Question 1

1. Identify the nets which can be used to make cubes (cut out copies of the nets and try it): (i) a chain of six squares with two adjacent squares at the left forming a step and four squares running to the right; (ii) six squares arranged as two offset rows of three; (iii) six squares arranged as a descending staircase of pairs; (iv) a cross net with four squares in a row and one square attached above and below the second square; (v) a T-shaped net with four squares in a row and two squares attached below the second square; (vi) a cross net with four squares in a row and one square attached above and below the third square. 2. Dice are cubes with dots on each face. Opposite faces of a die always have a total of seven dots on them. Here are two nets to make dice (cubes); the numbers inserted in each square indicate the number of dots in that box. Insert suitable numbers in the blanks, remembering that the number on the opposite faces should total to 7. In the first net, there is one blank square above the left end of a row of four squares; in that row the third square is 4 and the fourth is 5; a square marked 6 is below the fourth square. In the second net, the top row has 1 and 2; the square below 1 is 3; there are three blanks continuing down-left in a staircase. 3. Can this be a net for a die? Explain your answer. The given six-square net has 1 and 2 in the top row, 3 and 4 in the next offset row, and 5 and 6 in the bottom offset row. 4. Here is an incomplete net for making a cube. Complete it in at least two different ways. Remember that a cube has six faces. How many are there in the net here? (Give two separate diagrams. If you like, you may use a squared sheet for easy manipulation.) The incomplete net has three squares in one horizontal row. 5. Match the nets with appropriate solids: solids are (a) cube, (b) cylinder, (c) cone, (d) triangular pyramid; nets are (i) a square with four triangles attached, (ii) six squares in a cross, (iii) a rectangle with a circle at each end, (iv) a circle attached to a sector.

Accepted Answer

1. Nets in (ii), (iii), (iv), (vi) form cubes.
2. One completed first net is: 1 in the square above the left end; the horizontal row is 3, 2, 4, 5; 6 remains below 5. One completed second net is: top row 1, 2; middle row 5, 3; bottom row 4, 6.
3. No, because one pair of opposite faces will have 1 and 4 on them whose total is not 7, and another pair of opposite faces will have 3 and 6 on them whose total is also not 7.
4. Three faces are there in the net. Two completions are: keep the three-square row and add one square above the middle square, one below the middle square and one below that; or keep the three-square row, add one square to extend the row at one end, and attach one square above and one below that end square.
5. (a) (ii) (b) (iii) (c) (iv) (d) (i).

Question 2

1. Use isometric dot paper and make an isometric sketch for each one of the given shapes in Fig 13.15: (i) an oblique cuboid marked length 6, breadth 2 and height 3; (ii) a stepped solid on a square grid with base length 8 and marked edge lengths 3, 3, 2, 2 and 2; (iii) a slant-edged solid made from three joined rectangular faces on a square grid; (iv) a staircase-like solid on a square grid with repeated steps marked 2 and a final length marked 4. 2. The dimensions of a cuboid are 5 cm, 3 cm and 2 cm. Draw three different isometric sketches of this cuboid. 3. Three cubes each with 2 cm edge are placed side by side to form a cuboid. Sketch an oblique or isometric sketch of this cuboid. 4. Make an oblique sketch for each one of the given isometric shapes: (i) a vertical solid made from a tall cuboid with a lower cubical extension; (ii) a block-like solid with slanting top and side faces; (iii) and (iv) the remaining isometric solids shown in the exercise. 5. Give (i) an oblique sketch and (ii) an isometric sketch for each of the following: (a) A cuboid of dimensions 5 cm, 3 cm and 2 cm. (Is your sketch unique?) (b) A cube with an edge 4 cm long.

Accepted Answer

1. Draw each given oblique shape on isometric dot paper, keeping the same unit lengths: (i) a

6 \times 2 \times 3

cuboid; (ii) the same stepped solid with base length 8 and the marked 3-unit and 2-unit edges; (iii) the same three-faced slant-edged solid; (iv) the same staircase solid with 2-unit steps and a 4-unit lower edge.
2. Three valid sketches are obtained by taking the visible front face as

5 \text{ cm} \times 3 \text{ cm}

with depth 2 cm,

5 \text{ cm} \times 2 \text{ cm}

with depth 3 cm, and

3 \text{ cm} \times 2 \text{ cm}

with depth 5 cm.
3. The cuboid formed has dimensions

6 \text{ cm} \times 2 \text{ cm} \times 2 \text{ cm}

; draw an oblique or isometric cuboid of those dimensions.
4. Draw oblique sketches with the front faces true to shape and the receding edges slanting equally; each sketch must preserve the number of cubical/rectangular parts and their relative positions from the given isometric shape.
5. (a) Draw a cuboid of dimensions

5 \text{ cm}, 3 \text{ cm}, 2 \text{ cm}

in both oblique and isometric forms. The sketch is not unique because any of the three dimensions may be chosen as the visible length, breadth or height. (b) Draw a cube of edge

4 \text{ cm}

in both oblique and isometric forms, with all edges representing 4 cm.

Question 3

1. What cross-sections do you get when you give a (i) vertical cut (ii) horizontal cut to the following solids? (a) A brick (b) A round apple (c) A die (d) A circular pipe (e) An ice cream cone

Accepted Answer

1. (a) A brick: vertical cut - rectangle; horizontal cut - rectangle. (b) A round apple: vertical cut - circle-like section; horizontal cut - circle-like section. (c) A die: vertical cut - square or rectangle; horizontal cut - square. (d) A circular pipe: vertical lengthwise cut - rectangle; horizontal cut - circle or annulus. (e) An ice cream cone: vertical cut through its axis - triangle; horizontal cut - circle.

Question 4

1. A bulb is kept burning just right above the following solids. Name the shape of the shadows obtained in each case. Attempt to give a rough sketch of the shadow. (You may try to experiment first and then answer these questions). (i) A ball (ii) A cylindrical pipe (iii) A book. 2. Here are the shadows of some 3-D objects, when seen under the lamp of an overhead projector. Identify the solid(s) that match each shadow. (There may be multiple answers for these!) (i) A circle (ii) A square (iii) A triangle (iv) A rectangle. 3. Examine if the following are true statements: (i) The cube can cast a shadow in the shape of a rectangle. (ii) The cube can cast a shadow in the shape of a hexagon.

Accepted Answer

1. (i) A ball gives a circular shadow. (ii) A cylindrical pipe, placed as shown horizontally under the bulb, gives a rectangular shadow. (iii) A book gives a rectangular shadow.
2. (i) A circle can be the shadow of a ball, a cone or a cylinder. (ii) A square can be the shadow of a cube. (iii) A triangle can be the shadow of a cone or a triangular pyramid. (iv) A rectangle can be the shadow of a cuboid, book or cylinder.
3. (i) True. (ii) True.