Geometry: Branches and Formulas

Geometry is the branch of mathematics that deals with measurements, properties, as well as relationships of lines, points, angles, surfaces, solids, and a variety of shapes we see in everyday life.

• There are two-dimensional shapes and three-dimensional shapes in Euclidean geometry.
• Plane geometry covers 2D shapes such as triangles, squares, rectangles, circles, which are also called flat shapes.
• Solid geometry comprises 3D shapes such as a cube, cuboid, cone, etc. which are also called solids.
• Basic geometry is based on points, lines, and planes which are explained in coordinate geometry.

These are all parts of geometry. Geometry simply deals numerous shapes and figures, and their measurements.

Key Terms: Geometry, Plane Geometry, Three-dimensional shapes, Solids, Surface area, Volume, Angles, Polygons, Euclidean Geometry

---

What is Geometry?

[Click Here for Sample Questions]

Geometry is “a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids.” To put it in a simple way, it is the study of different types of shapes, figures and sizes both in Maths and in real life. We learn, in geometry, about different angles, transformations and similarities in the figures.

Geometry can be classified into three types:

• Euclidean
• Hyperbolic
• Elliptical

Read Also: Analytical Geometry

---

Branches of Geometry

[Click Here for Previous Year Questions]

The branches of geometry can be divided into:

• Algebraic geometry: It is a branch of mathematics, that studies the zeros of multivariate polynomials. Modern algebraic geometry deals with the use of abstract algebraic techniques, particularly commutative algebra, which helps to solve geometrical problems about these sets of zeros.
• Discrete geometry: It deals with the relative position of simple geometric objects, such as points, lines, triangles, circles and so on.
• Differential geometry: It covers the techniques of algebra and calculus for problem-solving, where the various problems include general relativity in physics etc.
• Euclidean geometry: It is the study of plane and solid figures that are based on axioms and theorems including points, lines, planes, angles, congruence, similarity and solid figures. It is vastly used in Computer Science, Modern Mathematics problem-solving. 
• Convex geometry: It deals with the convex shapes in Euclidean space using techniques of real analysis. It has a wide application in optimization and functional analysis in number theory.
• Topology: It is the study of the properties of space under continuous mapping. It is widely used in the consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches and hyperspace topologies. 

Branches of Geometry

---

Geometry Formulas

[Click Here for Sample Questions]

Geometry formulas are important to calculate the area, perimeter, volume, and different measures using the length, breadth, and height of different geometrical figures. By using the geometry formulas, we can calculate the physical characterises of the geometries easily. 

Some of the Geometry formulas include:

• Perimeter of a Square = 4 (Side)
• Perimeter of a Rectangle = 2(Length + Width)
• Area of a Rectangle = Length × Width
• Area of a Square = Side²
• Area of a Triangle = ½ × base × height
• Area of a Trapezoid = ½ × (base 1 + base 2) (base 1 + base 2) × height

Basic geometry formulas by using the mathematical constant π has been used:

• Area of Circle = A = π × r² 
• Circumference of Circle = 2πr
• Curved Surface Area of Cylinder (CSA)  = 2πrh
• Total Surface Area of Cylinder (TSA) = 2πr (r + h)
• Volume of a Cylinder = V = πr²h
• Volume of a Cone = V = ⅓ × πr²h
• Curved Surface Area of a Cone (CSA) =  πrl
• Total Surface Area of a Cone (TSA) = πr(r+l) = πr [r + √(h² + r²)]
• Surface Area of Sphere = S = 4πr²
• Volume of Sphere = V = 4/3 × πr³

Note: Here, r = Radius; h = Height. and, l = Slant height

Read More:

---

Euclidean Geometry

[Click Here for Previous Year Questions]

Euclidean geometry is the study of planes and solid figures based on axioms (a statement or proposition) and theorems. Some of the major concepts of Euclidean geometry comprise of Points and Lines, Euclid’s Axioms and Postulates, Geometrical Proof, and Euclid’s Fifth Postulate. The five postulates of Euclidean Geometry that help to define geometrical figures are:

• A straight line segment can be drawn from one given point to any other.
• A straight line can be extended indefinitely in either direction.
• A surface comes with only two dimensions: length and width.
• All right angles are proved to be congruent.
• Any two straight lines are considered to be infinitely parallel and are equidistant from each another at two points.

Euclid’s Axioms

Axiom 1: Objects or things that are equal to the same thing are equal to one another.

Consider that the area of a given rectangle is equal to a triangle's area, which is equal to a square's area. Thus, it can be said that the area of the triangle and the square are equivalent.

Axiom 2: In case the equals are added to equals, the wholes become equal.

Assume any given line segment AB, where AP is equivalent to QB. When PQ is used on each side, AP + PQ = QB + PQ, i.e. AQ = PB, as per the axiom 2.

Axiom 3: If equals are subtracted from equals, the remainder is equal.

In two rectangular shapes, ABCD and PQRS are of the same size. If triangle XYZ is removed from both rectangles, the areas of the remaining sections of the two triangles become identical.

Axiom 4: Things which coincide with one another are equal to one another.

By assuming a line segment AB with C in the middle. The line segment AB has been intersected by AC + CB. It can be said that AC + CB = AB.

Axiom 5: The whole is greater than the part.

Assume that AC is considered to be a portion of AB. Thus, it can be said that AB > AC.

Axioms 6: Things which are double of the same things are equal to one another.

Axiom 7: Things which are halves of the same things are equal to one another.

Non-Euclidean geometry

Non-Euclidean geometry can be expressed as a sub-discipline of geometry.

• Everything which does not follow Euclidean geometry is non-Euclidean geometry.
• It is used to explain spherical and hyperbolic geometry.
• Because spherical geometry has been termed non-euclidean, it is important to adjust real lengths, point locations, region area, and actual angles to convert it to Euclid's geometry or fundamental geometry.

---

Plane Geometry

[Click Here for Sample Questions]

Plane Geometry can be defined as the two-dimensional shapes which can be drawn on a piece of paper. All the two-dimensional figures have only two measures such as length and breadth. Some examples of plane figures are squares, triangles, rectangles, circles, and so on.

The important terminologies related to plane geometry are:

• Point: In geometry, the point is defined as a location and no size, and is represented by a dot.  Also, a point has no dimension, it only has a position.
• Line: A line is a group of points that extends infinitely in two directions. Moreover, it is straight with no curves and has no thickness. It is important to note that it is the combination of infinite points together to form a line. 
   

Components of Plane Geometry

• Angles: An angle, in a planar geometry, is the figure formed by two rays, known as the sides of the angle, sharing a common endpoint, called the vertex of the angle. There are four types of angles. 
  • Obtuse angle: Here the angle between the two rays is more than 90⁰.
  • Acute angle: Here the angle between the two rays is less than 90⁰.
  • Right angle: In a right angle, the angle between the two rays is exactly 90⁰.
  • Straight angle: The angle between the two rays is 180⁰ in a straight angle. 

Types of Angles

Polygons in Geometry

In geometry, a polygon is referred to as a flat or plane, two-dimensional closed shape with straight sides. The name ‘poly’ refers to multiple, that do not have curved sides. Polygons can be of two types: Regular Polygons – Polygons that have equal sides and angles are regular polygons. 

Types of Polygon

Depending on the number of sides, polygons can be classified into many types as given in the image below.

Polygon Types

Circles in Geometry

A circle can be referred to as the set of all points in the plane that are at a fixed distance (radius) from a fixed point (the center). Some of the terminilogies related to circle are:

• The line that joins a point on the circle to the center is called a radius.
• Line joining two points on the circle is called a chord.
• Diameter is a line that connects two points of the circle and passes through the centre. Diameter is the largest chord.
• A secant is a line that cuts the circle at any two points.

Circle

---

Solid Geometry

[Click Here for Previous Year Questions]

Solid geometry or stereometry is the study of three dimensions in Euclidean space and deals with 3-dimensional objects such as cubes, prisms, cylinders, and spheres. Moreover, it deals with three dimensions of the figure such as length, breadth, and height.

However, some solids do not have faces like spheres. Three-dimensional objects can be found all around us, which are obtained from the rotation operation of 2D shapes. The important properties of 3D shapes are:

• Faces: Faces are flat surfaces that geometric shapes are made up of. It is important to note that for any three-dimensional shape, the face should be a two-dimensional figure.
• Edges: It is referred to as the line segment on the boundary that joins one vertex to the other vertex. Meaning, it joins one corner point to the other, forming the skeleton of 3D shapes. 
• Vertices: A vertex is a point where the edges of the solid shape intersect at each other. In other words, we can say that, the point where the adjacent sides of the polygon meet, is called the vertex. In geometry, the number of vertices for different solid shapes are listed below:

In three-dimensional geometry, the following can be demonstrated:

• Skew Lines in Geometry: \(\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_1^2+b_1^2+c_1^2}} \)
• Angle between two lines: \(\dfrac{\vec b_1 . \vec b_2}{|\vec b_1| |\vec b_2|} \) (if \(\vec r = \vec a_1+ \lambda \vec b_1 \) and \(\vec r = \vec a_2+ \lambda \vec b_2 \))

---

Things to Remember

• Geometry is a mathematical branch dealing with measurements, properties, and the relationships of lines, points, angles, surfaces, solids, and other shapes.
• Geometry can be further classified into: Euclidean, Hyperbolic and Elliptical Geometry
• Euclidean geometry can be defined as the study of planes and solid figures based on axioms (a statement or proposition) and theorems.
• The branches of geometry are Algebraic geometry, Discrete geometry, Differential geometry, Euclidean geometry, Convex geometry, and Topology.
• The three important properties of 3-dimensional geometry are Faces, Edges and Vertices.

​Read More:

---