Class 7 — Mathematics (NCERT)

Visualising Solid Shapes

Class 7

✓By the end of this lesson students will be able to identify and differentiate between 2D and 3D shapes.
✓By the end of this lesson students will be able to understand and apply the terms faces, edges, and vertices of solid shapes.
✓By the end of this lesson students will be able to recognise and draw nets for common solid shapes like cubes, cuboids, and cylinders.
✓By the end of this lesson students will be able to draw the front, side, and top views of simple solid objects.

Key concepts

Solid Shapes (3D Shapes)

Solid shapes, also known as three-dimensional (3D) shapes, are objects that have length, breadth (or width), and height (or depth). Unlike 2D shapes which lie flat on a plane, 3D shapes occupy space. Common examples include cubes, cuboids, cylinders, cones, spheres, pyramids, and prisms.

Faces, Edges, and Vertices

Solid shapes are made up of specific components:\n* **Faces**: These are the flat surfaces of a solid shape. For example, a cube has 6 square faces.\n* **Edges**: These are the line segments where two faces of a solid shape meet. For example, a cube has 12 edges.\n* **Vertices**: These are the corners of a solid shape, where three or more edges meet. For example, a cube has 8 vertices.

Nets of Solid Shapes

A net is a two-dimensional (2D) pattern that can be folded along its lines to form a three-dimensional (3D) solid shape. Imagine unfolding a cardboard box; the flattened shape is its net. Each solid shape can have multiple different nets that can form the same 3D object. Understanding nets helps in visualising how 3D shapes are constructed from 2D components.

Views of Solid Shapes

When we look at a 3D object from different positions, it appears differently. These different appearances are called views. The most common views we consider are:\n* **Front View**: What the object looks like when viewed directly from the front.\n* **Side View**: What the object looks like when viewed directly from one of its sides (e.g., left side or right side).\n* **Top View**: What the object looks like when viewed directly from above.

Key facts to remember

1Solid shapes (3D shapes) have length, breadth, and height, occupying space.
2Faces are the flat surfaces, edges are where faces meet, and vertices are the corners of a solid shape.
3A net is a 2D pattern that can be folded to form a 3D solid.
4Many 3D shapes can have more than one possible net.
5The three main views of a 3D object are the front view, side view, and top view.
6Each view (front, side, top) shows a 2D representation of the 3D object from a specific perspective.

Worked examples

Example 1

Identify which of the following nets can be folded to form a cube. (Assume Net A is a standard cross shape with 6 squares, and Net B is 6 squares in a straight line).

I**Step 1: Understand the properties of a cube.** A cube has 6 identical square faces.

II**Step 2: Examine Net A.** Net A consists of 6 squares arranged in a 'cross' shape. If we consider the central square as the base, the four squares attached to its sides can be folded up to form the four lateral faces. The remaining square can be folded to form the top face. All faces will meet correctly without any overlaps or gaps.

III**Step 3: Examine Net B.** Net B consists of 6 squares arranged in a single straight line. If we try to fold this net, we will find that some faces will overlap, and it will not be possible to form a closed cube without missing faces or excessive overlaps.

IV**Step 4: Conclude.** Based on the visualisation of folding, Net A can form a cube, while Net B cannot.

Answer

Net A can be folded to form a cube.

Always visualise the folding process in your mind or by sketching to check if a net forms the desired solid shape.

Example 2

Draw the front view, side view (from the right), and top view of a cuboid with length 6 cm, breadth 4 cm, and height 3 cm.

I**Step 1: Determine the Front View.** When looking at the cuboid from the front, we observe its length and its height. Therefore, the front view will be a rectangle with dimensions 6 cm (length) by 3 cm (height).

II**Step 2: Determine the Side View (from the right).** When looking at the cuboid from the right side, we observe its breadth and its height. Therefore, the side view will be a rectangle with dimensions 4 cm (breadth) by 3 cm (height).

III**Step 3: Determine the Top View.** When looking at the cuboid from directly above, we observe its length and its breadth. Therefore, the top view will be a rectangle with dimensions 6 cm (length) by 4 cm (breadth).

Answer

Front View: A rectangle of 6 cm x 3 cm.\nSide View (Right): A rectangle of 4 cm x 3 cm.\nTop View: A rectangle of 6 cm x 4 cm.

Ensure your drawings clearly indicate the dimensions for each view.

Example 3

A solid shape has a net consisting of one circle and one sector of a circle. Identify the solid shape.

I**Step 1: Recall common solid shapes and their nets.**

II**Step 2: Consider the given components.** The net has one circular part and one part that is a sector of a circle (which forms a curved surface when folded).

III**Step 3: Match with known nets.**\n* A cube's net consists of squares.\n* A cuboid's net consists of rectangles.\n* A cylinder's net consists of two circles and one rectangle.\n* A cone's net consists of a circular base and a curved lateral surface which, when flattened, forms a sector of a circle.

IV**Step 4: Conclude.** The description perfectly matches the net of a cone.

Answer

The solid shape is a cone.

Familiarise yourself with the nets of various common 3D shapes.

Common mistakes

✗Confusing the number of faces, edges, and vertices for different solid shapes.
✗Incorrectly identifying nets that cannot form a specific solid shape due to overlaps or gaps when folded.
✗Drawing views without correct proportions or dimensions, or confusing which dimensions correspond to which view.
✗Misinterpreting the direction for front, side, or top views, leading to incorrect drawings.
✗Not understanding that a single solid can have multiple possible nets, assuming only one correct net exists.

Exam tips

★Always visualise the folding process carefully when dealing with questions on nets to ensure no overlaps or gaps.
★When drawing views, imagine yourself looking at the object from the specified direction (front, side, or top) and draw only what you would see from that exact angle.
★Use a ruler and a sharp pencil for neat and accurate drawings of views and nets, as precision is important.
★Practice drawing nets and views for various common solid shapes (cubes, cuboids, cylinders, cones, pyramids) to build confidence and accuracy.

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