Class 7 Mathematics Notes Chapter 15 (Visualising Solid Shapes) – Mathematics Book
Detailed Notes with MCQs of Chapter 15: Visualising Solid Shapes. This is a crucial chapter not just for your current class, but it builds a foundation for geometry and spatial reasoning often tested in various government exams. We need to understand how to represent and interpret three-dimensional objects in two dimensions.
Chapter 15: Visualising Solid Shapes - Detailed Notes
1. Introduction: Plane Shapes vs. Solid Shapes
- Plane Shapes (2-Dimensional or 2D):
- Have only two measurements: Length and Breadth.
- Can be drawn completely on a flat surface (a plane).
- Examples: Square, Rectangle, Circle, Triangle.
- They have area but no volume.
- Solid Shapes (3-Dimensional or 3D):
- Have three measurements: Length, Breadth, and Height (or Depth).
- Occupy space.
- Cannot be drawn completely on a flat surface while showing all features accurately without specific techniques.
- Examples: Cube, Cuboid, Sphere, Cylinder, Cone, Pyramid.
- They have surface area and volume.
2. Elements of 3D Shapes (Especially Polyhedrons)
Many solid shapes are made up of polygonal regions. These are called Polyhedrons (singular: Polyhedron). Examples: Cube, Cuboid, Prisms, Pyramids. Shapes like Cylinders, Cones, Spheres are not polyhedrons because they have curved surfaces.
For polyhedrons, we identify three key elements:
-
Faces (F): The flat surfaces of the solid shape. These are polygons.
- Example: A cube has 6 square faces. A cuboid has 6 rectangular faces (can include squares). A triangular pyramid (tetrahedron) has 4 triangular faces.
-
Edges (E): The line segments where two faces meet.
- Example: A cube has 12 edges. A cuboid has 12 edges. A triangular pyramid has 6 edges.
-
Vertices (V): The points where three or more edges meet (the corners).
- Example: A cube has 8 vertices. A cuboid has 8 vertices. A triangular pyramid has 4 vertices.
-
Euler's Formula for Polyhedrons: A fundamental relationship connecting Faces (F), Vertices (V), and Edges (E) for any polyhedron:
F + V - E = 2- Example (Cube): F=6, V=8, E=12. So, 6 + 8 - 12 = 14 - 12 = 2.
- Example (Triangular Pyramid): F=4, V=4, E=6. So, 4 + 4 - 6 = 8 - 6 = 2.
- This formula is a useful check and often appears in competitive exams.
3. Nets for Building 3D Shapes
- Definition: A net is a 2-dimensional shape (a pattern or layout) that can be folded along its edges to form a 3-dimensional solid shape.
- Purpose: Helps understand the relationship between 2D representations and 3D objects. Visualises how the faces of a solid fit together.
- Examples:
- Cube: A common net consists of six squares arranged in a cross shape (or other valid arrangements). There are 11 distinct nets for a cube.
- Cuboid: Similar to a cube, but uses rectangles (and possibly squares).
- Cylinder: A net consists of one rectangle (for the curved surface) and two circles (for the top and bottom bases).
- Cone: A net consists of one sector of a circle (for the curved surface) and one circle (for the base).
- Pyramid (Square Base): A net consists of one square (base) and four triangles attached to its sides.
- Key Skill: Identifying whether a given 2D pattern can actually fold into a specific 3D shape.
4. Drawing Solids on a Flat Surface
Representing 3D shapes on 2D paper requires specific techniques:
- Oblique Sketches:
- Simple way to give a visual impression of a 3D object.
- Easy to draw.
- Characteristics:
- The front face (and sometimes the opposite back face) is drawn true to shape and size.
- Edges going back from the front face (receding lines) are drawn typically at a 45° angle to the horizontal.
- Lengths of these receding edges are often drawn shorter than the actual proportional length to give a sense of perspective, but they are not accurately scaled.
- Does not maintain accurate proportions or angles of the solid.
- Isometric Sketches:
- Provides a more accurate representation of the solid's dimensions and proportions.
- Drawn on special isometric dot paper, which has dots arranged in equilateral triangles.
- Characteristics:
- Represents the object's length, width, and height accurately in proportion.
- Vertical lines are drawn vertically.
- Horizontal edges are drawn usually at a 30° angle to the horizontal baseline (following the dot grid).
- All lines parallel to the three main axes (length, width, height) are drawn to scale.
5. Visualising Solid Objects - Different Ways
Understanding a 3D object involves being able to visualise it from different perspectives or imagine its internal structure.
- (i) Slicing or Cutting (Cross-sections):
- Imagine cutting through a solid shape with a straight cut (like with a knife). The shape of the flat surface exposed by the cut is called the cross-section.
- The shape of the cross-section depends on:
- The solid shape itself.
- The angle/orientation of the cut (horizontal, vertical, slanted).
- Examples:
- Horizontal cut through a cylinder -> Circle.
- Vertical cut through a cylinder -> Rectangle.
- Vertical cut through a cone (through the apex) -> Triangle.
- Horizontal cut through a cone -> Circle.
- Cut through a cube parallel to a face -> Square.
- Diagonal cut through a cube -> Can be a rectangle or other polygons depending on the cut.
- (ii) Shadow Play:
- A 3D object casts a 2D shadow when light shines on it.
- The shape of the shadow depends on:
- The solid shape.
- The direction and type of the light source (e.g., torchlight overhead vs. from the side).
- The orientation of the object relative to the light.
- Examples (with light directly overhead):
- Sphere -> Circle shadow.
- Cube -> Square shadow.
- Cone (standing on base) -> Circle shadow.
- Cylinder (standing on base) -> Circle shadow.
- Examples (with light from the front):
- Cube -> Square shadow.
- Cone (standing on base) -> Triangle shadow.
- Cylinder (standing on base) -> Rectangle shadow.
- (iii) Viewing from Different Angles (Orthographic Projections):
- Looking at a 3D object from different standard positions gives different 2D views.
- The three most common views are:
- Front View: What you see looking directly at the front.
- Top View: What you see looking directly down from above.
- Side View: What you see looking directly from one side (usually specified as left or right).
- This is crucial in engineering and design drawings to represent a 3D object completely using 2D views.
- Example: For a simple house shape (cuboid with a triangular prism roof), the front view might show a rectangle with a triangle on top, the top view might show two rectangles (roof panels), and the side view might show a rectangle (wall) or a shape like a pentagon depending on the roof overhang.
Key Takeaways for Exams:
- Be able to differentiate between 2D and 3D shapes.
- Identify and count Faces, Edges, and Vertices of common polyhedrons.
- Know and apply Euler's formula (F + V - E = 2).
- Recognize nets for common solids (cube, cuboid, cylinder, cone, pyramid) and identify invalid nets.
- Understand the difference between oblique and isometric sketches and their characteristics.
- Predict the shape of cross-sections when a solid is sliced.
- Predict the shape of shadows cast by solids under different lighting conditions.
- Identify the Front, Top, and Side views of simple solid objects or arrangements.
Multiple Choice Questions (MCQs)
Here are 10 MCQs based on the concepts discussed:
-
Which of the following is a 3-Dimensional shape?
(a) Rectangle
(b) Circle
(c) Triangle
(d) Sphere -
A solid shape has 6 faces, 8 vertices, and 12 edges. What is this shape most likely?
(a) Triangular Pyramid
(b) Square Pyramid
(c) Cube
(d) Cone -
Which formula correctly relates the number of faces (F), vertices (V), and edges (E) of a polyhedron?
(a) F + E - V = 2
(b) F + V - E = 2
(c) V + E - F = 2
(d) F - V + E = 2 -
Which of the following 2D patterns can be folded to form a cube?
(a) A pattern with 5 squares.
(b) A pattern with 6 squares arranged in a 2x3 rectangle.
(c) A pattern with 6 squares forming a cross shape.
(d) A pattern with 7 squares. -
A drawing of a solid shape where the front face is drawn accurately, but receding edges are drawn at an angle (often 45°) and may not be to scale, is called:
(a) An Isometric Sketch
(b) A Net
(c) An Oblique Sketch
(d) A Top View -
What is the shape of the cross-section obtained when a cylinder is cut vertically through its centre?
(a) Circle
(b) Rectangle
(c) Triangle
(d) Oval -
If you shine a torchlight directly from the top onto a cone standing on its circular base, what shape will its shadow be?
(a) Triangle
(b) Square
(c) Circle
(d) Rectangle -
How many vertices does a square pyramid have?
(a) 4
(b) 5
(c) 6
(d) 8 -
Isometric sketches are typically drawn on:
(a) Plain paper
(b) Graph paper with squares
(c) Isometric dot paper
(d) Lined paper -
Looking directly down from above onto a solid object gives which view?
(a) Front View
(b) Side View
(c) Top View
(d) Bottom View
Answer Key:
- (d) Sphere
- (c) Cube (or Cuboid)
- (b) F + V - E = 2
- (c) A pattern with 6 squares forming a cross shape.
- (c) An Oblique Sketch
- (b) Rectangle
- (c) Circle
- (b) 5 (4 at the base, 1 apex)
- (c) Isometric dot paper
- (c) Top View
Study these notes carefully. Visualising shapes takes practice, so try drawing nets, sketches, and identifying views for different objects around you. Good luck!