Rectangle

A rectangle is a shape with four sides and four corners. The corners are all right angles. It follows that the lengths of the pairs of sides opposite each other must be equal.

People make many rectangular things, including most tables, boxes, books, and papers.

A rectangle with all four sides equal in length is called a square.

PROPERTIES OF RECTANGLE:

1. The opposite sides are equal.
2. All angles are equal.
3. Each angle is a right angle.
4. The diagonals are equal in length.
5. The diagonals bisect each other.

FORMULAS OF RECTANGLE:

 Area of Rectangle=length*breadth

                              =l*b sq.units.

Perimeter of Rectangle=2(l+b) units.

Square

A square is a shape with four equal sides and four corners that are all right angles (90 degrees). The diagonals of a square also cross at right angles. The angle between any diagonal and a side of a square is 45 degrees. A square has rotational symmetry of four. It has four lines of regular symmetry.

A square is a type of rectangle with all sides of equal length. However note that while a square is a type of rectangle, a rectangle does not necessarily need to be a square.

A square is also the 2-dimensional analogue of a cube.

PROPERTIES OF SQUARE:

1. All the angles are equal.                                                                    
2. All the sides are of equal length.
3. Each of the angle is a right angle.
4. The diagonal are of equal length and breadth.
5. The diagonals bisect each other at right angles.

FORMULAS:

  Area of square=side*side

                       A=a*a 

                         =a² sq.units

 Perimeter of square=4a units

Circle

A circle is a round, two-dimensional shape. All points on the edge of the circle are at the same distance from the centre.

The radius of a circle is a line from the centre of the circle to a point on the side. Mathematicians use the letter r for the length of a circle's radius. The centre of a circle is the point in the very middle.

The diameter (meaning "all the way across") of a circle is a straight line that goes from one side to the opposite and right through the centre of the circle. Mathematicians use the letter d for the length of this line. The diameter of a circle is equal to twice its radius (d equals 2 times r).

${\displaystyle d=2\ r}$

The circumference (meaning "all the way around") of a circle is the line that goes around the centre of the circle. Mathematicians use the letter C for the length of this line.

The number π (written as the Greek letter pi) is a very useful number. It is the length of the circumference divided by the length of the diameter (π equals C divided by d). As a fraction the number π is equal to about ²²⁄₇ or 335/113 (which is closer) and as a number it is about 3.1415926535.

	${\displaystyle \pi ={\frac {C}{d}}}$
${\displaystyle \therefore }$	${\displaystyle C=2\pi \,r}$

The area, a, inside a circle is equal to the radius multiplied by itself, then multiplied by π (a equals π times r times r).

${\displaystyle A=\pi \,r^{2}}$

Calculating π

π can be measured by drawing a large circle, then measuring its diameter (d) and circumference (C). This is because the circumference of a circle is always π times its diameter.

${\displaystyle \pi ={\frac {C}{d}}}$

π can also be calculated by only using mathematical methods. Most methods used for calculating the value of π have desirable mathematical properties. However, they are hard to understand without knowing trigonometry and calculus. However, some methods are quite simple, such as this form of the Gregory-Leibniz series:

${\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots }$

While that series is easy to write and calculate, it is not easy to see why it equals π. An easier to understand approach is to draw an imaginary circle of radius r centered at the origin. Then any point (x,y) whose distance d from the origin is less than r, calculated by the pythagorean theorem, will be inside the circle:

${\displaystyle d={\sqrt {x^{2}+y^{2}}}}$

Finding a set of points inside the circle allows the circle's area A to be estimated. For example, by using integer coordinates for a big r. Since the area A of a circle is π times the radius squared, π can be approximated by using:

${\displaystyle \pi ={\frac {A}{r^{2}}}}$

REFERENCE FROM WIKIPEDIA

Shapes

A shape is a geometric figure that can be described with mathematics. One way to classify shapes is to describe a bigger kind of shape that the shape is one of. For example, they can be classified by their different numbers of dimensions. Thus, circles are two-dimensional shapes so, like other 2D shapes, they will fit into a flat plane.

Three-dimensional objects like cubes will not fit inside a plane, because they are not flat. Four-dimensional shapes made of polygons are called polychorons, and shapes made of polygons of any dimension are polytopes.

Two shapes are said to be equal, if one can be changed into the other by turning, moving, growing, shrinking, or more than one of these in combination.

2D SHAPES:

• circles
• triangles
• ovals
• quadrilaterals
  • squares
  • rectangles
• octagons
• hexagons
• pentagons
• heptagon

These are two-dimensional shapes or flat plane geometry shapes. Their sides are made of straight or curved lines. They can have any number of sides. Plane figures made of lines are called polygons. Triangles and squares are examples of polygons.

3D SHAPES:

• spheres
• platonic solids
• cubes
• cones
• pyramids
• hemispheres
• cuboids
• cylinders
• prisms

These are three-dimensional shapes. Their sides are made of flat or curved surfaces.

REFERENCE FROM WIKIPEDIA

Three dimension shapes

The shapes which can be measured in 3 directions are called three-dimensional shapes. These shapes are also called solid shapes. Length, width, and height (or depth or thickness) are the three measurements of the three-dimensional shapes. They are different from 2D shapes because they have thickness. A number of examples can be found in everyday life. Some of them are:

Three Dimensional Shapes

The Rubik’s Cube is an example of a cube, the drum is a cylinder, the birthday cap is a cone and the orange is a sphere.

Faces, Edges and Vertices

Three-dimensional shapes have many attributes such as faces, edges and vertices. The flat surfaces of the 3D shapes are called the faces. The line segment where two faces meet is called an edge. A vertex is a point where 3 edges meet.

Let us consider a few shapes to learn about them.

Cube

• All edges are equal
• 8 vertices
• 12 edges
• 6 faces.

Cuboid

• 8 vertices
• 12 edges
• 6 faces

Prism

• 6 vertices
• 9 edges
• 5 faces – 2 triangles and 3 rectangles

Square Pyramid

• 5 vertices
• 8 edges
• 5 faces

Cylinder

• No vertex
• 2 edges
• 2 flat faces – circles
• 1 curved face

Cone

• 1 vertex
• 1 edge
• 1 flat face – circle
• 1 curved face

Sphere

• No vertex
• No edges
• 1 curved face
• 
  
• REFERENCE FROM   https://byjus.com/maths/three-dimensional-shapes/

Polygon

Definition of a Polygon

A polygon is any 2-dimensional shape formed with straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. The name tells you how many sides the shape has. For example, a triangle has three sides, and a quadrilateral has four sides. So, any shape that can be drawn by connecting three straight lines is called a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral.

Polygons

All of these shapes are polygons. Notice how all the shapes are drawn with only straight lines? This is what makes a polygon. If the shape had curves or didn't fully connect, then it can't be called a polygon. The orange shape is still a polygon even if it looks like it has an arrow. All the sides are straight, and they all connect. The orange shape has 11 sides.

I've mentioned a few polygons and have shown you a few common shapes. Here is a list of those in addition to several more:

Shape	# of Sides
Triangle	3
Square	4
Rectangle	4
Quadrilateral	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Nonagon	9
Decagon	10
n-gon	n sides

The last entry includes the general term for a polygon with n number of sides. Polygons aren't limited to the common ones we know but can get pretty complex and have as many sides as are needed. They can have 4 sides, 44 sides, or even 444 sides. The names would be 4-gon, or quadrilateral, 44-gon, and 444-gon, respectively. An 11-sided shape can be called an 11-gon.

Regular Polygons

A special class of polygon exists; it happens for polygons whose sides are all the same length and whose angles are all the same. When this happens, the polygons are called regular polygons. A stop sign is an example of a regular polygon with eight sides. All the sides are the same and no matter how you lay it down, it will look the same. You wouldn't be able to tell which way was up because all the sides are the same and all the angles are the same.

When a triangle has all the sides and angles the same, we know it as an equilateral triangle, or a regular triangle. A quadrilateral with all sides and angles the same is known as a square, or regular quadrilateral. A pentagon with all sides and angles the same is called a regular pentagon. An n-gon with sides and angles the same is called a regular n-gon.

Regular polygons

Here is a regular triangle, a regular quadrilateral, and a regular pentagon. Do you see how all the sides are the same and no matter how you flip it, it will look the same?

Angles of Regular Polygons

Regular polygons also have two different angles related to them. The first is called the exterior angle, and it is the measurement between the shape and each line segment when you stretch it out past the shape.

REFERENCE FROM  https://study.com/academy/lesson/what-is-a-polygon-definition-shapes-angles.html

Geometrical shapes

A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object.^[1] That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape.

Objects that have the same shape as each other are said to be similar. If they also have the same scale as each other, they are said to be congruent.

Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles, squares, and pentagons. Other shapes may be bounded by curves such as the circle or the ellipse.

Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional facesenclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and the sphere.

A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape.

REFERENCE FROM WIKIPEDIA

Mathematical notations

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, limiting mathematical discovery. Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language. Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog. Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas.

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.

Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

REFERENCE FROM WIKIPEDIA

Mathematics as Science

The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery". In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural sciencefollows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as biology, chemistry, or physics. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.

REFERENCE FROM WIKIPEDIA