Geometry is the branch of mathematics that deals with points, lines, shapes and space. It is one of the oldest branches of mathematics, used by ancient civilisations to build temples, measure fields and chart the stars. In our surroundings, we see geometrical shapes everywhere: the sun looks like a circle, a pencil box has the shape of a cuboid, and the edge of a table is straight like a line.
In this chapter, we learn the basic building blocks of geometry: the point, the line, the ray and the line segment. We also study curves, polygons, and the fundamental two-dimensional shapes such as triangles, quadrilaterals and circles. These basic ideas form the foundation for understanding more complex geometrical concepts in higher classes, including angles, symmetry and area.
Geometry also introduces important relationships between shapes. A point is a location with no size, a line extends infinitely in both directions, and a circle is a collection of all points at a fixed distance from a centre. We learn how to name these objects using capital letters and study the parts of a circle like the centre, radius, diameter, chord, arc and sector.
A point is a position or a location with no dimensions. It has no length, breadth or thickness. A point is named by a capital letter. For example, the point P, the point Q. When we mark a dot with a sharp pencil tip on paper, we represent a point.
A line segment is a part of a line with two endpoints. It has a definite length and can be measured. A line segment joining points A and B is written as AB. For example, the edge of a ruler or a stretched thread is a line segment.
A line is a straight path that extends endlessly in both directions. It has no endpoints. A line passing through points A and B is written as line AB. It is drawn with arrows at both ends to show that it extends infinitely.
A ray is a part of a line that has one endpoint and extends infinitely in one direction. If a ray starts at point O and passes through point A, we write it as ray OA. The end point O is called the initial point of the ray. Rays of sunlight and a laser beam are examples of rays.
A curve is a figure drawn on paper without lifting the pencil from the paper. It can be straight or bent. A curve that does not cross itself and has no breaks is called a simple curve. A curve that joins its starting and ending points is called a closed curve. A curve that does not join its endpoints is an open curve.
A polygon is a simple closed figure made up of line segments. The line segments that make a polygon are called its sides. The point where two sides meet is called a vertex. A polygon is named by the number of sides it has.
| Polygon | Number of Sides |
|---|---|
| Triangle | 3 |
| Quadrilateral | 4 |
| Pentagon | 5 |
| Hexagon | 6 |
| Heptagon | 7 |
| Octagon | 8 |
A triangle has three sides, three vertices and three angles. A quadrilateral has four sides, four vertices and four angles. Examples of quadrilaterals include the rectangle, square, parallelogram, rhombus and trapezium.
A triangle is a polygon with three sides. The three line segments that form a triangle are its sides, and the three corners are its vertices. The triangle with vertices A, B and C is written as triangle ABC. A triangle is named after its vertices. A triangle has three angles as well, and the sum of the three angles of a triangle is always 180 degrees.
A quadrilateral is a polygon with four sides, four vertices and four angles. The sum of all four angles of a quadrilateral is 360 degrees.
A circle is a simple closed curve made up of all the points in a plane that are at a fixed distance from a fixed point. The fixed point is called the centre of the circle, and the fixed distance is called the radius.
We can describe the position of a point relative to figures such as curves or polygons.
For example, in a circle, the points at a distance less than the radius from the centre lie in the interior, points exactly at the radius lie on the circle, and points farther than the radius lie in the exterior.
An angle is formed when two rays start from the same point. The common point is called the vertex of the angle, and the two rays are called the arms of the angle. An angle is named by the symbol angle, for example angle ABC means the angle with vertex B and arms BA and BC. Angles are measured in degrees. A complete angle around a point measures 360 degrees, a straight angle measures 180 degrees, and a right angle measures 90 degrees.
| Figure | Definition | Number of Sides |
|---|---|---|
| Triangle | Simple closed figure of 3 line segments | 3 |
| Quadrilateral | Simple closed figure of 4 line segments | 4 |
| Pentagon | Simple closed figure of 5 line segments | 5 |
| Hexagon | Simple closed figure of 6 line segments | 6 |
| Part of Circle | Description |
|---|---|
| Centre | Fixed point equidistant from all points of the circle |
| Radius | Centre to any point on the circle |
| Diameter | Through the centre, joining two points on the circle; twice the radius |
| Chord | Segment joining any two points on the circle |
| Arc | Part of the circumference |
Basic Geometrical Ideas introduces us to the fundamental vocabulary of geometry: points, lines, rays, segments, curves, polygons and circles. We learnt how to name and describe these objects, studied the special features of triangles, quadrilaterals and circles, and understood the relationships among their parts. This foundation will be used in the next chapters on elementary shapes, symmetry and practical geometry.