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AREA AND PERIMETER OF PLANE FIGURES math capsule

Today we are going to learn about the Area and Perimeter of Plane Figures .

These include various figures like, square , rectangle , quadrilateral , parallelogram , rhombus , trapezium , right angled triangle , isosceles triangle , scalene triangle, equilateral triangle , circle , circular ring , semi circle , quadrant of circle , area of sector , regular polygon , etc.

All these are important plane figures and today we are going to study about their area and perimeter and how we can compute them easily .

Tip : get a pen and paper and write each formula for five to six times

so that you may easily remember the formulae of area and perimeter of plane figures .

So, lets begin ,



Area of the plane figure is the amount of surface enclosed by its boundary . It is measured in square units.

SQUARE

Let each side of a square be a unit .

SQUARE


Then,
Perimeter of square = 4 (SIDE) = 4a units
Diagonal of square = √2 (SIDE)= a√2 units
Area of square = SIDE * SIDE = a² sq units = (diagonal)² / 2 = d² / 2
Side of square = √area = √a² sq units

RECTANGLE 

Let l and b be the length and breadth of a rectangle respectively, then 

RECTANGLE


Area of rectangle = Length * Breadth = l * b
Perimeter of rectangle = 2 (length + breadth)
Diagonal of rectangle = √{(length)² + (breadth)²}
Area of track = (L1 *B1 - L2*B2) sq units
RECTANGLE 1

QUADRILATERAL

Let ABCD is a quadrilateral in which DM = h1 and BN = h2 are perpendiculars on diagonal AC from other two vertices B and D, then 

Area of quadrilateral = Diagonal * (h1 + h2) / 2
                                  = AC * (DM + BN) / 2 sq units

QUADRILATERAL


PARALLELOGRAM

Let adjacent sides of a parallelogram are b and a and b is corresponding altitude (height) of side a.

Area of the parallelogram = (Base * Heigtht) = a * b sq units

PARALLELOGRAM


Perimeter of a parallelogram = 2 (Sum of adjacent side) = 2(a + b)  units

Each diagonal of a parallelogram divides it into two triangles of equal area.

RHOMBUS

Let the length of each sides of a rhombus is a and length of both diagonals are d1 and d2 , then

RHOMBUS


Area of rhombus = d1 * d2 / 2 sq units
Side of rhombus =[ √{d1² + d2²} ] / 2
=> 4a² = d1² + d2²
Perimeter of rhombus = 4 * side units

Diagonals of a rhombus bisect each other.


TRAPZEIUM

Let the length of parallel sides of a trapezium are a and  b and distance between them is h , then 

Area of trapezium = (Sum of parallel sides) * (Distance between them) / 2

TRAPEZIUM


= (AB + CD) * h / 2 = (a + b) * h sq units .


RIGHT ANGLED TRIANGLE 

A figure bounded by three straight lines is called a triangle.
Let perpendicular , base and hypotenuse of a right angled triangle (ABC) are p , b and h respectively then,

RIGHT ANGLED TRIANGLE


Perimeter of right angled triangle = AB + BC + CA = b + p + h units
Area of right angle triangle = Base * Altitude / 2


ISOSCELES TRIANGLE

Let sides of an isosceles triangle are a, b and b , then 
isosceles triangle


Perimeter of isosceles triangle = a + b + b = a + 2b units
Area of isosceles triangle = (s - b) (√s(s - a))
where, a = Base and b = Equal sides
Area of a right isosceles triangle , in which equal sides from right angle then
Area = a² / 2 sq units

SCALENE TRIANGLE

Let the sides of a triangle are a, b, c and h be the corresponding height to side a , then 

Perimeter of scalene triangle , 2s = a + b + c
Semi perimeter of scaler triangle = s = ( a + b + c ) / 2
Area of triangle = √s (s - a) (s - b) (s - c)                  [HERO'S FORMULA]
or area of triangle = a * h / 2 
scalene triangle


EQUILATERAL TRIANGLE 

Let a be the side of an equilateral triangle , then 

Height (altitude) of equilateral triangle = a√3 / 2
Area of equilateral triangle = a² √3 / 4
Perimeter of equilateral triangle = 3 * Side = 3a

equilateral triangle


 CIRCLE

Let the radius of a circle be r, then 

circle


Circumference of circle = 2𝝿r , also 2r =D
Area of circle = 𝝿r²
Distance covered be a wheel in one revolution = Circumference of the wheel


CIRCULAR RING

If 'R' and 'r' be outer and inner radii of a ring , then the area of ring = 𝝿(R² - r²) sq units
circular ring



SEMI CIRCLE

A diameter divides a circle into two equal parts . Each of these two arcs is called semi circle.
If r is the radius of a circle , then 

Area of semi circle = 𝞹r² / 2 sq units
Perimeter of semi circle = (𝞹r + 2r) units

semicircle


QUADRANT OF A CIRCLE

If r is the radius of a circle, then 
Perimeter of the quadrant = (circumference of a circle) / 4 + 2r
                                          = 2𝞹r / 4 + 2r
Area of the quadrant = (Area of circle) / 4
                                  = 𝞹r² / 4 sq units
If two diameters are perpendicular to each other , then they divides the circle into four quadrants.


AREA OF SECTOR

If Θ be the angle at the centre of a circle of radius r , then 
area of sector

Length of the arc PQ = 2𝞹rΘ / 360⁰
Area of sector OPRQO = 𝞹r²Θ / 360⁰
Area of minor segment PRQP = Area of sector OPRQO - Area of △OPQ
                                                 = 𝞹r²Θ - r² sin Θ                                                    360⁰      2
Area of major segment QSPQ = Area of circle = Area of minor segment PRQP

REGULAR POLYGON

Let a be the side of a regular polygon.
Then , 
Area of regular polygon = 5√3 a² / 4 sq units
Area of regular hexagon = 3√3 a² / 2 sq units
Area of regular octagon = 2 (√2 + 1) a² sq units

SOME USEFUL RESULTS

* area of room = length * breadth 

* area of 4 walls of a room = 2 (length + breadth) * height 

* radius of circumcircle of an equilateral triangle of side 'a'  = a / √3

* radius of incircle of triangle = ◭ / s , s = (a + b + c) / 2

* angle inscribed by minute hand in 60 min = 360⁰

* angle inscribed by hour hand in 12 h = 360⁰

* angle inscribed by minute hand in 1 min = 6⁰

* distance moved by a wheel in one revolution = circumference of the wheel

* If the length of a square / rectangle is increased by a% and the breadth is increased by y %, the net effect on the area is given by 

net effect = [x + y + xy /100] %

* If the length and breadth of a square / rectangle are increased by x% and the breadth is decreased by y% the net effect on the area is given by

net effect = [x - y - xy /100] %

* If the length and breath of a square / rectangle are decreased by x% and y% respectively, the net effect on the area is given by 

net effect = [-x  -y + xy /100] %

* If the side of a square / rectangle / triangle is doubled the area is increased by 300%, i.e. the area becomes four times of itself.

*If the radius of a circle is decreased by x%, the net effect on the area is (-x² / 100)% , i.e. the area is decreased by (x² / 100)%.

*If a floor of dimensions (l*b) m is to be covered by a carpet of width wm at the rate X rs per metre , then the total amount required is rs (Xlb/w)

*If a room of dimensions (l*b) m is to be proved with square tiles , then 

the side of the square tiel = HCF of l and b 

the number of tiles required = l*b / (HCF of l and b)²

* area of a square inscribed in a circle of radius r is 2r² and the side of a square inscribed in a circle of radius r is √2 r .

* area of the largest triangle inscribed in a semi circle of radius r is r².

Hope you liked our article on how to find the area and perimeter of plane figures.


EXERCISES

area and perimeter 1
area and perimeter 2

area and perimeter 3

area and perimeter 4

area and perimeter 5

area and perimeter 6

area and perimeter 7

area and perimeter 9

area and perimeter 10

area and perimeter 11

area and perimeter 12

area and perimeter 13

area and perimeter 14

area and perimeter 15

area and perimeter 16

area and perimeter 17

area and perimeter 18



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