Edges, Faces, and Vertices: Geometrical Shapes

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Vertices, edges, and faces are the building units that make up the three-dimensional and two-dimensional structures. Mathematical shapes and structures can be in the form of two-dimensional figures, which lie on a plane surface or three-dimensional structures, which occupy space and have mass.

Key Takeaways: Shapes, Structures, Three dimensional, Two dimensional, Edges, Faces, Vertices


What is a Geometrical Shape?

Shapes form an important part of mathematics. The study of shapes in mathematics is called geometry. In geometry, a shape is defined as an enclosed structure formed by lines. These lines enclosing an area are called the boundary or outline of the shape. We are constantly surrounded by shapes in our everyday life, they can be of several types including regular, irregular, open, closed, three-dimensional, or two-dimensional.

Geometric shapes

Geometric shapes

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Types of Geometrical Shapes

The geometrical shape can be of several types as follows.

Regular shape: Has a fixed and defined shape. Eg: Sphere, circle, square, triangle, etc.

Irregular Shape: Has an undefined shape. Eg: Cloud, cotton, etc.

Irregular shapes

Irregular shapes

Open Shape: These are the shapes that are not closed and have one open side, which is an incomplete outline.

Closed Shape: They have an enclosed and complete outline defining them. Eg: Square. Triangle, cube, cuboid, etc.

Two-Dimensional Shape: They are present on a plane surface. Eg: Square, circle, triangle, rectangle, etc.

Two dimensional shape

Two dimensional shape

Three-Dimensional Shape: These shapes occupy space and have mass. Eg: Cube, cuboid, prism, sphere, etc.

Three dimensional shape

Three dimensional shape

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Components of a mathematical structure

The regular geometric shapes are composed of three essential components which make them a defined structure. These three components are:

  • Vertices
  • Edges
  • Faces

Vertices

A Vertice is a point where two or more than two lines join together. Also known as the corner or the tip of a structure. More than one vertex is called vertices.

Vertices

Vertices

Example:

Cube → 8 vertices

Prism → 6 vertices

Cylinder → 0 vertices

Also Read: Distance formula

Edges

An edge is a point where two or more than two faces or surfaces join together. Edge is not a point but a line segment. It provides the framework for a geometrical structure. More than one edge is called edges.

Edges

Edges

Example:

Cube → 12 edges

Prism → 9 edges

Cylinder → 3 edges

Also Read: Section Formula

Faces

The face is the flat surface of the geometrical structure. A face is like a two-dimensional shape that makes up the three-dimensional structure.

Faces

Faces

Example:

Cube → 6 faces

Prism → 5 faces

Cylinder → 3 faces


Things to Remember

  • Vertices, edges, and faces are the building units that make up the three-dimensional and two-dimensional structures.
  • Geometrical shapes can be of several types, regular, irregular, open, closed, three-dimensional, or two-dimensional.
  • Vertice is a point where two or more than two lines join together.
  • An edge is a point where two or more than two faces or surfaces join together.
  • The face is the flat surface of the geometrical structure.

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Sample Questions

Ques: What are mathematical shapes? (2 marks)

Ans: Mathematical shapes are also called geometrical shapes. A geometrical shape is defined as an enclosed structure formed by lines. These lines enclosing an area are called the boundary of the shape or its outline. A geometrical shape can be of different types, regular, irregular, open, closed, three-dimensional, or two-dimensional shapes.

Ques: Explain the different types of geometrical shapes. (5 marks)

Ans. A geometrical shape can be of different types, regular, irregular, open, closed, three-dimensional, or two-dimensional shapes.

  1. Regular shapes: They have a fixed and defined shape, e.g., Sphere, circle, square, triangle, etc.
  2. Irregular shapes: They have a non-fixed and undefined shape, ex: Cloud, cotton, etc
  3. Open shapes: These are the shapes that are not closed and have one open side, which is an incomplete outline.
  4. Closed shapes: They have an enclosed and complete outline defining them, ex: Square. Triangle, cube, cuboid, etc
  5. Two-dimensional shapes: They are present on a plane surface, ex: Square, circle, triangle, rectangle, etc.
  6. Three-dimensional shapes: These shapes occupy space and have mass, ex: cube, cuboid, prism, sphere, etc.

Ques: What are the three properties of a three-dimensional structure? (2 marks)

Ans: The regular geometric shapes are composed of three essential components which make them a defined structure. These three components are:

  1. Vertices: Vertice is a point where two or more than two lines join together.
  2. Edges: Edge is a point where two or more than two faces or surfaces join together.
  3. Faces: The face is the flat surface of the geometrical structure.

Ques: What is a polyhedron? (2 marks)

Ans: A polyhedron is any three-dimensional geometrical shape that has a flat face and a fixed number of edges. A polyhedron is a solid regular structure.

Also Read: Nature of Roots of Quadratic Equation

Ques: How to correlate vertices, edges, and faces? (2 marks)

Ans: The vertices, faces, and edges can be correlated in a polyhedron using Euler’s formula. According to Euler’s formula:

F + V − E = 2

F = number of faces,

V = number of vertices, and

E = number of edges.

Ques: State the number of faces, edges, and vertices of the cuboid, and what is the nature of its faces. (2 marks)

Ans: A cuboid has the following parameters:

  • Faces: Six
  • Edges: Twelve
  • Vertices: Eight

The faces are rectangular and are two-dimensional in nature.

Ques: A polyhedron has 6 faces, and 12 edges, what are the number of vertices? (2 marks)

Ans: In order to determine the number of vertices we can use the given data and Euler’s Formula.

F + V − E = 2

F = 6

V = ?

E = 12

6+V-12=2

V-6=2

V= 8

Thus, the number of vertices is eight.

Also Read: Surface Areas and Volumes

Ques: What are the different views of a three-dimensional structure? (2 marks)

Ans: There are three types of view:

  • Side view: From one side of the shape
  • Top View: From the upper side of the shape
  • Front view

Ques. If three cubes of dimensions 2 cm × 2 cm × 2 cm are placed end to end, what would be the dimension of the resulting cuboid? (2 marks)

Ans. Length of the resulting cuboid = 2 cm + 2 cm + 2 cm = 6 cm

Breadth = 2 cm

Height = 2 cm

Visualising Solid Shapes Class 7 Extra Questions Maths Chapter 15 Q1

Hence the required dimensions = 6 cm × 2 cm × 2 cm.

Ques. What is an edge? (2 marks)

Ans. An edge is a point where two or more than two faces or surfaces join together. Edge is not a point but a line segment. It provides the framework for a geometrical structure. More than one edge is called edges.

Example:

Cube → 12 edges

Prism → 9 edges

Cylinder → 3 edges

Also Read:

CBSE X Related Questions

  • 1.
    If the zeroes of the polynomial $ax^2 + bx + \dfrac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

      • $\dfrac{1}{2}$
      • 2
      • -2
      • $-\dfrac{1}{2}$

    • 2.
      Find length and breadth of a rectangular park whose perimeter is \(100 \, \text{m}\) and area is \(600 \, \text{m}^2\).


        • 3.

          Directions: In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
          Choose the correct option from the following:
          (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
          (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
          (C) Assertion (A) is true, but Reason (R) is false.
          (D) Assertion (A) is false, but Reason (R) is true.

          Assertion (A): For any two prime numbers $p$ and $q$, their HCF is 1 and LCM is $p + q$.
          Reason (R): For any two natural numbers, HCF × LCM = product of numbers.

            • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
            • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
            • Assertion (A) is true, but Reason (R) is false.
            • Assertion (A) is false, but Reason (R) is true.

          • 4.
            Check whether the following system of equations is consistent or not. If consistent, solve graphically: \[ x - 2y + 4 = 0, \quad 2x - y - 4 = 0 \]


              • 5.
                Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)


                  • 6.
                    In a trapezium \(ABCD\), \(AB \parallel DC\) and its diagonals intersect at \(O\). Prove that \[ \frac{OA}{OC} = \frac{OB}{OD} \]

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