Geometry

In this chapter, students will learn about various shapes that they see in their surroundings and become familiar with their names. They will develop the ability to differentiate between these shapes.

But first, let's lay the foundation by exploring the fundamentals of geometry.

What is Geometry?

Geometry is a branch of mathematics that helps us understand the shapes, sizes, angles and measurements of things we see in our everyday lives. The word "geometry" comes from two ancient Greek words: ‘Geo’ which means ‘Earth’ and ‘Metron’ which means ‘Measurement’.

Geometric Shapes

Geometric shapes are the building blocks of geometry. These shapes can be found everywhere in our daily lives, from the objects in our homes to the natural world outside. Some of the geometric shapes are shown:

The concept of geometrical shapes encompasses a variety of shapes including one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) shapes. Here, " D " means "dimensional".

One Dimensional (1D) Shapes

One-dimensional (1D) shapes are basically just lines that have a length. They can be drawn with just a line and some curves. Points, straight lines, rays, parallel lines and curves are all examples of 1D shapes.

These one-dimensional shapes give us an idea of what geometry is all about and are useful for future studies on more intricate shapes.

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

Types of Lines

→ Straight Line: In geometry, we define a line as a straight and never-ending figure that goes on in both directions.

It is the shortest possible path between any two places.

For example, the bus moves in a straight line.

There are three types of straight lines:

1. Slanting Line: This is a straight line that is straight but leans or slants in another direction.

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

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

→ Curved Line: A curved line is a non-straight line that is composed of curves. The following examples of curved lines are provided:

For example, the bus moves along a curved line.

→ Parallel Lines: Parallel lines are defined as lines that occupy the same plane and are always at the same distance from each other. They may be horizontal or vertical in length. The parallel lines are shown as:

For example, all the imaginary lines on the earth are parallel to each other. [Imaginary lines are the Arctic circle, Antarctic circle, equator, Tropic of Cancer and Capricorn.]

→ Intersecting Lines: Intersecting lines are lines that intersect at a point. These two or more intersecting lines share exactly one point. The point at which the intersecting lines meet is called the point of intersection. The intersecting lines are shown as:

For example, the roads intersect each other.

→ Perpendicular Lines: Perpendicular lines are lines that intersect at a right angle, resulting in an L-shaped shape. Intersecting horizontal and vertical lines indicates that the lines are perpendicular to each other.

The following perpendicular lines show that m is perpendicular to l or l is perpendicular to m:

For example, a shop is perpendicular to the ground.

→ Line Segment: A line segment is a line that has two ends. It is made up of all the points on the line that are between the two ends. The length of a line segment can be measured but not that of a line.

→ Ray: A ray is a straight line that only has one end and runs forever in one direction.

For example, a ray of light comes from a torch.

→ Angle:An angle is the result of the joining of two rays at a single point. This point is referred to as a vertex while the two rays of light are referred to as arms. The symbol '∠' is used to represent the angle.

Angles can be measured using a protractor with common measurements being 30°, 45°, 60°, 90°, 100°, 120°, 150° and 180°.

Types of Angles

Angles come in different shapes and sizes. Let's learn about some of the types of angles:

1. Acute Angle: An acute angle is an angle greater than 0° but less than 90°. It is smaller than a right angle.

2. Right Angle: A right angle is an angle exactly equal to 90°. It looks like the letter 'L.'

3. Obtuse Angle: An obtuse angle is an angle greater than 90° but less than 180°. An obtuse angle is bigger than a right angle but not as big as a full turn.

4. Straight Angle: A straight angle is an angle exactly equal to 180°. It is like a straight line.

5. Reflex Angle:A reflex angle is greater than 180° but less than 360°. A reflex angle is larger than a straight angle but not as big as going all the way around in a circle.

6. Complete Angle: A complete angle is an angle exactly equal to 360°. It is also known as a full angle. When you go all the way around in a circle, you make a full angle. It's like going in a complete circle.

Two Dimensional (2D) Shapes

Two-dimensional shapes are flat figures that you can draw on a piece of paper. They possess two distinct attributes: length and breadth. So, two-dimensional shapes are flat figures made of straight sides that meet at vertices, creating angles. We use these shapes to describe all sorts of things in our world, from squares and circles to triangles and rectangles. These shapes take the form of closed figures, illustrated as:

Let's learn about them:

Sides: Two-dimensional shapes are made up of straight lines. These lines are the sides of the shape.

Vertices: The places where the sides of a shape meet are called vertices. You can think of vertices as the corners of the shape.

Angles: When two sides of a shape meet at a corner (vertex), they create an angle. Angles show us how the sides of the shape are positioned.

For example, if you have a square, it has four angles because four sides come together at each corner.

In order to gain a better understanding of the plane figure, it is important to consider the use of certain prefixes, such as Tri – 3, Quad – 4, Penta – 5, Hexa – 6, Hepta – 7, Octa – 8 and many more.

Certainly, here is a simplified explanation of different plane figures:

→ Triangle: It is a closed figure that has three sides. It is made of three straight lines. The sum of the three angles in the triangle is 180°.

Now, let's know more about the different types of triangles:

Classification of Triangles Based on Angles:

1. Acute Triangle: An acute triangle is a triangle where all three angles are less than 90 degrees.

2. Right Triangle: A right triangle has one special angle that's exactly 90 degrees. It is a triangle with one corner forming a perfect 'L' shape.

3. Obtuse Triangle: An obtuse triangle has one angle that is larger than 90 degrees.

Classification of Triangles Based on Sides:

1. Equilateral Triangle: An equilateral triangle is a triangle where all three sides are the same length. It is like a perfect triangle with equal sides and all the angles inside are the same too.

2. Isosceles Triangle: In an isosceles triangle, two sides are the same length while the third side is different. In this triangle, angles equal to equal sides are equal.

3. Scalene Triangle: A scalene triangle is a triangle where all three sides are different lengths and all angles are not equal.

Understanding these classifications helps us describe and identify different types of triangles we come across in our everyday lives. Triangles are fascinating shapes with lots of variety!

→ Quadrilateral: It is a closed shape that has four straight lines such as a rectangle, a square, a parallelogram, a kite, a rhombus or a trapezium.

Most common types of quadrilateral:

1. Square: A square is a shape with four equal sides and four right angles. 
2. Rectangle: A rectangle is a shape with two equal sides and two right angles.

→ Pentagon: It is a closed shape that has five straight lines.
→ Hexagon: It is a closed shape that has six straight lines.
→ Heptagon: It is a closed shape that has seven straight lines.
→ Octagon: It is a closed shape that has eight straight lines.
→ Circle: A circle is a closed shape that looks round and doesn't have any corners or edges. It is a flat shape that is curved.

Terms related to a circle:

1. Radius of a circle: The distance from the centre to the boundary of a circle is called the radius.
2. Diameter of a circle: The distance from one point on a circle through the centre to another point on the circle is called a diameter. It is twice the radius and the longest distance across the centre. Hence, the length of the radius is half the length of the diameter.

A circle with a centre, radius and diameter is shown:

These shapes are all around us and they help us describe and understand the world of figures and objects.

Three Dimensional (3D) or Solid Shapes

Three-dimensional (3D) shapes are like physical objects that have three attributes - length, breadth and height. We often see 3D shapes like cubes, cuboids, cones and cylinders which help us understand things in our world that aren't flat like pictures but have different heights or depths.

These shapes are shown as:

These 3D shapes are all around us and understanding them helps us describe and work with objects in the real world.

Example 1: How many straight lines are present in the figure given below?

a) 17
b) 19
c) 21
d) 22

Answer: b) 19

Explanation: The straight lines are marked with numbers, shown as

Example 2: What is the radius of the circle given below?

a) 27 cm
b) 27.5 cm
c) 28 cm
d) 28.5 cm

Answer: d) 28.5 cm

Explanation: Diameter is twice the radius. Hence, the length of the radius is half the length of the diameter.

Radius = Diameter 2 = 57 ÷ 2 
                               = 28.5 cm

Example 3:  Identify the type of solid figure given below.

a) Prism
b) Pyramid
c) Tetrahedron
d) Rectangular prism

Answer: b) Pyramid

Explanation: The given figure is a pyramid.A pyramid has a square base and triangular faces that come together.