Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Area Related to Plane Figures

Chapter 12 Class 10

Area related the combination of plane figures like triangles, circles and quadrilaterals and important results related to the topic. Basic points related to the plane figures class 10

Main points to be discussed here 

  • Plane Figure (Triangle, Quadrilaterals and Circles)
  • Area and perimeter of triangles, Heron's Formula to find the area of triangle
  • Different types of quadrilaterals and formulas of finding their perimeter and area.
  • Area and perimeter of circle, semi-circle and quadrant.
  • Length of arc, area of sector and area of segment of a circle.

Two dimensional figures are called plane figures. Like  Triangle, Quadrilateral, and Circle  all are called plane figures. 
For example: Table top is a rectangle, when a sphere is cut by a plane then a circle is formed.
We can draw the plane figures but cannot hold these in our hands.

Triangle is a plane figure which has three sides, three vertices and three edges. Triangle is a type of polygon.
\[Perimeter\: of\: triangle=sum \: of \: all\: sides\]\[Perimeter\: of\: Equilateral \: Triangle=3\times side\]\[Area\: of \: Triangle=\frac{1}{2}\times \: base\times height\]\[Area\: of \: Equilateral \: \: Triangle=\frac{\sqrt{3}}{4}\times \left ( side \right )^{2}\]
This formula is specially used when all three sides of triangle are given.
Where  s = semi-perimeter of triangle and a, b, c  are the sides of triangle
Quadrilateral is a plane figure which have four sides, four angles and four vertices. It is a type of polygon with four sides. Mainly quadrilaterals are of 6 types. 
Like : Trapezium, Parallelogram, Rectangle, Rhombus, Square and Kite.

\[Perimeter\: of\: Quadrilateral=sum \: of \: all\: sides=Perimeter\: of\:Parallelogram\]\[Area \: of\: Quadrilateral= Base\times Height\]\[Area \: of\: Parallelogram= Base\times Height\]\[Perimeter\: of\: Rectangle =2(Length + Breadth)\]\[Area \: of\: Rectangle= Length\times Breadth\]\[Area \: of\: Rhombus= \frac{1}{2}\times d_{1}\times d_{2}\]\[Perimeter\: of\: Square =4\times Side=Perimeter\: of\: Rhombus\]\[Area \: of\: Square= \frac{1}{2}\times d_{1}\times d_{2}\]

\[Area \: of\: Square=side\times side\]

Circle is the collection of all points in a plane which are at equidistant from the fixed point. Fixed point is called the centre of the circle and the fixed distance is called the radius of the circle.

\[Perimeter \:or\:circumference\:of \:the\: circle =2\pi r\]\[Area \:of \:a\: circle=\pi r^{2}\]

\[Circumference\: \: of\: semi-circle=\pi r\]\[Perimeter\: \: of\: semi-circle=\pi r+2r\]\[Area\: \: of\: semi-circle=\frac{1}{2}\pi r^{2}\]\[Circumference\: \: of\: Quadrant=\frac{1}{2}\pi r\]\[Perimeter\: \: of\: Quadrant=\frac{1}{2}\pi r+2r\]\[Area\: \: of\: quadrant\: of\: the\: circle=\frac{1}{4}\pi r^{2}\]

The area between the radius and the arc is called sector.

\[Area\: \: of\: the \: sector\: of\: the\: circle=\frac{\theta }{360}\pi r^{2}\]


The area between the chord and arc is called segment.

\[Area\: \: of\: the \: segment \: of\: the\: circle=\frac{\theta }{360}\pi r^{2}-\frac{1}{2}r^{2}sin\theta\]

Regular hexagon is a plane figure which has six sides, six angles and six vertices. It is the type of polygon with 6 sides. In regular hexagon all sides are equal and all angles are equal. A regular Hexagon can be divided into six equilateral triangles.

\[Perimeter\: of\: regular\: hexagone=6\times side \: of\: regular\: hexagon\]\[Area\: of\: regular\: hexagone=6\times \frac{\sqrt{3}}{4}\times (side)^{2}\]

Important Note:-
For minute hand of the clock
Angle made by minute hand in 1 hour( 60 minute) = 360 degree
Angle made by minute hand in one minute = 360 / 60 =  6 degree

For hour  hand of the clock
Angle made by hour hand in one 12 hours = 360 degree
Angle made by hour hand in 1 hour (60 minutes) = 360/12 = 30 degree
Angle made by hour hand in one minute = 30/60 = 1/2 degree\[Sin120^{o}=Sin(90+30)^{o}=Cos30^{o}=\frac{\sqrt{3}}{2}\]



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