Geometry: Definition, Types and Formulas for Various Geometrical Objects

Geometry is the branch of mathematics in which we study different kinds of figures (dimensionless, one dimensional, two dimensional and three dimensional) and their properties. Questions related to geometry judge the visual ability along with the analytical skill of a candidate. The various types of shapes in geometry enable us to understand the different figures in mathematics as well as our day to day life objects. Also, with the help of different geometric terms, you can find the area, perimeter, angle, sides, volume and other parameters of a given shape. These topics are frequently asked in examinations like SSC JE, and SSC CGL, followed by banking exams like SBI PO, SBI Clerk, IBPS PO, IBPS Clerk, etc.

In this article, you will learn about the key concepts of geometry followed by shapes like triangles, quadrilaterals, polygons, circles, and their types with images. The topic also covers the basic geometry formulas followed by the advanced ones. Read the article thoroughly to clear all the doubts regarding the same.

Branches of Geometry

Let us start the discussion with the different branches of geometry and learn about each of them.

Euclidean Geometry: Euclid’s geometry or the Euclidean Geometry deals with the study of geometrical shapes both two-dimensional and three-dimensional, along with the relation between these figures in terms of lines, and angles, congruence points, and surfaces. Euclid’s book on elements gave an introduction to axioms and different postulates for solid & plane figures that helped in describing geometric shapes.

Non-Euclidean Geometry: As the name suggests, it is the branch of geometry that includes everything that does not fall under Euclidean geometry. It is also known as spherical geometry and hyperbolic geometry.

Algebraic Geometry: It is a branch of geometry which deals with curves or surfaces and includes linear & polynomial algebraic equations that are used for solving the sets of zeros.

Projective Geometry: Projective geometry is a branch of geometry that deals with the connections between geometric shapes that are a result of the projection of the object onto another surface.

Discrete Geometry or Combinatorial Geometry: It is something which relates to the study of the geometric objects which are discrete by nature such as lines, triangles, points, circles etc.

Differential Geometry: It is related to general relativity in physics and uses algebra techniques and calculus for solving problems.

Topology: It deals with the properties of space under continuous mapping. It has applications in various fields such as metric spaces, initial & final structure, continuity, proximal continuity, proximal spaces etc.

Also, read about Hyperbola here.

Dimensions of Geometry

In the previous header we saw the geometry basics branches, let us now understand the different dimensions of geometry in mathematics. In mathematics, objects can be categorised into no dimension objects, one-dimensional, two-dimensional and three-dimensional objects.

Non-Dimensional Geometry

A point can be visualised as a single spot or a place on a plane. It is usually specified by a dot that has no real size or shape. Hence point geometry has no dimension or we can say that it has the only position.

One Dimensional Geometry

The line is straight and the briefest distance between two points. That is we can say that the number of points when connected makes a line geometry. As lines only possess length and no width, therefore it is counted in one-dimensional shapes/objects. The different types and terms related to lines are as follows:

Types	Definition	Representation
Straight Line	A figure formed by joining two or more collinear points is called a straight line. The length of the straight line is not finite.
Line Segment	When we join two fixed points by a straight line, then the figure so formed is called a line segment. The length of a line segment is finite i.e. it can be measured.
Ray	A straight line originating from one point is called a ray.
Intersecting Lines	When two or more straight lines meet at a point then these lines are called intersecting lines.
Transversal Lines	A line is said to be transversal if it intersects two or more lines.
Parallel Lines	Two or more lines are said to be parallel lines if they do not intersect each other.
Perpendicular Lines	Two lines are said to be perpendicular lines if the angle between them is 90°.
Concurrent Lines	Two or more lines are said to be concurrent if they all pass through a fixed point.

Angles and Types of Angles in Geometry

When two rays originate from a fixed point, then the amount of rotation from one ray to another ray is called the angle between the rays or the angle between lines. The rays are called arms of the angle. Angles are generally measured in degrees or radians. The different types of geometry angles with their definition and representation are as follows:

Types	Definition	Representation
Acute Angle	An angle whose measurement is between 0° to 90° is called an acute angle. Here, 0° <∠AOB< 90°
Right Angle	An angle whose measurement is 90° is called the right angle. Here, ∠AOB = 90°
Obtuse Angle	An angle whose measurement is in between 90° to 180° is called an obtuse angle. Here, 90° <∠AOB< 180°
Straight Angle	An angle whose measurement is equal to 180° is called a straight angle.
Reflex Angle	An angle whose measurement is between 180° to 360° is called a reflex angle.
Complete Angle	An angle whose measurement is 360° is called a complete angle.

Two-Dimensional Geometry

Plane geometry or two-dimensional geometry includes flat figures like circles, rectangles, triangles and other polygons. The two-dimensional objects can be drawn on paper and they hold both length and width. The different two-dimensional objects with definitions and images are as follows:

Polygons

A polygon in maths is said to be a two-dimensional shape. There are different names as per the sides:

Triangle: a polygon formed by three line segments.

Quadrilateral: a polygon formed by four line segments.

Similarly, there are different other types of polygons. Refer to the below image for the same.

Learn more about Mensuration in 2D.

Triangle

A triangle is a polygon with 3 vertices say P, Q, and R which is represented as △PQR. The different types of triangles are listed in the below table.

Types of Triangles based on the Sides

Triangle can be classified into the following groups on the basis of the side of the triangle.

Type	Definition	Representation
Scalene Triangle	A triangle whose all sides are different in length is called a scalene triangle.
Isosceles Triangle	A triangle whose two sides are equal is called an isosceles triangle.
Equilateral Triangle	A triangle whose all the sides are equal is called an equilateral triangle.

Also, read about Area of a Triangle here.

Types of Triangles based on the Angles

Triangles in geometry can also be classified into the following groups on the basis of the measurement of the angle of a triangle.

Type	Definition	Representation
Acute Angled Triangle	A triangle whose all angles are less than 90° is called an acute angle triangle.
Right Angled Triangle	A triangle whose one angle is 90° is called a right-angled triangle.
Obtuse Angled Triangle	A triangle whose one angle is more than 90° is called an obtuse-angled triangle.

Check out this article on the properties of triangles.

Quadrilaterals

A figure formed by a four-line segment is called a quadrilateral. The sum of angles of any quadrilateral is 360°. That is if the four angles of a quadrilateral are ∠A, ∠B, ∠C, ∠D. Then ∠A + ∠B + ∠C + ∠D = 360°.

The different types of quadrilaterals are listed in the below table.

Type	Definition	Representation
Parallelogram	A quadrilateral is said to be a parallelogram whose opposite sides are parallel and equal to each other.
Rectangle	A parallelogram is said to be a rectangle if the measurement of each angle is 90°.
Square	A parallelogram is said to be a square if all sides are equal and the measurement of each angle is 90°.
Rhombus	A parallelogram is said to be a rhombus if all sides are equal and the angle between the diagonal is 90°.
Trapezium	A quadrilateral is said to be a trapezium if it’s one pair of opposite sides is parallel to each other.

The other types of polygons are:

Type	Definition	Representation
Pentagon	A polygon formed by five line segments.
Hexagon	A polygon formed by six line segments.
Heptagon	A polygon formed by seven line segments.
Octagon	A polygon formed by eight line segments.
Nonagon	A polygon formed by nine line segments.
Decagon	A polygon formed by ten line segments.

Also, learn about Parabola here.

Circles

It is the locus of all the points which lie in a plane such that their distances from a fixed point are always constant. The fixed point is called the center of the circle. A circle of center O and radius r is shown below.

The different parts of a circle are listed below:

Parts of Circle	Definition	Representation
Radius of a Circle	The distance between the center and any point present on the circumference of the circle is called the radius of a circle. Usually, the radius of a circle is denoted by r or R.
Chord of a Circle	The line segment formed by joining any two points present on the circumference of the circle is called the chord in the geometry of the circle.
Diameter of a Circle	The longest chord of a circle is called the diameter of a circle. OR The chord that passes through the center of the circle is called the diameter of the circle.
Secant of Circle	A line that intersects a circle in two distinct points is called a secant of the circle.
Tangent of a Circle	A line that touches a circle at only one point is called a tangent of the circle.
Sector of a Circle	A region of a circle bounded by two radii and an arc is called a sector of a circle.

Also, learn about the Area of a Circle.

Three-Dimensional Geometry

Solid geometry or three-dimensional geometry includes objects like cubes, cuboids, prisms, cylinders, cones and spheres. All these objects hold a length, width, and height. The important characteristics of solid geometry are edges, faces and vertices. Let us understand each of them in detail.

Edges

Edges in a 3D shape join one corner point to another corner point. That is it is the line segment on the boundary that connects one vertex to the other one. They serve as the junction of two faces.

Faces

A face for 3D objects refers to the flat or curved surface of an object. For a 3D object, the face is in 2D format. A particular shape can have multiple faces.

Vertices

A point in 3D objects where two or more lines meet is said to be the vertex. Or one can understand vertices as the point of intersection of edges.

Check out this article on Mensuration 3D.

Measurement in Geometry

So far we read about the geometry definition, different branches, dimensions and related shapes with their images. Let us now understand the geometry formulas under the header measurement in geometry. This involved formula for the length, area, perimeter and volume calculation of different objects.

Measurement in Two-Dimensional Geometry

The area and perimeter are determined using the length and bread of different geometrical objects. Some of the important geometry symbols formulas related to 2D are listed below.

Plane Geometry Objects	Formula	Representation
Triangle	Perimeter, P = a + b + c Area \(A=\frac{1}{2}bh\) Here, a, b and c are the sides of a triangle and h is the height.
Right Triangle	Perimeter = \(a+b+\sqrt{\left(a^2+b^2\right)}\) Here, ‘a’ and ‘b’ are the sides and ‘c’ is the hypotenuse. Area = \(A=\frac{1}{2}bh\)
Circle	Circumference = 2πr Area = \(πr^2\) Here, r is the radius of the given circle.
Rectangle	Perimeter = 2(l + w) Area = lw Here, l is the length and w is the width of the given rectangle.
Parallelogram	Perimeter, P = 2(a + b) Area, A = bh Here, a and b are the adjacent sides of a parallelogram, and h is the height.
Square	Perimeter, P = 4a Area, A = \(a^2\)

Analytic geometry in two-dimensional deals with the section formula, distance formula, the centroid of a triangle, the midpoint formula and so on. All these are determined using knowledge of coordinate geometry concepts.

Similarity and Congruency in Geometry

When two figures hold an identical shape or an equal angle but do not have the same size are said to be similar. If two figures have the same shape and size, then they are said to be congruent.

Measurement in Three-Dimensional Geometry

The three-dimensional geometry objects or shapes as per the name are represented using 3 coordinates namely x, y and z. The formula of some important and relevant solid geometry are listed below.

Solid Geometry Objects	Formula	Representation
Cube	Lateral area of a cube: \(4a^2\) Total surface area of a cube=\(6a^2\) Volume, V = \(a^3\)
Cuboid	Surface Area, A = 2(lb + bh + hl) Volume, V = lbh
Cone	Total Surface Area, A = πr(r+l)= \( πr^2 + πrl\) Curved Surface Area = πrl Volume, V = \(\frac{1}{3}\pi r^2h\)
Cylinder	Total Surface Area, A = \(2πrh + 2πr^2\) Curved Surface Area = 2πrh Volume, V = \(πr^2h\)
Sphere	Surface Area, A = \(4πr^2\) Volume, V = \(\frac{4}{3}\pi r^3\)

Know more about Family of Lines here.

Some other related concepts of solid geometry are listed below.

Direction Cosines of a Line

If a directed line say ‘R’ departing through the origin creates angles α, β and γ with x, y and z-axes, respectively, then the cosine of these angles, i.e., cos α, cos β and cos γ is termed as direction cosines of the given directed line ’R’. Direction cosines of the line joining two points: \(P(x_1,y_1,z_1)\text{ and }Q(x_2,y_2,z_2)\) is given by the formula:

\(\frac{x_2-x_1}{PQ},\ \frac{y_2-y_1}{PQ},\ \frac{z_2-z_1}{PQ}\)

Also, read about the General Equation of a Line.

Direction Ratios of a Line

The directional ratios of a given line are the digits or numbers that are proportional to the direct cosines of that line. If a, b, c as the direction ratios of a line and l, m and n be the direction cosines then:

\(l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\pm\frac{b}{\sqrt{a^2+b^2+c^2}},n=\pm\frac{c}{\sqrt{a^2+b^2+c^2}}\)

Equation of Line in 3-D Geometry

If l, m, n are the direction cosines of a line passing through the point \(\left(x_1,y_1,z_1\right)\), then the equation of the line is as follows:

\(\left(\frac{\left(x-x_1\right)}{l},\ \frac{\left(y-y_1\right)}{m},\ \frac{\left(z-z_1\right)}{n}\right)\)

The equation of a line passing through the two points say; \(\left(x_1,y_1,z_1\right)\) and \(\left(x_2,y_2,z_2\right)\) . Then the equation of the line is given by the formula:

\(\left(\frac{x-x_1}{x_2-x_1},\ \frac{y-y_1}{y_2-y_1},\ \frac{z-z_1}{z_2-z_1}\right)\)

Angle Between Two Lines

The angle θ between two lines whose direction ratios are proportional to \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) respectively is given the formula:

\(\cos\ \theta=\ \left|\frac{\left(a_1a_2+b_1b_2+c_1c_2\right)}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|\)

We hope you found this article regarding Geometry informative and helpful, and please do not hesitate to contact us with any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation.

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