4 More about 3-D Figures Case Study 4.1 Symmetry of 3-D Figures 4.2 Nets of 3-D Figures 4.3 2-D Representations of 3-D Figures 4.4 Points, Lines and Planes of 3-D Figures 4.5 Further Exploration on 3-D Figures Chapter Summary

Case Study Can you help them to make a box using cardboard? Here is some cardboard. Do you know how to make a box using cardboard? We haven’t got enough boxes for packing the goods. Can you help them to make a box using cardboard?

Case Study As shown in the figure below, imagine that we cut a box along the orange edges and unfold the box. If we copy the unfolded plane figure onto a piece of cardboard and reverse the above procedure, we can fold the piece of cardboard into a box.

4.1 Symmetry of 3-D Figures A. Reflectional Symmetry (a) Introduction A 3-D object is said to have reflectional symmetry if a plane can divide the object into 2 identical parts which are mirror images of each other. Such a plane is called the plane of reflection. Planes of reflection

4.1 Symmetry of 3-D Figures A. Reflectional Symmetry Some objects have more than one plane of reflection. The figure below shows 3 of the 4 planes of reflection of a regular triangular prism.

4.1 Symmetry of 3-D Figures A. Reflectional Symmetry (b) Cube A cube is a regular solid with 6 identical squares as faces. There are 9 different planes of reflection in a cube. The planes of reflection cut through 4 faces. The planes of reflection cut through 2 edges and 2 faces.

4.1 Symmetry of 3-D Figures A. Reflectional Symmetry (c) Regular Tetrahedron A regular tetrahedron is a solid with 4 identical equilateral triangles as faces. There are 6 different planes of reflection in a regular tetrahedron. The planes of reflection are formed by an edge and the mid-point of its opposite edge.

4.1 Symmetry of 3-D Figures B. Rotational Symmetry Introduction A 3-D figure is said to have rotational symmetry if it repeats itself more than once when it is rotated about a line in one complete revolution. Such a line is called the axis of rotational symmetry. The above cuboid has 2-fold rotational symmetry or rotational symmetry of order 2 about the line PQ.

4.1 Symmetry of 3-D Figures B. Rotational Symmetry Some objects have more than one axis rotational symmetry. 2-fold rotational symmetry 5-fold rotational symmetry

4.1 Symmetry of 3-D Figures B. Rotational Symmetry (b) Cube There are 13 different axes of rotational symmetry in a cube. 4-fold rotational symmetry 3-fold rotational symmetry 2-fold rotational symmetry

4.1 Symmetry of 3-D Figures B. Rotational Symmetry (c) Regular Tetrahedron There are 7 different axes of rotational symmetry in a regular tetrahedron. 2-fold rotational symmetry 3-fold rotational symmetry

4.2 Nets of 3-D Figures A net is a 2-D pattern which can be folded into a solid. 1. There are many different nets for a cube. 2. Each edge will coincide with exactly one other edge of the net to form a solid. 3. There are totally 11 different nets for a cube.

Example 4.1T 4.2 Nets of 3-D Figures Solution: The figure shows an open box. Draw a net of the box. Solution: If we cut along the sides and unfold the box, we can obtain its net. There are 4 faces in the open box.

Example 4.2T 4.2 Nets of 3-D Figures Solution: The figure shows a net of a cube. When it is folded into a cube, what is the face opposite to face ‘A’? Solution: Top Front Side The face opposite to face ‘A’ is face ‘E’.

4.3 2-D Representations of 3-D Figures A. 3-D Objects in 2-D Views If we look at an object from different directions, we can obtain different views. For example, the following shows different views of an object below. They are the front, top and side views.

Example 4.3T 4.3 2-D Representations of 3-D Figures Solution: A. 3-D Objects in 2-D Views Example 4.3T Draw the front view, top view and side view of the solid shown. Solution:

B. Sketch 3-D Objects from 2-D Views 4.3 2-D Representations of 3-D Figures B. Sketch 3-D Objects from 2-D Views We can sketch 3-D objects from their 2-D views. You may draw the solid on a piece of isometric-grid paper or oblique-grid paper. When sketching the solid, we may 1. first sketch the solid from the front view, 2. next sketch the solid according to the top view and the side view, 3. then check the ratio of the lengths of the sides of the solid. 4. Lastly, find the 3 views from the sketch to check whether they match the given 2-D information.

Example 4.4T 4.3 2-D Representations of 3-D Figures Solution: B. Sketch 3-D Objects from 2-D Views Example 4.4T The following shows the front view, top view and side view of a solid. Sketch the solid. Solution:

C. Limitation of Plane Figures 4.3 2-D Representations of 3-D Figures C. Limitation of Plane Figures There is a limitation in using 2-D representation of a solid to figure out the actual solid. For example, for the front view and side view as shown, we can obtain many possible solids, such as: Therefore, the more the views of an object that are given, the better the shape of the solid we can obtain.

4.4 Points, Lines and Planes of 3-D Figures In 2-dimensional space, we learnt that 1. 2 non-parallel lines L1 and L2 would intersect at a point O with an angle q as shown in the figure, 2. the perpendicular distance between a point P and a line L3 is their shortest distance, denoted by PQ as shown in the figure.

A. Angle between a Line and a Plane 4.4 Points, Lines and Planes of 3-D Figures A. Angle between a Line and a Plane (a) Projection of a Point on a Plane In the figure, O is a point on the plane p and V is a point not on the plane. If VO is perpendicular to any lines on the plane (say L1 and L2), then O is called the projection of point V on the plane. VO is the shortest distance between point V and the plane. In the top view of the figure, point V coincides with its projection O.

A. Angle between a Line and a Plane 4.4 Points, Lines and Planes of 3-D Figures A. Angle between a Line and a Plane (b) Angle between a Line and a Plane In the figure, A is a point on the plane p and the line AB does not lie on the plane. If C is the projection of B on the plane, then AC is the projection of AB on the plane. q is the angle between AB and the plane. 1. In the top view of the figure, line AB coincides with its projection AC. 2. If a line not on the plane is parallel to the plane, then we cannot find the point of intersection.

Example 4.5T 4.4 Points, Lines and Planes of 3-D Figures Solution: A. Angle between a Line and a Plane Example 4.5T The figure shows a cube ABCDHEFG. Write down the angles between line DF and the following planes. (a) Plane CDHG (b) Plane BCGF Solution: (a) ∵ G is the projection of F on plane CDHG. ∴ The angle between line DF and plane CDHG is FDG. (b) ∵ C is the projection of D on plane BCGF. ∴ The angle between line DF and plane BCGF is DFC.

B. Angle between Two Planes 4.4 Points, Lines and Planes of 3-D Figures B. Angle between Two Planes In the figure, 2 non-parallel planes p1 and p2 intersect and they meet at a line AB which is called the line of intersection. To find the angle between 2 planes p1and p2, 1. first construct a perpendicular line PQ of AB on plane p1, where Q is a point on AB, 2. then through Q, construct a perpendicular line QR of AB on plane p2. 3. As a result, the required angle is PQR. 2 planes are parallel if they do not intersect even after they are extended.

Example 4.6T 4.4 Points, Lines and Planes of 3-D Figures Solution: B. Angle between Two Planes Example 4.6T The figure shows a cuboid ABCDHEFG. Write down the angles between the following planes. (a) Planes ADHE and EFGH (b) Planes ADGF and BCGF Solution: (a) ∵ EH is the line of intersection of the 2 planes. ∴ The angle between planes ADHE and EFGH is DHG or AEF. (b) ∵ FG is the line of intersection of the 2 planes. ∴ The angle between planes ADGF and BCGF is DGC or AFB.

Example 4.7T 4.4 Points, Lines and Planes of 3-D Figures Solution: B. Angle between Two Planes Example 4.7T The figure shows a regular tetrahedron ABCD. P, Q, R, S, T and U are the mid-points of the edges. Write down the angles between (a) planes PCD and BCD, (b) planes ABD and ACD. Solution: (a) ∵ CD is the line of intersection of the 2 planes, and PT and BT are perpendicular to CD. ∴ The angle between planes PCD and BCD is PTB. (b) ∵ AD is the line of intersection of the 2 planes, and CR and BR are perpendicular to AD. ∴ The angle between planes ABD and ACD is CRB.

4.5 Further Exploration on 3-D Figures A. Euler’s Formula The relationship between the numbers of vertices, edges and faces of a polyhedron is as follows. V  E  F  2 This is called Euler’s formula, where V is the number of vertices, E is the number of edges and F is the number of faces.

Example 4.8T 4.5 Further Exploration on 3-D Figures Solution: A. Euler’s Formula Example 4.8T The figure shows the net of a polyhedron. (a) What polyhedron can the net be folded into? (b) Find the number of vertices (V), the number of edges (E) and the number of faces (F) of the polyhedron. (c) Does the Euler’s formula hold? Solution: (a) Triangular prism (b) V  6, E  9, F  5 (c) V  E  F  6  9  5  2 ∴ Euler’s formula holds.

B. Duality of Regular Polyhedra 4.5 Further Exploration on 3-D Figures B. Duality of Regular Polyhedra In Book 1B Chapter 8, we learnt that there are 5 regular polyhedra: Figure Name Regular Tetrahedron Regular Hexahedron Regular Octahedron Regular Dodecahedron Regular Icosahedron When the number of vertices and the number of faces of 2 polyhedra are reversed, the 2 polyhedra are called dual polyhedra. For example: Regular Hexahedron: V  8, E  12, F  6 Regular Octahedron: V  6, E  12, F  8

B. Duality of Regular Polyhedra 4.5 Further Exploration on 3-D Figures B. Duality of Regular Polyhedra For a dual pair, each vertex of a polyhedron touches the mid-point of one of the faces of the other polyhedron as shown in the following figure. Apart from regular hexahedron and regular octahedron, can you find another dual pair?

Chapter Summary 4.1 Symmetry of 3-D Figures 1. Reflectional Symmetry A 3-D figure is said to have reflectional symmetry if a plane, that is the plane of reflection, can divide the figure into 2 identical parts which are mirror images of each other. The figure shows a plane of reflection of a binder clip. 2. Rotational Symmetry A 3-D figure is said to have rotational symmetry if it repeats itself more than once when it is rotated about a line, that is the axis of rotational symmetry, in one complete revolution. The figure shows an axis of rotational symmetry of an hourglass.

Chapter Summary 4.2 Nets of 3-D Figures 1. A net is a 2-D pattern which can be folded into a solid. 2. There may be different nets for the same 3-D figure.

Chapter Summary 4.3 2-D Representations of 3-D Figures 1. If we look at an object from different directions, we can obtain different views. 2. We can identify 3-D objects from their 2-D views.

Chapter Summary 4.4 Points, Lines and Planes of 3-D Figures 1. If VO is perpendicular to any lines (say L1 and L2) on a plane, then O is called the projection of V on the plane. 2. If C is the projection of B on a plane, then AC is the projection of AB on the plane, and q is the angle between AB and the plane. 3. PQR is the angle between 2 planes p1 and p2, where AB is the line of intersection of the planes, PQ  AB and QR  AB.

Chapter Summary 4.5 Further Exploration on 3-D Figures 1. Euler’s Formula For a polyhedron, V  E  F  2, where V is the number of vertices, E is the number of edges and F is the number of faces. 2. Duality of Regular Polyhedra For dual polyhedra, each vertex of one of the polyhedron touches the mid-point of one of the faces of the other polyhedron.