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 .

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 

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

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

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

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

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

= (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,

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 

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 

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

 CIRCLE

Let the radius of a circle be r, then 

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

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

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 

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

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