Geometry

Geometry is the study of different types of shapes, figures and sizes in Maths or in real life. In geometry, we learn about different angles, transformations and similarities in the figures.

Plane Geometry deals with flat shapes which can be drawn on a piece of paper. These include lines, circles & triangles of two dimensions. Plane geometry is also known as two-dimensional geometry.

All the two-dimensional figures have only two measures such as length and breadth. It does not deal with the depth of the shapes. Some examples of plane figures are square, triangle, rectangle, circle, and so on.

The important terminologies in plane geometry are:

• Point
• Line
• Angles

Point

A point is a precise location or place on a plane. A dot usually represents them. It is important to understand that a point is not a thing, but a place. Also, note that a point has no dimension; preferably, it has the only position.

Line

The line is straight (no curves), having no thickness and extends in both directions without end (infinitely). It is important to note that it is the combination of infinite points together to form a line. In geometry, we have a horizontal line and vertical line which are x-axis and y-axis respectively.

Line Segment – If a line has a starting and an endpoint then it is called a Line Segment.

Ray – If a line has a starting point and has no endpoint is called Ray.

Incidence Axioms on Lines

• A line contains infinitely many points
• An infinite number of lines can be drawn to pass through a given point
• One and only one line can be drawn to pass through two given points A and B.

Collinear PointsThree or more points are said to be collinear, if there is a line which contains them all.In the above figure; P, Q, R are collinear points.

INTERSECTING LINES
Two lines having a common point are called intersecting lines. The point common to two given lines is called their point of intersection. In the figure, the lines AB and CD intersect at a point O.

PARALLEL LINES
Two lines l and m in a plane are said to be parallel, if they have no point in common and is written as l || m. The distance between two parallel lines always remains the same.

CURVES
Curves can be defined as figures that flow smoothly without a break. A line is also a curve, and is called a straight curve.
Simple curves
Curves that do not intersect themselves are called simple curves.

Open curves
Curves whose end points do not meet are called open curves.

Closed curves
Curves whose end points join to enclose an area are called closed curves.

For a closed curve, we can identify three regions:

The interior of the curve: These points are in the interior of the closed curve.
Boundary of the curve: These points are on the boundary of the closed curve.
Exterior of the curve: These points are in the exterior of the closed curve.
The interior of a curve together with its boundary is called its “region”.

Angles in Geometry

An angle is made up of two rays starting from a common end point.

In this figure BA  and BC rays have one common end point, that is, B. The rays BA and BC are called the arms or sides of the angle. The common end point B is the vertex of the angle.
We name the above angle as BAC.

Types of Angle

Acute Angle – An Acute angle (or Sharp angle) is an angle smaller than a right angle ie. it can range between 0 – 90 degrees.

Obtuse Angle – An Obtuse angle is more than 90 degrees but is less than 180 degrees.

Right Angle – An angle of 90 degrees.

Straight Angle – An angle of 180 degrees is a straight angle, i.e. the angle formed by a straight line

Polygons in Geometry

A plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

The name ‘poly’ refers to multiple. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon polygon.

General Formula for Sum of internal Angles of a polygon – (n - 2) x 180

Types of Polygon

The types of polygons are:

• Triangles
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon

Triangle : A 3-sided polygon whose sum of internal angles always sums to 180 degrees.

Types Triangle :

• Equilateral Triangle – Has 3 equal sides and angles.
• Isosceles triangle –  Has 2 equal sides and angles.
• Scalene triangle – Has all the 3 unequal sides and angles.

Quadrilateral :  A 4-sided polygon with four edges and four vertices. Sum of internal angles is 360 degrees.

• Square – Has 4 equal sides and vertices which are at right angles.
• Rectangle – Has equal opposite sides and all angles are at right angles.
• Parallelogram – has two pairs of parallel sides. The opposite sides & opposite angles are equal in measure.
• Rhombus – Has all the four sides to be of equal length. However, they do not have its internal angle to be 90 degrees
• Trapezium – Has one pair of opposite sides to be parallel.

Pentagon : A plane figure with five straight sides and five angles

Hexagon : A plane figure with six straight sides and six angles

Heptagon : A plane figure with seven sides and seven angles

Octagon : A plane figure with eight straight sides and eight angles.

Nonagon : A plane figure with nine straight sides and nine angles.

Decagon : A plane figure with ten straight sides and ten angles.

Circle in Geometry

A circle is formed by a point moving at the same distance from a fixed point. The fixed point is the centre of the circle. A circle is also a simple closed curve however, it does not have any sides or angles.

The fixed point O is the centre of the circle.
The fixed distance OP = OQ is the radius of the circle.
The distance around the circle is its circumference.

Circumference
The line that forms the boundary of a circle is called its circumference. The part enclosed by the circumference of a circle is called the interior of the circle. The part left outside the circle is said to be the exterior of the circle. Some points may lie on the circumference of the circle.
A line segment that joins the centre of the circle and a point on the circumference is called the radius of the circle. The radius of a circle is half of the diameter.
Chord
A chord is a line segment joining two points that lie on a circle.
Diameter
A chord passing through the centre of the circle is called its diameter. A diameter is the longest chord of a circle.
Arc
An arc is a part of the circumference of a circle.

Sector
The part of a circle enclosed by two radii and an arc is called a sector.
Segment
The part of a circle that is enclosed by a chord and an arc is called a segment of the circle.

Semi-circle
A diameter of a circle divides it into two halves. Each half is called a semi-circle.

Similarity and Congruency in Geometry

Similarity – Two figures are said to be similar if they have the same shape or have an equal angle but do not have the same size.

Congruence –  Two figures are said to be Congruent if they have the same shape and size. Thus, they are totally equal.

Solid Geometry (Three-dimensional geometry)

Solid Geometry deals with 3-dimensional objects like cubes, prisms, cylinders & spheres. It deals with three dimensions of the figure such as length, breadth and height. But some solid solids do not have faces (e.g. sphere).

Solid geometry is the study of three dimensions in Euclidean space. The objects which are around us are three-dimensional. All the three-dimensional shapes are obtained from the rotation operation of 2D shapes. The important attributes of 3D shapes are:

• Faces
• Edges
• Vertices

Edges

An edge is defined as the line segment on the boundary that joins one vertex to the other vertex. It means that it joins one corner point to the other. It forms the skeleton of 3D shapes.  In other words, it can be defined as the faces, that meet in the straight line is called edge. Following are the list of edges for the different solid shapes:

Triangular Prism : 9

Cube : 12

Rectangular prism : 12

Pentagonal Prism : 15

Hexagonal Prism : 18

Triangular Pyramid : 6

Square Pyramid : 8

Pentagonal Pyramid : 10

Hexagonal Pyramid : 12

Faces

We know that all the geometric shapes are made up of flat surface called faces. It is a flat surface enclosed by the edges. For any three-dimensional shapes, the face should be a two-dimensional figure. The list of the number of faces for different solid shapes are given below:

Triangular Prism : 5

Cube : 6

Rectangular prism : 6

Pentagonal Prism : 7

Hexagonal Prism : 8

Triangular Pyramid : 4

Square Pyramid : 5

Pentagonal Pyramid : 6

Hexagonal Pyramid : 7

Vertices

A vertex is defined as the point where the edges of the solid figure meet at each other. In other words, it can be said that, the point where the adjacent sides of the polygon meet. The vertex is the corners where the edges meet. The number of vertices for different solid shapes in  geometry is as follows:

Triangular Prism : 6

Cube : 8

Rectangular prism : 8

Pentagonal Prism : 10

Hexagonal Prism : 12

Triangular Pyramid : 4

Square Pyramid : 5

Pentagonal Pyramid : 6

Hexagonal Pyramid : 7