ML Aggarwal Class 7 Maths Solutions Chapter 15 Visualising Solid Shapes

Exercise 15.1

Question 1. Match the following:
(i) \( \rightarrow \) (f)
(ii) \( \rightarrow \) (e)
(iii) \( \rightarrow \) (b)
(iv) \( \rightarrow \) (a)
(v) \( \rightarrow \) (c)
(vi) \( \rightarrow \) (d)
Answer: The matchings show the relationship between three-dimensional objects and their two-dimensional flat versions. When you look at a cube from different angles or unfold it, you get different shapes - rectangles for sides, a cross pattern when completely unfolded, or squares for certain views.
In simple words: Solid shapes match to flat shapes. When you unfold a 3D object or see it from the side, it looks like a different 2D shape.

Exam Tip: Remember that each view of a solid shape produces a specific flat shape - practice unfolding cubes and recognizing which flat pattern matches which 3D object.

Question 2. Which of the following form a cube?
(i) Does not form cube
(ii) Forms cube
(iii) Forms cube
(iv) Forms cube
(v) Does not form cube
(vi) Forms cube
Answer: To determine whether a flat pattern creates a cube, you must check if all six faces will connect properly without overlapping. Some patterns work - like (ii), (iii), (iv), and (vi) - because when you fold them, every edge lines up correctly and no faces cover each other. Other patterns, like (i) and (v), fail because certain faces would overlap or leave gaps when folded.
In simple words: A cube net works only if you can fold it so all pieces fit together without gaps or overlaps. Some patterns do this, and some do not.

Exam Tip: Test each net by mentally folding it - if you picture any two faces trying to occupy the same space, that net cannot form a cube.

Question 3. Draw the net for each cube shown:
(i) [Net pattern with squares labeled 1-6]
(ii) [Net pattern with squares labeled 1-6]
Answer: For the first cube, the net displays the six faces arranged in a cross formation with one square on top, four squares in a horizontal line, and one square extending downward. For the second cube, the net shows three squares in the bottom row and three squares in the top row, arranged in an offset pattern where the top set sits above and to the side of the bottom set. Each net, when folded, creates a complete cube with all faces connected at their edges.
In simple words: A cube net is the flat version of a cube. It shows all six square faces laid out flat so that when you fold them up, they create a cube.

Exam Tip: Always verify your net by checking that each pair of faces shares an edge and that no two faces would overlap when the net is folded up into a cube.

Question 4. Can the given figure be a net for a die? Explain your answer.
Answer: No, this figure cannot be a net for a die. When examining the arrangement, certain faces are positioned such that they would overlap or fail to align correctly when folded into a cube. Specifically, looking at faces labeled 1, 4, and corresponding opposite faces, their placement violates the requirement that opposite faces on a die must be separate and non-overlapping. For instance, if you attempt to fold the pattern, faces 1 and 4 would try to occupy the same space. Additionally, the sum of opposite faces on a standard die must equal 7, and this net does not support that configuration - the arrangement does not permit 1 and 6, 2 and 5, or 3 and 4 to be properly opposite to each other.
In simple words: This net does not work for a die because some faces would overlap when you fold it, and opposite faces would not be in the correct positions.

Exam Tip: For die nets, remember that opposite faces must sum to 7 - so 1 must be opposite 6, 2 opposite 5, and 3 opposite 4 - and the net must allow these faces to be truly opposite when folded.

Exercise 15.2

Question 1. (i) Height 2 cm
[Rectangular prism with dimensions 5 cm × 2 cm × 2 cm shown in 3D perspective]
Answer: This is a rectangular prism (also called a cuboid) with a length of 5 cm, a width of 2 cm, and a height of 2 cm. In this case, because the width and height are equal at 2 cm each, the shape appears as a slightly flattened box when viewed in three dimensions. The drawing shows the prism in an oblique or three-quarter view, which reveals the top face, front face, and right side face simultaneously, giving a clear sense of the object's depth and solidity.
In simple words: This box is 5 cm long, 2 cm wide, and 2 cm tall. It looks like a flat rectangular block because the width and height are the same size.

Exam Tip: When sketching a rectangular prism in three dimensions, make sure the three visible faces (top, front, side) are clearly drawn and the dimensions are labeled on each edge.

Question 1. (ii) Height 3 cm
[Rectangular prism with dimensions 5 cm × 2 cm × 3 cm shown in 3D perspective]
Answer: This is a rectangular prism with a length of 5 cm, a width of 2 cm, and a height of 3 cm. The shape is taller than the first example because the height measurement has increased to 3 cm. The three-dimensional drawing again uses an oblique view to display the top, front, and side faces, allowing you to perceive the box's length, width, and vertical extent all at once.
In simple words: This box is 5 cm long, 2 cm wide, and 3 cm tall. It is taller than the previous box because the height is larger.

Exam Tip: Notice how increasing the height dimension changes the overall appearance - taller prisms appear more elongated vertically in a three-dimensional sketch.

Question 1. (iii) Height 5 cm
[Rectangular prism with dimensions 5 cm × 2 cm × 5 cm shown in 3D perspective]
Answer: This is a rectangular prism measuring 5 cm in length, 2 cm in width, and 5 cm in height. Because the length and height are both 5 cm, the shape now appears more square-like when viewed from certain angles. The three-dimensional representation continues to show the top, front, and right side faces in an oblique projection, making the proportions and spatial arrangement evident.
In simple words: This box is 5 cm long, 2 cm wide, and 5 cm tall. The length and height are the same size, making it look taller and more square-like than the previous boxes.

Exam Tip: When two dimensions of a prism are equal, the shape takes on a distinctive appearance - pay attention to how this affects the overall visual balance of your 3D drawing.

Exercise 15.2 (continued)

Question 2. (a) A cube with an edge 4 cm long - Oblique sketch
[3D cube with all edges labeled 4 cm]
Answer: An oblique sketch of a cube shows three faces - the front, top, and right side - arranged so you can see the cube's three-dimensional form in a single image. For a cube with a 4 cm edge, all visible edges are labeled as 4 cm to indicate that every side of the cube has the same length. The sketch uses parallel lines to represent edges that would normally run into the distance, creating the illusion of depth while keeping the drawing flat on paper.
In simple words: An oblique drawing shows a cube from an angle where you see three faces at once. All edges are 4 cm because a cube has all equal sides.

Exam Tip: In an oblique sketch, the front face is drawn as a true square, while the receding edges are drawn at an angle (usually 45 degrees) and often at half their true length to create a realistic appearance.

Question 2. (ii) Isometric view
[3D cube shown in isometric perspective]
Answer: An isometric view presents the cube using three axes that are equally spaced (each separated by 120 degrees) to create a balanced, three-dimensional representation. In this type of drawing, all edges are drawn to their true length, and all angles are measured consistently. This method results in a cube that appears very regular and symmetrical, with no distortion of size as objects recede into the distance - making isometric drawings particularly useful for technical and engineering drawings where accurate proportions must be maintained.
In simple words: An isometric drawing shows all three dimensions equally. The three edges you see are drawn at equal angles and to their real lengths, making the cube look very balanced and true to size.

Exam Tip: Isometric drawings preserve all measurements accurately - they are preferred when you need precise representations of objects in technical contexts.

Question 3. (a) Cuboid with length 6 cm, breadth 4 cm and height 9 cm - Oblique sketch
[3D rectangular prism with dimensions labeled]
Answer: An oblique sketch of this cuboid displays the front face as a true rectangle, while the receding edges are drawn at an angle to suggest depth. The three visible dimensions - length 6 cm, breadth 4 cm, and height 9 cm - are labeled on the appropriate edges. This particular cuboid is relatively tall compared to its length and width, which is evident in the sketch. The oblique technique makes this tall, narrow box appear clearly three-dimensional while remaining easy to draw on a flat surface.
In simple words: This box is 6 cm long, 4 cm wide, and 9 cm tall. An oblique sketch shows it from an angle so you see three faces and can tell how tall it really is.

Exam Tip: For tall cuboids, ensure your height edge is drawn significantly longer than the length and breadth edges so the proportions are visually accurate.

Question 3. (b) Isometric sketch
[3D rectangular prism shown in isometric perspective]
Answer: The isometric representation of this cuboid uses three axes at 120-degree angles to each other, with all edges drawn to their true length. The three dimensions - 6 cm, 4 cm, and 9 cm - are preserved exactly as measured, so the drawing accurately reflects the cuboid's actual proportions. The isometric view makes the cuboid's height particularly visible, as all three dimensions are treated with equal visual weight in the drawing.
In simple words: An isometric sketch of this box shows all three measurements equally. Every edge is drawn to its real size, so you see exactly how long, wide, and tall it truly is.

Exam Tip: In isometric drawings, measure carefully along each axis - each dimension should be drawn at its full true length along its respective axis line.

Exercise 15.3

Question 1. How many faces, edges, and vertices does each solid have?
(i) 30 faces
(ii) 27 edges
(iii) 46 vertices
Answer: These numbers describe the components of various three-dimensional solids. A cube, for instance, has 6 faces (all square), 12 edges (where two faces meet), and 8 vertices (corner points). More complex polyhedra have different counts - a triangular pyramid has 4 faces, 6 edges, and 4 vertices, while a rectangular prism has 6 faces, 12 edges, and 8 vertices. The specific values given (30, 27, 46) would correspond to more intricate solid shapes formed by combining multiple simple shapes or creating shapes with many sides. To find the number of faces, edges, and vertices of any solid, you can use Euler's formula: Faces + Vertices - Edges = 2.
In simple words: Every solid shape has faces (the flat sides), edges (where two sides meet), and vertices (the corners). Different shapes have different numbers of each.

Exam Tip: Always use Euler's formula to check your count: add faces and vertices, then subtract edges - the answer should always equal 2 for any closed solid shape.

Question 2. Match each object to its vertical and horizontal cross-section:
(a) A brick \( \rightarrow \) (i) Vertical cut: Rectangle, (ii) Horizontal cut: Rectangle
(b) A round apple \( \rightarrow \) (i) Vertical cut: Circle, (ii) Horizontal cut: Circle
(c) A die \( \rightarrow \) (i) Vertical cut: Square, (ii) Horizontal cut: Square
(d) A circular pipe \( \rightarrow \) (i) Vertical cut: Circle, (ii) Horizontal cut: Rectangle
(e) An ice cream cone \( \rightarrow \) (i) Vertical cut: Triangle, (ii) Horizontal cut: Circle
(f) A square pyramid \( \rightarrow \) (i) Vertical cut: Triangle, (ii) Horizontal cut: Square
Answer: When you slice through a solid object in different directions, the shape you see depends on both the object's form and the direction of the cut. A vertical cut passes through the object from top to bottom, while a horizontal cut goes from side to side at a constant height. A brick cut vertically produces a rectangle, and cutting it horizontally also produces a rectangle. A sphere, like an apple, always yields a circle no matter which direction you cut it. A cube or die gives a square in both directions. A circular pipe shows a circle when cut from top to bottom but a rectangle when sliced horizontally. An ice cream cone displays a triangle when cut vertically (showing the cone's pointed profile) but a circle when sliced horizontally (showing the circular opening). A square pyramid yields a triangle when cut vertically and a square when cut horizontally through its middle.
In simple words: When you cut through a shape, you get different flat shapes depending on how you cut it. Vertical cuts and horizontal cuts usually give different results because you are cutting at different angles.

Exam Tip: Visualize each object and mentally slice it - draw the cross-section shape for both directions to master this skill.

Question 3. Match the views of the given solids:
(a) Column 1 (shape): (1) Side View, (2) Top View, (3) Front View
Column 2 (shape): (i) Side View
Column 3 (shape): (ii) Top View
Column 4 (shape): (iii) Front View
(b) The views show different perspectives of simple three-dimensional objects like cubes, cylinders, and prisms.
Answer: Each solid object appears different depending on which direction you observe it from. The side view shows the profile of the object as seen from the left or right. The top view displays how the object looks when you look straight down from above. The front view shows the object's appearance when viewed directly from the front. For a cube, all three views show a square. For a cylinder, the front and side views display a rectangle, while the top view shows a circle. For a rectangular prism that is not a cube, the three views may all be different rectangles depending on the prism's proportions. Matching the correct view to the correct direction requires careful spatial reasoning - you must imagine rotating the object and determining which face or outline you would see from each angle.
In simple words: Every object looks different from the front, side, and top. You have to match each view to the right direction by imagining how the shape looks from each angle.

Exam Tip: Practice drawing three views of simple objects - this trains your spatial imagination and helps you solve view-matching problems quickly.

Question 4. (i) Draw the three views of a stepped solid:
[Three separate 2D projections of a stepped solid - front view, side view, and top view shown as rectangular outlines with internal divisions]
Answer: The stepped solid, when viewed from different directions, reveals three distinct orthogonal projections. The front view shows a profile where the steps are visible as horizontal lines, creating a shape that appears to have different heights across its width. The side view similarly reveals the stepping pattern from the lateral perspective, displaying the ascending or descending arrangement of the steps. The top view presents the outline of the object as it would appear when viewed from directly above, showing the footprint and the horizontal arrangement of the steps. Each view is drawn as a two-dimensional representation using only horizontal and vertical lines, with internal divisions indicating where the steps occur. All three views together allow someone to reconstruct the three-dimensional shape mentally or in a drawing.
In simple words: When you look at a stepped shape from the front, side, and top, you see three different flat pictures. Each picture shows where the steps are from that direction.

Exam Tip: For stepped solids, draw each view carefully with all internal lines showing where steps begin and end - these internal divisions are crucial for understanding the structure.

Question 4. (ii) Draw the three views of another stepped solid:
[Three separate 2D projections of a different stepped solid configuration]
Answer: This second stepped solid, depending on its specific configuration, may have its steps arranged in a different pattern or orientation compared to the first example. The front, side, and top views will again show rectangular outlines with internal divisions marking the step positions. The exact appearance of each view depends on whether the steps ascend in a particular direction, descend, or form a more complex arrangement. For instance, if the steps form an L-shape or zigzag pattern, the three views will reflect that unique configuration. As with the first stepped solid, the three orthogonal projections together convey complete spatial information about the object's form.
In simple words: This stepped shape also has three views. Depending on how the steps are arranged, each view looks different from the first example.

Exam Tip: Compare your three views for consistency - if a step appears in the front view, it should also appear in the side or top view in a way that makes spatial sense.