0.0, "y": -10.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_81": {"position": {"x": 0.0, "y": -10.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_90": {"position": {"x": 0.0, "y": 0.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_99": {"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": 10, "ly": 10, "lz": 10}, "Cuboid_108": {"position": {"x": 10.0, "y": -10.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_117": {"position": {"x": 10.0, "y": -10.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_126": {"position": {"x": 10.0, "y": 0.0, "z": -10.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_135": {"position": {"x": 10.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_144": {"position": {"x": -20.0, "y": 10.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Cuboid_153": {"position": {"x": -10.0, "y": -10.0, "z": 30.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 10, "ly": 10, "lz": 10}, "Point3D_Unit\u00e9 de volume": {"type": "Point", "position": {"x": -10.0, "y": 5.0, "z": 35.0}}, "Label_for_Point3D_Unit\u00e9 de volume": {"type": "Label", "position": {"x": -10.0, "y": 5.0, "z": 35.0}, "text": "Unit\u00e9 de volume", "position_offset": {"x": 4.8, "y": 0.0, "z": 0.0}}}}

On pourra faire pivoter la figure en cliquant dessus.

Exercice 4 : Calcul d'une arête d'un cube à partir de son volume

Le volume d'un cube est de \(216 dam^{3}\).
Quelle est la longueur de ses arêtes ?

(On donnera le résultat en précisant l'unité)

Exercice 5 : Reconnaître le patron d'un pavé

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, 16.25]], "scale": [20, 20]}, "label": [[[0.0, 10.0], "10 mm", "left", {}], [[33.0, 5.0], "10 mm", "above", {}], [[4.5, 15.0], "9 mm", "above", {}], [[23.5, 5.0], "9 mm", "above", {}], [[14.0, 5.0], "10 mm", "above", {}], [[0.0, 2.5], "5 mm", "left", {}]], "line": [[[38.0, 0.0], [38.0, 5.0], {"subtype": "segment"}], [[0.0, 5.0], [9.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [0.0, 5.0], {"subtype": "segment"}], [[19.0, 5.0], [28.0, 5.0], {"subtype": "segment"}], [[28.0, 0.0], [28.0, 5.0], {"subtype": "segment"}], [[9.0, 5.0], [9.0, 15.0], {"subtype": "segment"}], [[0.0, 5.0], [0.0, 15.0], {"subtype": "segment"}], [[19.0, 0.0], [28.0, 0.0], {"subtype": "segment"}], [[28.0, 5.0], [38.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [9.0, 0.0], {"subtype": "segment"}], [[0.0, 15.0], [9.0, 15.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 5.0], {"subtype": "segment"}], [[9.0, 0.0], [9.0, 5.0], {"subtype": "segment"}], [[9.0, 5.0], [19.0, 5.0], {"subtype": "segment"}], [[9.0, 0.0], [19.0, 0.0], {"subtype": "segment"}], [[28.0, 0.0], [38.0, 0.0], {"subtype": "segment"}]]}
B.

{"init": {"range": [[-1.25, 25.0], [-1.0, 29.25]], "scale": [20, 20]}, "line": [[[0.0, 9.0], [0.0, 14.0], {"subtype": "segment"}], [[19.0, 14.0], [24.0, 14.0], {"subtype": "segment"}], [[0.0, 23.0], [9.0, 23.0], {"subtype": "segment"}], [[19.0, 23.0], [19.0, 28.0], {"subtype": "segment"}], [[9.0, 23.0], [19.0, 23.0], {"subtype": "segment"}], [[9.0, 14.0], [9.0, 23.0], {"subtype": "segment"}], [[0.0, 28.0], [9.0, 28.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 9.0], {"subtype": "segment"}], [[0.0, 9.0], [9.0, 9.0], {"subtype": "segment"}], [[24.0, 14.0], [24.0, 23.0], {"subtype": "segment"}], [[0.0, 14.0], [9.0, 14.0], {"subtype": "segment"}], [[0.0, 23.0], [0.0, 28.0], {"subtype": "segment"}], [[9.0, 23.0], [9.0, 28.0], {"subtype": "segment"}], [[19.0, 9.0], [19.0, 14.0], {"subtype": "segment"}], [[9.0, 9.0], [19.0, 9.0], {"subtype": "segment"}], [[9.0, 28.0], [19.0, 28.0], {"subtype": "segment"}], [[9.0, 9.0], [9.0, 14.0], {"subtype": "segment"}], [[19.0, 14.0], [19.0, 23.0], {"subtype": "segment"}], [[9.0, 14.0], [19.0, 14.0], {"subtype": "segment"}], [[9.0, 0.0], [9.0, 9.0], {"subtype": "segment"}], [[9.0, 0.0], [19.0, 0.0], {"subtype": "segment"}], [[19.0, 23.0], [24.0, 23.0], {"subtype": "segment"}]], "label": [[[9.0, 4.5], "9 mm", "left", {}], [[21.5, 23.0], "5 mm", "above", {}], [[14.0, 28.0], "10 mm", "above", {}], [[0.0, 25.5], "5 mm", "left", {}], [[9.0, 18.5], "9 mm", "left", {}], [[0.0, 11.5], "5 mm", "left", {}], [[4.5, 28.0], "9 mm", "above", {}]]}
C.

{"init": {"range": [[-1.25, 26.0], [-1.0, 39.25]], "scale": [20, 20]}, "label": [[[5.0, 19.0], "10 mm", "above", {}], [[10.0, 24.0], "10 mm", "left", {}], [[0.0, 14.5], "9 mm", "left", {}], [[12.5, 38.0], "5 mm", "above", {}], [[20.0, 19.0], "10 mm", "above", {}], [[10.0, 5.0], "10 mm", "left", {}], [[10.0, 33.5], "9 mm", "left", {}]], "line": [[[10.0, 19.0], [10.0, 29.0], {"subtype": "segment"}], [[10.0, 19.0], [15.0, 19.0], {"subtype": "segment"}], [[0.0, 19.0], [10.0, 19.0], {"subtype": "segment"}], [[10.0, 0.0], [15.0, 0.0], {"subtype": "segment"}], [[10.0, 0.0], [10.0, 10.0], {"subtype": "segment"}], [[0.0, 10.0], [0.0, 19.0], {"subtype": "segment"}], [[10.0, 29.0], [10.0, 38.0], {"subtype": "segment"}], [[10.0, 38.0], [15.0, 38.0], {"subtype": "segment"}], [[0.0, 10.0], [10.0, 10.0], {"subtype": "segment"}], [[10.0, 29.0], [15.0, 29.0], {"subtype": "segment"}], [[10.0, 10.0], [15.0, 10.0], {"subtype": "segment"}], [[15.0, 19.0], [25.0, 19.0], {"subtype": "segment"}], [[15.0, 29.0], [15.0, 38.0], {"subtype": "segment"}], [[15.0, 0.0], [15.0, 10.0], {"subtype": "segment"}], [[10.0, 10.0], [10.0, 19.0], {"subtype": "segment"}], [[15.0, 10.0], [15.0, 19.0], {"subtype": "segment"}], [[15.0, 19.0], [15.0, 29.0], {"subtype": "segment"}], [[25.0, 10.0], [25.0, 19.0], {"subtype": "segment"}], [[15.0, 10.0], [25.0, 10.0], {"subtype": "segment"}]]}
D.

{"init": {"range": [[-1.25, 29.0], [-1.0, 29.25]], "scale": [20, 20]}, "line": [[[0.0, 0.0], [0.0, 9.0], {"subtype": "segment"}], [[9.0, 23.0], [19.0, 23.0], {"subtype": "segment"}], [[19.0, 0.0], [19.0, 9.0], {"subtype": "segment"}], [[19.0, 9.0], [19.0, 14.0], {"subtype": "segment"}], [[9.0, 9.0], [19.0, 9.0], {"subtype": "segment"}], [[9.0, 9.0], [9.0, 14.0], {"subtype": "segment"}], [[9.0, 0.0], [9.0, 9.0], {"subtype": "segment"}], [[19.0, 28.0], [28.0, 28.0], {"subtype": "segment"}], [[0.0, 23.0], [9.0, 23.0], {"subtype": "segment"}], [[28.0, 23.0], [28.0, 28.0], {"subtype": "segment"}], [[19.0, 23.0], [19.0, 28.0], {"subtype": "segment"}], [[9.0, 14.0], [9.0, 23.0], {"subtype": "segment"}], [[0.0, 28.0], [9.0, 28.0], {"subtype": "segment"}], [[0.0, 9.0], [9.0, 9.0], {"subtype": "segment"}], [[0.0, 23.0], [0.0, 28.0], {"subtype": "segment"}], [[9.0, 23.0], [9.0, 28.0], {"subtype": "segment"}], [[19.0, 23.0], [28.0, 23.0], {"subtype": "segment"}], [[0.0, 0.0], [9.0, 0.0], {"subtype": "segment"}], [[9.0, 28.0], [19.0, 28.0], {"subtype": "segment"}], [[19.0, 14.0], [19.0, 23.0], {"subtype": "segment"}], [[9.0, 14.0], [19.0, 14.0], {"subtype": "segment"}], [[9.0, 0.0], [19.0, 0.0], {"subtype": "segment"}]], "label": [[[0.0, 4.5], "9 mm", "left", {}], [[14.0, 28.0], "10 mm", "above", {}], [[0.0, 25.5], "5 mm", "left", {}], [[9.0, 18.5], "9 mm", "left", {}], [[23.5, 28.0], "9 mm", "above", {}], [[9.0, 11.5], "5 mm", "left", {}], [[4.5, 28.0], "9 mm", "above", {}]]}

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