Introduction to Solid Shapes and Basic Terms

Understanding Three Dimensional Shapes (Solids)

In the study of geometry, we classify shapes based on the number of dimensions they occupy. So far, we have primarily discussed plane figures or Two-Dimensional (2D) shapes, which are shapes that lie completely on a flat surface (a plane) and can be described using only two measurements: length and breadth (or width).

Examples of 2D shapes include squares, rectangles, triangles, circles, quadrilaterals, and other polygons. These shapes have properties like perimeter (the distance around the boundary) and area (the measure of the surface enclosed), but they have no thickness or depth.

Shapes that occupy space are called solid shapes or Three-Dimensional (3D) shapes. These shapes exist in three dimensions and require three measurements to describe their size:

• Length
• Breadth (or Width)
• Height (or Depth or Thickness)

Unlike 2D shapes, 3D shapes have volume (the amount of space they occupy or contain) and surface area (the total area of all the surfaces that enclose the solid).

Key Differences between 2D and 3D Shapes

Understanding the distinction between 2D and 3D shapes is fundamental:

Examples of 3D Shapes in Real Life

Our physical world is filled with objects that are three-dimensional. Recognizing the basic geometric solid shapes in everyday objects helps in understanding their properties:

• A standard building brick or a book often approximates a cuboid.
• A perfectly formed dice or a Rubik's cube is a good example of a cube.
• A football, a marble, or a globe is a representation of a sphere.
• A soup can, a drum, or a piece of pipe is cylindrical in shape, representing a cylinder.
• An ice cream cone or a birthday cap is typically shaped like a cone.
• The Egyptian pyramids are famous examples of pyramids.
• A Toblerone chocolate bar or certain tents can be shaped like a prism.

In mensuration of solids, we learn how to calculate the surface area and volume of these standard 3D shapes. Understanding the dimensions (length, breadth, height) and the basic characteristics of these shapes is the first step.

Basis Terms Related to Solid Figures (Faces, Edges, Vertices)

To accurately describe and analyze solid figures, especially those with flat surfaces, we use specific terminology to refer to their different components. The most fundamental terms for describing polyhedrons (solids with flat polygonal faces) are faces, edges, and vertices.

Definitions of Key Terms

• Face:

  A face is a flat surface that forms part of the boundary of a solid object. For polyhedrons, faces are polygons (such as squares, rectangles, triangles). Solids with curved surfaces, like cylinders and cones, also have flat faces (their bases) in addition to their curved surfaces.

• Edge:

  An edge is a line segment where two faces of a solid figure meet. It forms the boundary between two adjacent faces. Edges are always straight lines in polyhedrons.

• Vertex (plural: Vertices):

  A vertex is a point where three or more edges of a solid figure meet. It is essentially a corner point of the solid. For a cone, the single pointed tip is also referred to as a vertex (or apex), even though multiple edges don't meet there in the same way as in a polyhedron.

Illustration using a Cuboid

A cuboid (or rectangular prism) is a solid figure bounded by six rectangular faces. It is a common example used to illustrate the terms faces, edges, and vertices.

Let's count the faces, edges, and vertices of a cuboid:

• Faces (F): A cuboid has a top face, a bottom face, a front face, a back face, a left side face, and a right side face. All are rectangular. So, a cuboid has 6 faces.
• Edges (E): The edges are where the faces meet. There are 4 edges around the top face, 4 edges around the bottom face, and 4 vertical edges connecting the top vertices to the bottom vertices. Counting carefully (and avoiding double-counting), a cuboid has $4 + 4 + 4 = 12$ edges.
• Vertices (V): The vertices are the corner points. There are 4 vertices on the top face and 4 vertices on the bottom face. So, a cuboid has 8 vertices.

Euler's Formula for Polyhedrons

For any convex polyhedron (a solid with flat faces where a line segment connecting any two points on the surface lies entirely inside or on the surface), there exists a fascinating relationship between the number of faces (F), vertices (V), and edges (E). This relationship is given by Euler's Formula (named after the Swiss mathematician Leonhard Euler):

Let's verify Euler's formula for the cuboid using the counts we just determined:

• Number of Faces (F) = 6
• Number of Vertices (V) = 8
• Number of Edges (E) = 12

Substitute these values into Euler's formula:

The formula holds true for the cuboid, as $6 + 8 - 12 = 2$.

Euler's formula is a powerful tool in the study of polyhedrons and applies to various shapes like prisms (triangular prism, pentagonal prism, etc.), pyramids (triangular pyramid, square pyramid, etc.), and Platonic solids.

Note: The standard form of Euler's formula $F + V - E = 2$ specifically applies to simple convex polyhedrons. Solids with curved surfaces (like spheres, cylinders, cones) or non-convex polyhedrons require modified versions or topological considerations.

Types of Solid Figures (Prisms, Pyramids, Cylinder, Cone, Sphere)

Solid shapes, or three-dimensional figures, can be classified into various types based on their geometric properties. These properties include the nature of their surfaces (flat or curved), the shape of their bases, and whether they taper to a single point (an apex).

Common Solid Figures

1. Prism

A prism is a polyhedron (a solid with flat faces) that has two identical and parallel polygonal bases. These bases are connected by side faces, which are typically rectangles or parallelograms. The shape of the base polygon determines the name of the prism.

• Definition: A polyhedron with congruent and parallel bases, and parallelogram lateral faces.
• Properties:
  • It has two bases that are congruent polygons in parallel planes.
  • The lateral faces (side faces) are parallelograms. In a right prism, the lateral faces are rectangles and are perpendicular to the bases.
  • The number of lateral faces is equal to the number of sides of the base polygon.
• Types (named by the shape of the base):
  • Triangular Prism: Bases are triangles. It has 3 rectangular lateral faces.
  • Rectangular Prism (Cuboid): Bases are rectangles. It has 4 rectangular lateral faces. A rectangular prism where all faces are squares is called a cube.
  • Pentagonal Prism: Bases are pentagons. It has 5 rectangular lateral faces.
  • General n-gonal Prism: Bases are n-sided polygons. It has n rectangular lateral faces.
• Faces, Edges, Vertices (for an n-sided base prism):
  • Number of Faces (F): $n$ (lateral faces) + 2 (bases) = $\mathbf{n+2}$
  • Number of Vertices (V): $n$ (on one base) + $n$ (on the other base) = $\mathbf{2n}$
  • Number of Edges (E): $n$ (around the top base) + $n$ (around the bottom base) + $n$ (connecting the bases) = $\mathbf{3n}$

  Let's verify Euler's Formula ($F + V - E = 2$) for an n-gonal prism:

  $(n+2) + (2n) - (3n) = n+2+2n-3n = 3n+2-3n = 2$

  The formula holds true for prisms.

2. Pyramid

A pyramid is a polyhedron that has one polygonal base and triangular lateral faces that meet at a single common point called the apex or vertex. The shape of the base polygon determines the name of the pyramid.

• Definition: A polyhedron with one polygonal base and triangular lateral faces meeting at a point (apex).
• Properties:
  • It has a single polygonal base.
  • The lateral faces are triangles that meet at the apex.
  • The number of lateral faces is equal to the number of sides of the base polygon.
  • In a right pyramid, the apex is directly above the center of the base.
• Types (named by the shape of the base):
  • Triangular Pyramid (Tetrahedron): Base is a triangle. It has 3 triangular lateral faces. A tetrahedron where all four faces are equilateral triangles is called a regular tetrahedron.
  • Square Pyramid: Base is a square. It has 4 triangular lateral faces.
  • Rectangular Pyramid: Base is a rectangle. It has 4 triangular lateral faces.
  • Pentagonal Pyramid: Base is a pentagon. It has 5 triangular lateral faces.
  • General n-gonal Pyramid: Base is an n-sided polygon. It has n triangular lateral faces.
• Faces, Edges, Vertices (for an n-sided base pyramid):
  • Number of Faces (F): $n$ (lateral faces) + 1 (base) = $\mathbf{n+1}$
  • Number of Vertices (V): $n$ (on the base) + 1 (apex) = $\mathbf{n+1}$
  • Number of Edges (E): $n$ (around the base) + $n$ (connecting base vertices to apex) = $\mathbf{2n}$

  Let's verify Euler's Formula ($F + V - E = 2$) for an n-gonal pyramid:

  $(n+1) + (n+1) - (2n) = n+1+n+1-2n = 2n+2-2n = 2$

  The formula holds true for pyramids.

3. Cylinder

A cylinder is a solid shape that is not a polyhedron because it has curved surfaces. A typical cylinder has two identical, parallel circular bases connected by a curved lateral surface.

• Definition: A solid with two congruent and parallel circular bases connected by a curved surface.
• Properties:
  • It has 2 flat faces (the circular bases) and 1 curved lateral surface.
  • It has 2 circular edges where the curved surface meets the bases.
  • It has no vertices in the traditional sense of corners where edges meet, although sometimes the centers of the bases are relevant points.
  • A right circular cylinder is one where the line segment connecting the centers of the bases is perpendicular to the bases. This is the most common type studied.
• Faces: $\mathbf{2}$ (flat) + $\mathbf{1}$ (curved)
• Edges: $\mathbf{2}$ (circular)
• Vertices: $\mathbf{0}$

4. Cone

A cone is a solid shape that is not a polyhedron because it has a curved surface. It has one circular base and a curved lateral surface that tapers smoothly from the boundary of the base to a single point called the apex or vertex.

• Definition: A solid with a circular base and a curved surface connecting the base boundary to a single point (apex).
• Properties:
  • It has 1 flat face (the circular base) and 1 curved lateral surface.
  • It has 1 circular edge where the curved surface meets the base.
  • It has 1 vertex (the apex).
  • A right circular cone is one where the apex is directly above the center of the circular base. This is the most common type studied. The line segment from the apex to any point on the circumference of the base is called the slant height.
• Faces: $\mathbf{1}$ (flat) + $\mathbf{1}$ (curved)
• Edges: $\mathbf{1}$ (circular)
• Vertices: $\mathbf{1}$ (apex)

5. Sphere

A sphere is a perfectly round solid geometric object. It is the three-dimensional analogue of a circle in two dimensions. A sphere is defined as the set of all points in three-dimensional space that are equidistant from a fixed central point. This distance is called the radius.

• Definition: A solid where all points on its surface are equidistant from a central point.
• Properties:
  • It has only one continuous curved surface.
  • It has no flat faces.
  • It has no edges.
  • It has no vertices.
• Faces: $\mathbf{1}$ (curved)
• Edges: $\mathbf{0}$
• Vertices: $\mathbf{0}$

Summary Table of Common Solid Shapes

Here is a table summarizing the key characteristics of the common solid figures discussed:

This table provides a quick reference for distinguishing between these common 3D shapes based on their defining features and counts of faces, edges, and vertices (where applicable).