Lesson 16: Introduction to 3D Shapes

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1Summary 重點摘要

3D shapes have faces, edges, vertices. Cuboid has 6 faces, 12 edges, 8 vertices. Learn to identify different 3D shapes.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: CuboidA cuboid has 6 faces, 12 edges and 8 vertices. The six faces are top, bottom, front, back, left and right, with opposite faces exactly the same. Everyday examples include a cardboard box, a book and a phone box. Why count faces and edges in order? Because counting at random invites repeats and omissions; ticking off edges here and there, one soon forgets which are done. The right way is to count the faces in the order top, bottom, front, back, left, right, and the edges by the three directions of length, width and height, 4 in each, 12 in total. A common error is counting 8 edges, losing one direction's 4; a check with Euler's formula afterwards is the safest finish.
2. Block 2: CubeA cube is a special cuboid: all 6 faces are equal squares and all 12 edges are the same length. Everyday examples are a dice and a Rubik's cube. It still has 6 faces, 12 edges and 8 vertices, and the only difference from a cuboid is whether every face and edge is exactly equal. Why call the cube a special cuboid? Because it satisfies every feature of a cuboid, merely adding the demand that all faces and edges be equal, just as a square is a special rectangle. A common error is thinking the cube's face count differs from the cuboid's; the counts of faces, edges and vertices are identical, and only the equality of shapes and lengths differs.
3. Block 3: CylinderA cylinder has 2 flat faces, the two equal circles at the top and bottom, plus 1 curved surface wrapping the side. It has no vertices, and the rims of the two circles count as curves rather than edges. Everyday examples are a tin can, a paper roll and a cup. Why no vertices? Because a vertex is a sharp corner where several edges meet, while the cylinder's side meets each circular face in a smooth curve with no corner at all. To identify one, look for flat circles combined with a curved surface. A common error is calling the face count 1 or 3; it is 2 flat faces plus 1 curved surface, and the absence of vertices also sets it apart from cones.
4. Block 4: ConeA cone has 1 circular base and 1 curved surface, with a single sharp vertex at the top. The surface narrows from the base up to the vertex, forming the cone shape. Everyday examples are an ice cream cone, a heap of sand and a rocket nose. Why only one vertex? Because the curved surface narrows from the whole rim of the base to a single point and never separates again. Note that it has only one vertex and one flat face, unlike the cylinder, which has no vertex and two flat faces. A common error is mixing up the cone with the cylinder, or giving the cone two bases; remember cones have one base and one apex while cylinders have two bases and no apex.
5. Block 5: SphereA sphere has only 1 curved surface, with no flat faces, no edges and no vertices. From every direction it looks like a circle, and it can roll any way at all. Everyday examples are a ball, a basketball and a globe. Why can it roll in any direction? Because every point of its surface is the same distance from the centre, perfectly symmetric, with no flat face to block a direction. It is the only common solid made entirely of one curved surface, which is its most distinctive trait. A common error is saying the sphere has one circular face; a circle is a flat figure while the sphere's surface is curved, and the accurate statement is one curved surface.
6. Block 6: Euler's Formula (Simple)Solids hide an interesting rule: faces + vertices - edges = 2, called Euler's formula. Test it on a cuboid: 6 + 8 - 12 = 2, exactly right, and a cube fits too. Why does the rule exist? Because it reflects a deep connection among the faces, edges and vertices of a solid; the formula holds for solids without holes, and secondary school will study it further, so for now use it to verify the counting. For example, counting a solid gives 5 faces, 6 vertices and 9 edges; substituting, 5 + 6 - 9 = 2, so the count is right. A common error is reversing the order of the operations; remember to add the vertices first, subtract the edges, and check that the result is 2.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• 3D shapes have faces, edges, vertices
• Cuboid: 6 faces, 12 edges, 8 vertices
• Cube: same as cuboid, but all squares
• Cylinder: 2 circular faces + 1 curved
• Sphere: only 1 curved surface