CBSE Class 8 Maths Exemplar Chapter 6: Visualising the Solid Shapes - NCERT Solutions

The correct answer is (c) A sphere.

A polyhedron is a three-dimensional solid figure whose faces are all flat polygons. It has straight edges and sharp corners called vertices. Shapes like cuboids, triangular prisms, and square prisms are all made up of flat polygonal faces and straight edges, thus they are polyhedrons.

A sphere, however, is a perfectly round geometrical object in three-dimensional space. It has a curved surface and does not have any flat faces, straight edges, or vertices. Therefore, a sphere is not a polyhedron.

The correct answer is (a) 3 triangles.

A polyhedron is a solid figure that must enclose a volume. To form a closed solid, a polyhedron must have at least four faces.

• (a) 3 triangles: Three triangles can only form a flat shape (like a triangle itself if joined edge-to-edge) or an open structure, but they cannot enclose a 3D space.
• (b) 8 triangles: Eight triangles can form a polyhedron, for example, an octahedron.
• (c) 4 triangles: Four triangles can form a tetrahedron, which is the simplest polyhedron.
• (d) 1 pentagon and 5 triangles: A pentagonal pyramid has a pentagon as its base and five triangles as its lateral faces, forming a polyhedron.

Therefore, 3 triangles are insufficient to form a polyhedron.

The correct answer is (c) Cube.

A regular polyhedron, also known as a Platonic solid, is a convex polyhedron whose faces are congruent regular polygons and the same number of faces meet at each vertex.

• A cube has 6 faces, all of which are congruent squares, and 3 faces meet at each vertex. Thus, it is a regular polyhedron.
• A cuboid has rectangular faces, which are not necessarily regular polygons (unless it's a cube).
• A triangular prism has triangular bases and rectangular sides. The faces are not all congruent regular polygons.
• A square prism has square bases but rectangular sides, and the faces are not all congruent.

Therefore, a cube is the only regular polyhedron among the given options.

The correct answer is (a) Rectangle.

A two-dimensional (2D) figure is a shape that has only length and breadth, existing on a flat plane. It has no thickness or depth.

• Rectangle: A rectangle is a flat shape with four sides and four right angles. It has length and breadth, making it a 2D figure.
• Square Prism, Cone, Sphere: These are three-dimensional (3D) figures. They have length, breadth, and height (or depth), and occupy space.

Therefore, a rectangle is the only two-dimensional figure listed.

The correct answer is (c) Octagon.

A pyramid is a polyhedron that has a polygonal base and triangular faces that meet at a point called the apex. The base of a pyramid must be a polygon.

• Octagon: An octagon is a polygon with 8 sides. Therefore, it can serve as the base of a pyramid (an octagonal pyramid).
• Line segment, Circle, Oval: These shapes cannot be the base of a pyramid because they are not polygons. A line segment is 1D, and a circle and oval are curved shapes.

Thus, an octagon is a valid base for a pyramid.

The correct answer is (c) Sphere.

A vertex is a point where two or more edges meet. Let's examine the given shapes:

• Pyramid: A pyramid has an apex, which is a vertex where all the triangular faces meet.
• Prism: A prism has vertices at the corners of its bases.
• Cone: A cone has one vertex at its apex.
• Sphere: A sphere is a perfectly round 3D object. It has a continuous curved surface and no flat faces, edges, or vertices.

Therefore, a sphere is the 3D shape that does not have any vertices.

The correct answer is (a) Polyhedron.

A polyhedron is a solid figure composed of a finite number of flat polygonal faces. The boundaries of these faces are line segments, which are called edges. These edges meet at points called vertices.

• Polyhedron: By definition, the edges of a polyhedron are line segments.
• Cone: A cone has a circular base and a curved lateral surface that tapers to a point (apex). Its edge is the circle forming the base, which is not a line segment.
• Sphere: A sphere has a continuous curved surface and no edges or vertices.
• Polygon: A polygon is a 2D figure, not a solid.

Therefore, a solid having only line segments as its edges is a polyhedron.

The correct answer is (c) 8.

We can use Euler's formula for polyhedrons, which relates the number of faces (F), vertices (V), and edges (E) of a convex polyhedron. The formula is:

$$F\; +\; V\; -\; E\; =\; 2$$

Given in the problem:

• Number of Faces, $F\; =\; 5$
• Number of Vertices, $V\; =\; 5$

We need to find the number of Edges, $E$.

Substitute the given values into Euler's formula:

$5\; +\; 5\; -\; E\; =\; 2$

Simplify the equation:

$10\; -\; E\; =\; 2$

To solve for $E$, rearrange the equation:

$10\; -\; 2\; =\; E$

$E\; =\; 8$

Therefore, the number of edges in this shape is 8.