Geometric Shapes and Solids

In this chapter, students will gain knowledge about the different shapes present in their environment and become acquainted with their respective names. Let's establish a solid groundwork by exploring the basics of geometry. The term "geometry" originates from two ancient Greek terms: 'Geo,' signifying 'Earth,' and 'Metron,' signifying 'Measurement.'

Geometric Shapes and Solids

Geometric shapes and solids are the foundational elements of geometry and they are used in our everyday lives whether within the items within our households or in the natural world that surrounds us.

Examples of geometric shapes are shown:

Different types of geometrical shapes are one-dimensional (1D) shapes, two-dimensional (2D) shapes and three-dimensional (3D) shapes. "D" stands for "dimensional" here.

One Dimensional (1D) Shapes

The shapes in 1D are basically lines which have a length. There's only a line and some curves that can be drawn. Examples of 1D shapes are points, straight lines, rays, angles, parallel lines, perpendicular lines and curves.

Let’s discuss more about the one-dimensional figures.

Types of Lines

Straight Line: In geometry, a line is defined as a straight and never-ending line that runs in both directions. This is the shortest possible way from one place to another.

There are three types of straight lines:

Slanting Line: A slanting line is a straight line that remains straight but slants in a different direction.

Horizontal Line: A horizontal line is a straight line that extends from left to right and right to left.

Vertical Line: A vertical line is a straight line that extends from top to bottom and bottom to top.

Curved Line: A curved line is a line that is not straight and is formed by curves. The following examples of curved lines are given below:

Parallel Lines: Parallel lines are the lines that are in the same plane and are always at the same distance from each other. The length can be horizontal or vertical. The following parallel lines are shown:

Intersecting Lines: Intersecting lines are the lines which intersect at a point. The two or more overlapping lines which share a single point of contact. A point of intersection is the place where the intersecting lines meet. The intersection lines are shown as follows:

Perpendicular Lines: Perpendicular lines are lines that meet at 90°, creating an L-shaped look. The point where horizontal and vertical lines intersect signifies their perpendicular relationship. Perpendicular lines are shown as

Line Segment: Line segments are lines with two ends. It is made up of any points on a line between the two sides. It is possible to measure the length of a line segment but not the length of a line.

Ray:  A ray is a segment of a straight line that originates from one endpoint and extends infinitely in the other direction. This starting point is also referred to as the initial point.

Angle: An angle is formed when two rays converge at a single point. This point of convergence is called the vertex, and the two extending rays are termed arms. The symbol '∠' is employed to denote an angle.

Angles are measured using a protractor with measurements being 30°, 45°, 60°, 90°, 120°, 150°, 180°, 270° and 360°. It is shown as

Types of Angles

Types of Angles are as follows:

1. Acute Angle:  The acute angle is between 0° and 90° which means it is smaller than the right angle.

2. Right Angle:  The right angle is exactly 90° and resembles the shape of the letter 'L.'

3. Obtuse Angle:  The obtuse angle is between 90° and 180°. It is larger than the right angle but smaller than a half-turn.

4. Straight Angle:  The straight angle of 180° is exactly the same as a straight line.

5. Reflex Angle:  The reflex angle ranges from 180° to less than 360°. It is bigger than a straight angle but not as extensive as a full circle.

6. Complete Angle: The complete angle is exactly 360°. It is called a full angle, too. If you are creating a full circle, it creates a complete angle. It is like completing the entire circle.

Two-Dimensional (2D) Shapes

Two-dimensional shapes are flat figures that can be drawn on a sheet of paper. They are defined by two key characteristics: length and breadth (width). These shapes consist of straight lines (sides or edges) that intersect at vertices which form angles. They serve as a framework for a wide range of objects in our surroundings which are squares, circles, triangles, rhombus, rectangles, pentagons, hexagons, etc. These shapes are presented as enclosed figures.

Let's learn more about 2D shapes:

Specific prefixes are used such as Tri- for 3, Quad- for 4, Penta- for 5, Hexa- for 6, Hepta- for 7, Octa- for 8, and so on.

Polygons: Polygons are fundamental geometric shapes that consist of straight-line segments connected end-to-end to form a closed figure.

Polygons can be classified into various types based on the number of sides they have. Some common types of polygons are as follows:

1. Triangle: A polygon with three sides and three vertices.

2. Quadrilateral: A polygon with four sides and four vertices.

The most common types of quadrilaterals are

a. Rectangle

→ A rectangle has four right angles (90° each).
→ Opposite sides are parallel and equal in length.

b. Square

→ A square is a special type of rectangle where all four sides are of equal length.
→ All angles are right angles (90° each).

c. Parallelogram

→ A parallelogram has opposite sides that are parallel.
→ Opposite angles are equal.

d. Rhombus

→ A rhombus is a parallelogram with all sides of equal length.
→ Opposite angles are equal.
→ Diagonals bisect each other at right angles.

3. Pentagon: A polygon with five sides and five vertices.

4. Hexagon: A polygon with six sides and six vertices.

Below, you'll find a simplified explanation of various plane figures:

Triangle

A triangle is a closed figure with three sides. These sides are made of straight lines. In the triangle, the sum of the three angles is 180°.

Let's learn more about the different types of triangles:

Classification of Triangles Based on Sides:

Triangles can be classified into different categories based on the lengths of their sides. The primary classifications of triangles based on sides are as follows:

Equilateral Triangle

→ All three sides of an equilateral triangle are of equal length.
→ Each angle in an equilateral triangle measures 60°.

Isosceles Triangle

→ An isosceles triangle has two sides of equal length.
→ The angles opposite the equal sides are equal.
→ The third side is known as the base which is of a different length.

Scalene Triangle

→ In a scalene triangle, all three sides have different lengths.
→ The angles within a scalene triangle can have varying measures.

Classification of Triangles Based on Angles:

Triangles can be classified into different categories based on the measures of their internal angles. The primary classifications of triangles based on angles are as follows:

Acute Triangle

→ An acute triangle has all three angles measuring less than 90°.
→ An equilateral triangle is an example of an acute triangle whose angle measures 60°.
→ It is also known as an acute-angled triangle.

Right Triangle

→ A right triangle contains one angle that measures exactly 90° (a right angle).
→ The other two angles are acute whose sum is equal to 90°.
→ The interior angles of an isosceles right triangle are 90°, 45° and 45°.
→ It is also known as a right-angled triangle.

Obtuse Triangle

→ An obtuse triangle contains one angle that measures more than 90°.
→ The interior angles of an isosceles obtuse triangle are 120°, 30° and 30°.
→ It is also known as an obtuse-angled triangle.

Circle

A circle is a two-dimensional closed shape with all points in the shape equidistant from a single point called the centre. This equidistant distance is known as the radius. Unlike polygons, it lacks corners or edges.

Circle with centre, radius and diameter is shown:

Terms related to a circle:

1. Radius: The radius of a circle is the distance from the center to any point on the circle's boundary. It is represented by the letter "r".
2. Diameter: The diameter of a circle is a straight line segment that passes through the center and connects two points on the circle's boundary. It is equal to twice the radius.

Diameter = 2 × radius

In real-life uses, circles are found in objects such as wheels, clock faces, planets and many other natural and man-made structures.

Three Dimensional (3D) or Solid Shapes

Three dimensional shapes have the characteristics of real objects with three distinct dimensions: length, width and height. These shapes allow us to distinguish objects of depth and height in our world from flat representations such as cube, cuboid, cone or cylinder.

The 3D shapes is shown as:

Certainly! Here are some familiar types of 3D shapes:

Cube: A cube is a box-shaped where all sides are identical squares. It has 6 equal square faces, 12 edges, and 8 vertices (corners).

Cuboid: A cuboid is box-shaped and its sides are rectangles. It has 6 faces, 12 edges, and 8 vertices.

Perimeter of Various Shapes

It is the total distance around the boundary of a two-dimensional object.

Perimeter = Sum of sides

Units of measurement for perimeter are centimetres (cm), metres (m), inches (in), feet (ft), etc.

Area of Various Shapes

Area is the space occupied by a two-dimensional object. It signifies the region enclosed within the perimeter of a flat figure.

This is measured in units of measurement such as square centimetres (cm²) and square metres (m²).

Volume of Various Shapes/Solids

Volume of various shapes is the amount of space enclosed within a three-dimensional object.

This is measured in units of measurement such as cubic centimetres (cm3) and cubic metres (m3).

Example 1: What is the area of a given shaded region?

a) 127.25 cm²
b) 127.75  cm²
c) 137 cm²
d) 147 cm²

Answer: d) 147 cm²

Explanation: Number of shaded squares = 11 + (½ × 2) 
                                                              = 11 + 1
                                                               = 12

Area of 1 shaded square = 3.5 × 3.5 = 12.25 cm²
Area of 12 shaded squares = 12 × 12.25 cm² = 147 cm²

Example 2: What is the volume of a given dice of edge 2.2 cm?

a) 9.65 cm3
b) 10.25 cm3
c) 10.65 cm3
d) 12.25 cm3

Answer: c) 10.65 cm3

Explanation: Edge means side.

Volume of cube = Side x Side x Side = 2.2 ×  2.2 ×  2.2
                                                      = 10.648 cm³
                                                       = 10.65 cm³ [Round off to nearest tens]