Visualising Solid Shapes

Welcome dear students! Today we are going to learn about Visualising Solid Shapes from Class 7 Maths.

In this chapter, you will classify figures you have seen in terms of what is known as dimension. In our day to day life, we see several objects like books, balls, ice cream cones and so on around us which have different shapes. One thing common about most of these objects is that they all have some length, breadth and height or depth. That is, they all occupy space and have three dimensions. Hence, they are called 3-D shapes. Do you remember some of the 3-D shapes, that is, solid shapes, we have seen in earlier classes? Let us look at a matching activity. Figure 13.1 shows six shapes. Shape (i) is a cuboid. Shape (ii) is a cylinder. Shape (iii) is a cube. Shape (iv) is a sphere. Shape (v) is a pyramid. Shape (vi) is a cone. Try to identify some objects shaped like each of these in your daily life.

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By a similar argument, we can say figures drawn on paper which have only length and breadth are called 2-D, or plane, figures. We have also seen some 2-D figures in the earlier classes. Let us match the 2-D figures with their names as shown in Figure 13.2. Figure (i) is a circle. Figure (ii) is a rectangle. Figure (iii) is a square. Figure (iv) is a quadrilateral. Figure (v) is a triangle. Note that we can write 2-D in short for 2-dimension and 3-D in short for 3-dimension.

Now let us move on to faces, edges and vertices. Do you remember the faces, vertices and edges of solid shapes, which you studied earlier? Here you see them for a cube in Figure 13.3. The 8 corners of the cube are its vertices. The 12 line segments that form the skeleton of the cube are its edges. The 6 flat square surfaces that are the skin of the cube are its faces. Can you see that the 2-D figures can be identified as the faces of the 3-D shapes? For example, a cylinder has two faces which are circles, and a pyramid has triangles as its faces. We will now try to see how some of these 3-D shapes can be visualised on a 2-D surface, that is, on paper. In order to do this, we would like to get familiar with 3-D objects closely. Let us try forming these objects by making what are called nets.

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Take a cardboard box. Cut the edges to lay the box flat. You have now a net for that box. A net is a sort of skeleton outline in 2-D, which, when folded, results in a 3-D shape. Here you got a net by suitably separating the edges. Is the reverse process possible? Here is a net pattern for a box in Figure 13.5. Copy an enlarged version of the net and try to make the box by suitably folding and gluing together. The box is a solid. It is a 3-D object with the shape of a cuboid. Similarly, you can get a net for a cone by cutting a slit along its slant surface, as shown in Figure 13.6. You have different nets for different shapes. Copy enlarged versions of the nets given in Figure 13.7 and try to make the 3-D shapes indicated. The first net forms a cube. The second net forms a cylinder. The third net forms a cone. We could also try to make a net for making a pyramid like the Great Pyramid in Giza in Egypt, shown in Figure 13.8. That pyramid has a square base and triangles on the four sides. See if you can make it with the given net in Figure 13.9.

Let us complete Table 13.1 in your textbook together. The table asks you to fill in the number of Faces, Edges, and Vertices for different solids. For a cube, the number of faces is 6, the number of edges is 12, and the number of vertices is 8. For a cone, there are 2 faces, consisting of 1 flat circular base and 1 curved surface, 1 edge where the curved surface meets the base, and 1 vertex at the top. For a cylinder, there are 3 faces, consisting of 2 flat circular bases and 1 curved surface, 2 edges where the curved surface meets each base, and 0 vertices. For a square pyramid, there are 5 faces, consisting of 1 square base and 4 triangular sides, 8 edges, and 5 vertices.

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Now it is time for Exercise 13.1. Question one asks you to identify the nets which can be used to make cubes. The nets are shown as six different arrangements of six squares, labeled (i) through (vi). To solve this, you must mentally fold each arrangement along the shared edges. A valid cube net will fold into a closed box with exactly six faces, no overlaps, and no gaps. As you test each one, remember that if two faces try to occupy the same space when folded, or if a face is missing, it is not a valid net. Cut out copies and physically fold them to verify your answers.

Question two is about dice. Dice are cubes with dots on each face. Opposite faces of a die always have a total of 7 dots on them. The exercise shows two nets with some squares containing numbers and others left blank. To solve this, first identify which squares will become opposite faces when the net is folded into a cube. In standard nets, faces separated by exactly one square in a straight line will be opposite. Once you identify the opposite pairs, fill in the blanks so that each pair adds up to 7. For example, if a face shows 1, its opposite must be 6. If it shows 3, its opposite must be 4. Apply this rule systematically to both nets to find the missing numbers.

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Question three asks: Can this be a net for a die? Explain your answer. The net shown has a row of four squares with numbers 1, 2, 3, and 4 in order, with one square above the 2 and one square below the 3. When you fold this net, the face with 1 dot and the face with 3 dots will end up opposite each other. Their sum is 1 + 3 = 4. The face with 2 dots and the face with 4 dots will also be opposite, summing to 2 + 4 = 6. Since opposite faces on a standard die must always total 7, and neither of these pairs does, this arrangement cannot be a valid net for a standard die.

TRY THESE Here you find four nets in Figure 13.10. A tetrahedron is a pyramid with a triangular base and three triangular faces, making four triangular faces in total. There are two correct nets among them to make a tetrahedron. Look at each net carefully. Visualise folding them along the shared edges. A valid tetrahedron net will close perfectly with all four triangles meeting at a single apex without overlapping. Test each arrangement mentally or by cutting them out to discover which two successfully form the shape.

Question four gives an incomplete net for making a cube. It shows exactly four squares arranged in a single straight row. Remember that a cube has six faces, so you need to add two more squares. You can complete it in at least two different ways. One method is to attach one square above the second square in the row, and one square below the third square. Another valid method is to attach one square above the first square, and one square above the fourth square. Both arrangements will fold into a cube. Draw these two separate diagrams on your squared sheet and verify by folding.

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Question five asks you to match the nets with appropriate solids. Look at each net and identify its base shape and side faces. Net (a) has a rectangle with two identical circles attached to its opposite longer sides. This matches solid (ii), the cylinder. Net (b) has six identical squares arranged in a T-shape. This matches solid (i), the cube. Net (c) has a sector of a circle attached to the curved edge of a full circle. When folded, the sector forms the slanted surface and the circle forms the base, matching solid (iii), the cone. Net (d) has a square with four identical triangles attached to each of its sides. This matches solid (iv), the square pyramid. Match them in your book accordingly.

Play this game You and your friend sit back to back. One of you reads out a net to make a 3-D shape, describing the arrangement of squares, triangles, or other polygons, while the other attempts to copy it and sketch or build the described 3-D object. This helps you visualise the folding process mentally and improves your spatial reasoning.

Moving on to section 13.4: Drawing Solids on a Flat Surface. Your drawing surface is paper, which is flat. When you draw a solid shape, the images are somewhat distorted to make them appear three dimensional. It is a visual illusion. You will find here two techniques to help you. First, 13.4.1 Oblique Sketches. Here is a picture of a cube in Figure 13.11. It gives a clear idea of how the cube looks when seen from the front. You do not see certain faces. In the drawn picture, the lengths are not equal, as they should be in a real cube. Still, you are able to recognise it as a cube. Such a sketch of a solid is called an oblique sketch.

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How can you draw such sketches? You need a squared paper. Initially practising to draw on these sheets will later make it easy to sketch them on a plain sheet. Let us attempt to draw an oblique sketch of a 3 × 3 × 3 cube, as shown in Figure 13.12. Step one: Draw the front face. It is a square of 3 units by 3 units. Step two: Draw the opposite face. Sizes of the faces have to be same, but the sketch is somewhat offset from step one. Step three: Join the corresponding corners. Step four: Redraw using dotted lines for hidden edges. It is a convention. The sketch is ready now. In the oblique sketch above, note that the sizes of the front faces and its opposite are same, and the edges, which are all equal in a cube, appear so in the sketch, though the actual measures of edges are not taken so. You could now try to make an oblique sketch of a cuboid, remembering the faces in this case are rectangles. Note that you can draw sketches in which measurements also agree with those of a given solid. To do this we need what is known as an isometric sheet.

Let us try to make a cuboid with dimensions 4 cm length, 3 cm breadth and 3 cm height on a given isometric sheet. This leads us to 13.4.2 Isometric Sketches. Have you seen an isometric dot sheet? Such a sheet divides the paper into small equilateral triangles made up of dots or lines. To draw sketches in which measurements also agree with those of the solid, we can use isometric dot sheets. Let us attempt to draw an isometric sketch of a cuboid of dimensions 4 × 3 × 3, as shown in Figure 13.13. Step one: Draw a rectangle to show the front face. Step two: Draw four parallel line segments of length 3 starting from the four corners of the rectangle. Step three: Connect the matching corners with appropriate line segments. Step four: This is an isometric sketch of the cuboid. Note that the measurements are of exact size in an isometric sketch. This is not so in the case of an oblique sketch.

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Let us look at Example one. Here is an oblique sketch of a cuboid in Figure 13.14(i). Draw an isometric sketch that matches this drawing. Solution: Observe the dimensions from the oblique sketch. Identify the length, breadth, and height. On the isometric dot paper, draw the front face as a rectangle matching the length and height. Then draw lines of the breadth length at the standard isometric angle from each corner. Finally, connect the ends to complete the back face. The measurements are taken care of exactly, as shown in Figure 13.14(ii).

Now for Exercise 13.2. Question one asks you to use isometric dot paper and make an isometric sketch for each one of the given shapes in Figure 13.15. For each shape, identify the base and the height. Draw the base on the dot grid following the triangular lines, then extend vertical lines upward from each vertex to the required height, and connect the top vertices to complete the solid. Question two: The dimensions of a cuboid are 5 cm, 3 cm and 2 cm. Draw three different isometric sketches of this cuboid by changing which dimension is vertical, which is horizontal, and which represents depth. Question three: Three cubes each with 2 cm edge are placed side by side to form a cuboid. The resulting cuboid has length 2 + 2 + 2 = 6 cm, breadth 2 cm, and height 2 cm. Sketch it using either oblique or isometric technique, ensuring the proportions match.

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Question four: Make an oblique sketch for each one of the given isometric shapes in the exercise. For each shape, identify the front face, draw it as a rectangle or square, offset the back face diagonally, join the corresponding corners, and use dotted lines for any edges that would be hidden from view. Question five: Give an oblique sketch and an isometric sketch for each of the following. Part a: A cuboid of dimensions 5 cm, 3 cm and 2 cm. For the oblique sketch, draw a 5 by 3 rectangle, offset the back face, join, and use dotted lines. For the isometric sketch, use the dot paper to draw exact 5, 3, and 2 unit lengths. Part b: A cube with an edge 4 cm long. Oblique sketch: draw a 4 by 4 square, offset back face, join. Isometric sketch: draw exact 4 unit edges on dot paper.

Let us move to 13.4.3 Visualising Solid Objects. Sometimes when you look at combined shapes, some of them may be hidden from your view. Take some cubes and arrange them as shown in Figure 13.16. The figure shows three cubes stacked in a corner, with two on the bottom and one on top. Now ask your friend to guess how many cubes there are when observed from the view shown by the arrow mark. Try to guess the number of cubes in the following arrangements in Figure 13.17. For each arrangement, count the visible cubes first, then carefully add the hidden cubes that must be present underneath to support the upper layers. Such visualisation is very helpful. Suppose you form a cuboid by joining such cubes. You will be able to guess what the length, breadth and height of the cuboid would be.

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Let us look at Example two. If two cubes of dimensions 2 cm by 2 cm by 2 cm are placed side by side, what would the dimensions of the resulting cuboid be? Solution: As you can see in Figure 13.18 when kept side by side, the length is the only measurement which increases. It becomes 2 + 2 = 4 cm. The breadth = 2 cm and the height = 2 cm.

TRY THESE Question one: Two dice are placed side by side as shown in Figure 13.19. Look at the top faces of each die. Remember that in a standard die, the sum of numbers on opposite faces is 7. To find the total on the faces opposite to the visible top faces, subtract each top face number from 7, then add those two results together. Part b: Apply the same rule for the second pair of visible faces. Find their opposites by subtracting from 7, and calculate the sum. Question two: Three cubes each with 2 cm edge are placed side by side to form a cuboid. Try to make an oblique sketch and say what could be its length, breadth and height. The length is 2 + 2 + 2 = 6 cm. The breadth is 2 cm. The height is 2 cm. To sketch it obliquely, draw a front rectangle of 6 by 2 units, offset the back face, join corners, and use dotted lines for hidden edges.

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Now let us see how an object which is in 3-D can be viewed in different ways. Section 13.5.1: One Way to View an Object is by Cutting or Slicing. Slicing game. Here is a loaf of bread in Figure 13.20. It is like a cuboid with a square face. You slice it with a knife. When you give a vertical cut, you get several pieces. Each face of the piece is a square. We call this face a cross section of the whole bread. The cross section is nearly a square in this case. Beware. If your cut is not vertical you may get a different cross section. Think about it. The boundary of the cross section you obtain is a plane curve. Do you notice it? A kitchen play. Have you noticed cross sections of some vegetables when they are cut for the purposes of cooking in the kitchen? Observe the various slices and get aware of the shapes that result as cross sections.

Play this Make clay or plasticine models of the following solids in Figure 13.21 and make vertical or horizontal cuts. Draw rough sketches of the cross sections you obtain. Name them wherever you can. The solids shown are a cone, a cylinder, a cube, and a sphere. For each solid, predict the shape before cutting, then verify by slicing and observing the exposed face.

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Exercise 13.3. Question one asks what cross sections do you get when you give a vertical cut and a horizontal cut to the following solids. To solve this, imagine the cutting plane passing through the centre of each solid. For a brick, a vertical cut through the length and height gives a rectangle, and a horizontal cut through the length and breadth also gives a rectangle. For a round apple, both vertical and horizontal cuts through the centre give circles. For a die, both cuts give squares. For a circular pipe, a vertical cut gives a rectangle showing the hollow interior, and a horizontal cut gives a ring or annulus. For an ice cream cone, a vertical cut gives a triangle, and a horizontal cut gives a circle.

Section 13.5.2: Another Way is by Shadow Play. Shadow play. Shadows are a good way to illustrate how 3-D objects can be viewed in 2-D. Have you seen a shadow play? It is a form of entertainment using solid articulated figures in front of an illuminated back drop to create the illusion of moving images. It makes some indirect use of ideas in Mathematics. You will need a source of light and a few solid shapes for this activity. Keep a torchlight, right in front of a cone, as shown in Figure 13.23. What type of shadow does it cast on the screen? It casts a triangle. The solid is 3-D. What is the dimension of the shadow? The shadow is 2-D. If, instead of a cone, you place a cube in the above game, what type of shadow will you get? It will cast a square or rectangle depending on the angle. Experiment with different positions of the source of light and with different positions of the solid object. Study their effects on the shapes and sizes of the shadows you get.

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Here is another funny experiment that you might have tried already. Place a circular plate in the open when the Sun at the noon time is just right above it, as shown in Figure 13.24(i). What is the shadow that you obtain? You obtain a circular shadow. Will it be same during forenoons? During evenings? During forenoons and evenings, the sun is at an angle, so the shadow becomes an ellipse or oval shape, as shown in Figures 13.24(ii) and 13.24(iii). Study the shadows in relation to the position of the Sun and the time of observation.

Exercise 13.4. Question one: A bulb is kept burning just right above the following solids. Name the shape of the shadows obtained in each case. Part a: A ball casts a circular shadow. Part b: A cylindrical pipe casts a rectangular shadow when viewed from the side. Part c: A book casts a rectangular shadow. Question two: Here are the shadows of some 3-D objects, when seen under the lamp of an overhead projector. Identify the solid that match each shadow. Part a: A circle shadow could be cast by a sphere, a cylinder viewed from the top, or a cone viewed from the top. Part b: A square shadow could be cast by a cube or a square pyramid viewed from directly above. Part c: A triangle shadow could be cast by a cone or a triangular pyramid. Part d: A rectangle shadow could be cast by a cuboid, a cylinder viewed from the side, or a triangular prism. Question three: Examine if the following are true statements. Part a: The cube can cast a shadow in the shape of a rectangle. This is true. If the cube is tilted relative to the light source, its projection becomes a rectangle. Part b: The cube can cast a shadow in the shape of a hexagon. This is false. Under standard parallel or point lighting, a cube cannot project a regular hexagonal shadow.

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Section 13.5.3: A Third Way is by Looking at it from Certain Angles to Get Different Views. One can look at an object standing in front of it or by the side of it or from above. Each time one will get a different view, as illustrated in Figure 13.25. Here is an example of how one gets different views of a given building in Figure 13.26. The front view shows the main facade with windows and door. The side view shows the depth and side walls. The top view shows the roof layout. You could do this for figures made by joining cubes, as shown in Figure 13.27. The front view shows the height and width of the stacked cubes. The side view shows the depth and height. The top view shows the width and depth layout. Try putting cubes together and then making such sketches from different sides.

Play this game for different views Arrange a few cubes to form a structure. Ask your friend to look at it from the front, side, and top, and sketch each view. Then switch roles to verify your understanding of different perspectives. This activity strengthens your ability to translate 3-D objects into 2-D representations.

TRY THESE Question one: For each solid, the three views are given. Identify for each solid the corresponding top, front and side views. To do this, imagine looking directly down for the top view, straight ahead for the front view, and from the right or left for the side view. Match the resulting 2-D shapes to the given diagrams. Question two: Draw a view of each solid as seen from the direction indicated by the arrow. Simply project the visible faces onto a flat plane in the direction of the arrow, ignoring any hidden faces, and draw the outline of what you see.

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What have we discussed? Let us review the key points. One: The circle, the square, the rectangle, the quadrilateral and the triangle are examples of plane figures. The cube, the cuboid, the sphere, the cylinder, the cone and the pyramid are examples of solid shapes. Two: Plane figures are of 2-D and the solid shapes are of 3-D. Three: The corners of a solid shape are called its vertices. The line segments of its skeleton are its edges. And its flat surfaces are its faces. Four: A net is a skeleton outline of a solid that can be folded to make it. The same solid can have several types of nets. Five: Solid shapes can be drawn on a flat surface realistically. We call this 2-D representation of a 3-D solid. Six: Two types of sketches of a solid are possible. An oblique sketch does not have proportional lengths. Still it conveys all important aspects of the appearance of the solid. An isometric sketch is drawn on an isometric dot paper. In an isometric sketch of the solid the measurements kept proportional. Seven: Visualising solid shapes is a very useful skill. You should be able to see hidden parts of the solid shape. Eight: Different sections of a solid can be viewed in many ways. One way is to view by cutting or slicing the shape, which would result in the cross section of the solid. Another way is by observing a 2-D shadow of a 3-D shape. A third way is to look at the shape from different angles. The front view, the side view and the top view can provide a lot of information about the shape observed.

Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]