Fill in the blanks:
(i) A solid made up of polygonal regions is called .....
(ii) Polyhedrons which are not convex are called .....
(iii) A polyhedron is said to be ..... if all of its faces are regular polygons and the same number of faces meet at each vertex.
(iv) A polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape is called a .....
(v) A prism whose base and top are congruent hexagon is called a .....
(vi) A polyhedron whose all lateral faces are triangles is called a .....
(vii) A triangular pyramid is also known as .....
(viii) A rectangular prism is also known as .....
(ix) The polygonal regions forming a polyhedron are called .....
Answer
(i) A solid made up of polygonal regions is called polyhedron.
(ii) Polyhedrons which are not convex are called concave polyhedrons.
(iii) A polyhedron is said to be regular polyhedron if all of its faces are regular polygons and the same number of faces meet at each vertex.
(iv) A polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape is called a prism.
(v) A prism whose base and top are congruent hexagon is called a hexagonal prism.
(vi) A polyhedron whose all lateral faces are triangles is called a pyramid.
(vii) A triangular pyramid is also known as tetrahedron.
(viii) A rectangular prism is also known as cuboid.
(ix) The polygonal regions forming a polyhedron are called faces.
State which of the following statements are true (T) or false (F):
(i) A cylinder is a polyhedron.
(ii) All the prisms and pyramids are polyhedrons.
(iii) A tetrahedron is the only pyramid which can be a regular polyhedron.
(iv) The line segments where the faces of a polyhedron meet are called edges.
(v) F + E = V + 2 is called Euler's formula.
(vi) In any prism number of faces is 2 more than number of sides of polygonal base.
(vii) In any pyramid number of edges is twice the number of sides of polygonal base.
(viii) An octagonal prism has 18 vertices.
(ix) All pyramids are prisms.
(x) Lateral faces of a pyramid are triangles.
Answer
(i) False
Reason:
A polyhedron is a solid made up of polygonal regions. A cylinder has a curved surface and two circular ends, so it is not made up of polygonal regions.
(ii) True
Reason:
The base, the top and the lateral faces of a prism, and the base and the lateral faces of a pyramid, are all polygonal regions. So, every prism and every pyramid is a polyhedron.
(iii) True
Reason:
In a tetrahedron all the four faces are congruent equilateral triangles and 3 faces meet at each of its 4 vertices. In every other pyramid, the base is not congruent to the lateral faces and the number of faces meeting at the apex is not the same as the number of faces meeting at a vertex of the base.
(iv) True
Reason:
The line segments where the faces of a polyhedron meet are called edges and the points where the edges of a polyhedron meet are called vertices.
(v) False
Reason:
Euler's formula states that in any polyhedron F + V = E + 2.
(vi) True
Reason:
A prism has one lateral face along each side of its polygonal base together with 2 more faces, which are its base and its top.
(vii) True
Reason:
A pyramid has one edge along each side of its polygonal base and one more edge joining each vertex of the base to the apex.
(viii) False
Reason:
The number of vertices in a prism is twice the number of sides of the polygonal base. So, an octagonal prism has 8 × 2 = 16 vertices.
(ix) False
Reason:
In a prism the base and the top are congruent polygons and the lateral faces are parallelograms in shape, whereas a pyramid has only one polygonal base and its lateral faces are triangles meeting at a common vertex.
(x) True
Reason:
A pyramid is a polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex.