2-D Geometry

Square : 

In Geometry, Square refers to a two-dimensional figure with four equal sides and 4 equal right angles. it is a regular quadrilateral. A square with vertices P ,Q,R & S would be denoted as PQRS.

The Square is additionally both a Rhombus (Equal Sides) and a Rectangle (Equal Angles) and so consists of properties which are present in both these shapes. In other words,

Square = Rectangle ( all angles are same ) + Rhombus (all sides are same)

Bisection of a square by a diagonal leads to two right triangles.

If the length of the side of a square is s, then the area of the square is s2, or “s squared.” From this relation formulated the algebraic use of the term square, which denotes a product that results from multiplying any algebraic expression by itself.

Formula for Calculating Area of a Square:

A =s2

The area of a square is that a product of the length of its sides.

Perimeter of a Square is P = 4s

Quadrilateral: Square

Rectangle :

A rectangle is a flat shape with straight sides where every angle is a right angle and the opposite sides are parallel and of equal length.

A  Rectangle normally refers to a quadrilateral with four right angles. A simple rectangle is a special case of a parallelogram, which has two pairs of parallel opposite sides. For Example, a square is a special case of rectangle.

The area and perimeter of the Rectangle can be calculated by using the following formulas:


Area  = length x width

A = l.w
p = 2l + 2w

Where,
a = area
l = length
w = width
p = perimeter

Quadrilateral: Rectangle

Triangle :

Triangle is one of the basic shapes of geometry and is formed by connecting three points not in a straight line but by straight line segments. It is a geometric figure having three angles and three sides. A triangle is also a closed figure with three sides. A triangle has vertices A, B, and C is denoted ABC.

There are seven types of triangles depending upon their shapes and the degree of their angles. Triangles have some Unique properties which are common to all types. The three most common properties are:

1. The sum of interior angles of a triangle is always equal to 180°.

2. The sum of exterior angles of a triangle is always equal to 360°.


3. The sum of any two side is always greater than remaining side.

The area and the perimeter of a triangle can be calculated by using the following geometric formula:


Area of a triangle :
A = ½ × Base × Height

Perimeter of a triangle :
P = Sum of three sides

triangle

Equilateral Triangle :

An equilateral triangle is the one which has three equal sides and is a regular polygon, in this case with three sides. It also has three equal angles and therefore can be also termed as an equiangular triangle. All the features and properties described for regular polygons apply to an equilateral triangle.

Properties:

All three angles of an equilateral triangle are must be 60°.

Since the angles are the same and the internal angles of any triangle always sum 180°, each is 60°.

Parallelogram :

Parallelogram is a quadrilateral which has both pairs of opposite sides parallel and equal in length. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are equal.  It is similar to other quadrilaterals like the square and the rhombus.

Properties of a Parallelogram:

Any side can be considered as a base for calculating the area


The altitude or height of the parallelogram is perpendicular from the base to the opposite side


The area of a parallelogram can be calculate by multiplying a base by the corresponding altitude


The perimeter of a parallelogram is the distance around it or the sum of its sides

In the figure, angles are the same, and angles are the same

The Area and Perimeter of the Parallelogram can be derived by using the following formula:

Area: A = bh

where  b = base 

 and

h = altitude ( perpendicular height from the base )

Quadrilateral: parallelogram

Trapezoid :

Trapezoid is a quadrilateral which has at least one pair of parallel sides. It is called Trapezium in Britain and Trapezoid in North America. The trapezoid has 2 sides that are parallel lines.

Basic properties of Trapezoid:

Every trapezoid has two bases
Every trapezoid have two legs
The height of the trapezoid is the perpendicular distance from one base to the other

Trapezoid

Trapezium/Trapezoid Formula:

Area of Trapezium = ½×(a + b)×h

where
a, b = sides, h = height

Perimeter of Trapezium = sum of all sides 
p = a + b + c + d
where
a, b, c, d = sides

Circle : 

According to the Euclidean Geometry, a circle is a simple shape consisting of those points in a plane which is equidistant from its centre. The common distance of the points of a circle from its center is named its radius.

Circle

Components of a Circle:

A circle may be a shape with all points an equivalent distance from its center.

A circle is named by its center
The distance across a circle through the middle is named the diameter.

Every diameter may be a chord, but not every chord may be a diameter 
The radius of a circle is that the distance from the middle of a circle to any point on the circle. 
chord may be a line segment that joins two points on a curve. In geometry, a chord is usually wont to describe a line segment joining two endpoints that lie on a circle.
Area of Circle can be calculated by using the following formula:

Area :

 A = r2

A sector of a circle :

In the simplest form of definition, sector of a circle is a pie shaped portion of the area of the circle. It is a region bound by two radii and an arc lying between two radii.

Area of Circle Sector: (with central angle )

    If the angle  is in degrees, then area = (θ /360) x r2

    If the angle  is in radians, then area = (( θ/(2PI)) x r2

sector of circle

Ellipse :

An ellipse is defined by two points, each called a focus. If we take any point on the ellipse, the sum of the distances to the focus points is constant. The size of the ellipse is determined by the sum of these two distances. The sum of these distances is equal to the length of the major axis (the longest diameter of the ellipse).

AB = Major axis

CD = Minor axis

F1 and F2 are the focus of ellipse

The area of an ellipse is A=pi ab.

ellipse

Annulus :

Annulus can be defined as the area of the ring-shaped space between two concentric circles that define it.

The area of Annulus can be derived by using the formula mentioned underneath:

         where:

                                          R is the radius of the outer circle

                                          H  is the radius of the inner ‘hole’

                                          π  is Pi, approximately 3.142

This simplifies a little to:

  A = ( R2 – r2)

Polygon :

The word Polygon is derived from the Latin word polygonum and the Greek word polygonan. In the field of geometry, a regular polygon is a typical plane figure that is bound by a closed path or circuit. All the sides of a regular polygon are of equal length and all the interior angles of the same measurement.

Some of the well known polygons are the triangles, quadrilaterals, hexagon, pentagon etc.

regular polygon

For a regular polygon with n sides of length s, the world is given by:

Regular Pentagon

Regular Polygon Formulas :

A =(1/2) ap

p = perimeter ( sum of all sides )

n = number of sides
s = length of a side
r = apothem (radius of inscribed circle)