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Math Formulas | Hexagonal Pyramid



A Hexagonal Pyramid is a pyramid with a Hexagonal base.

Area of Base [ A ] = [6/2]a.s = 3.a.s
 
Surface Area of Pyramid 
= 3.a.s + 3.s.l = A + 3.s.l
 
Volume of Pyramid = a.b.h 


Note :
a = apothem,
l =  length,  b = side, 
h = height and s = slant height


Example

Formula Math Smart Hexagonal Pyramid Example :
Question : Find the surface area and volume of a hexagonal pyramid with the given apothem length 2, side 3, height 4 and the slant height 5.

Answer : Find the area of the base.
Area of the base [ A ] = 3as = 3 x 2 x 3 = 18.

Answer : Find the surface area of pyramid.
Surface Area of Pyramid = A + 3sl = 18 + [3 . 3 .5] = 18 + 45 = 63.

Answer : Find the volume of pyramid.
Volume of Pyramid = a.b.h = 2 . 3 . 4 = 24.

That is example will clearly illustrates how to calculate the Volume, Surface Area of a Formula Math Smart Hexagonal Pyramid .

Math Formulas | Cone

Smart Math Formula, A cone is a three-dimensional geometric shape consisting of all line segments joining a single point to every point of a two-dimensional figure.

Slant height of Cone 
[ l ] = Sqrt [r² + h²]
 

Volume of Cone = [1/3] πr²h

Curved Surface Area [ CSA ] of Cone = πrl
Total Surface Area [ TSA ] of Cone = πr [ l + r ]


Note :
r = radius, l = slant height, h = height


Example 

Question : Find the volume, curved surface and total surface area of a cone with the given radius 3 and height 4.

Answer : Find the slant height.
Slant height [ l ] = Sqrt [r² + h²] = Sqrt [3² + 4²]= Sqrt [9 + 16]
=Sqrt [25] = 5.

Answer : Find the volume.
Volume = [1/3]πr²h = [1/3] x 3.14 x 3² x 4 = 0.33 x 113.04 = 37.68.

Answer : Find the curved surface area [ CSA ].
CSA = πrl = 3.14 x 3 x 5 = 47.1.

Answer : Find the total surface area [ TSA ].
TSA = πr [l + r] = 3.14 x 3[5 + 3] = 3.14 x 3(8) = 3.14 x 24 = 75.36.

That is example will clearly illustrates how to calculate the Volume, Curved Surface and Total Surface Area of a Smart Math Formula Cone.

Math Formulas | Volum Of Cube

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Smart Math Formula, A cube has a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Cube Math Formula :
Volume of Cube = s³

                         Side of Cube     = ³√v
                         Surface Area of Cube = 6s²
Note :
s = side
                                             

Example

Question : Find the volume, surface area and side of a cube with the given side 5.

Answer : Find the volume.
Volume = s³ = 5³ = 125.

Answer : Find the surface area.
Surface Area = 6.s² = 6 . 5² = 6 . 25 = 150.

Answer : Find the Side.
s=³√v = ³√125 = 5


That smart formula cube example will clearly illustrates how to calculate the Volume, Surface Area and Side of a Cube.

Math Formulas Basic | Circle

Smart Math Formula Circle
Smart Math Formula Circle,  A circle is a certain distance from the point to the so-called rotating circle, whose name is the diameter (d) is the diameter and the radius [ r ] is half of the diameter.

Wide circle = πr ²
Diameter of circle = 2r
Circumference of circle= 2πr= πd
                                      Sector wide = πr x [θ/360]

Note :
r = radius , d = diameter , ∏ = 3.14 or 22/7 [ if radius and diameter can divided 7 ]
  

Example

Question : Find the area, diameter and circumference of a circle with the given radius 3.

Answer: Find the area.
Area = πr² = 3.14 x 3² = 3.14 x 9 = 28.26.

Answer: Find the diameter.
Diameter = 2r = 2 x 3 = 6.

Answer: Find the circumference.
Circumference = πd = 3.14 x 6 = 18.84.

Question : Find the area of sector with the given radius 3 and theta 30.

Answer: Find the area.
Area =πr²(θ/360) = 3.14 x 3² x (30/360) = 3.14 x 9 0.083 = 2.35.

That is example will clearly illustrates how to calculate the Area, Diameter and Circumference of a Circle, Area of Sector with Smart Math Formula way.

Math Formula Basic | Kite

Smart Math Formula KiteSmart Math Formula, A kite is a quadrilateral with two pairs of adjacent sides that are congruent. The diagonals of a kite are perpendicular.
 


Smart Diagonal Method : Area of Kite = ½ . d1 . d2
Smart Trigonometry Method : Area of Kite = a . b . Sin C
Perimeter of Kite = 2[a + b]
 

Note :
a = length, b = breadth, d1, d2 are diagonals

                                              

Example

Question : Find the area of a kite with the given diagonals 2, 4 using Smart Diagonal Method.
Answer : Find the area.
Area = ½ . d1 . d2 = 0.5 . 2 . 4 = 4.


Question : Find the area of a kite with the given length 2 and breadth 3 using Smart Trigonometry Method.
Answer : Find the area.
Area = a . b . Sin C = 2 . 3 . Sin(33) = 6 . 1 = 6.


Question : Find the perimeter of a kite with the given length 2 and breadth 3.
Answer : Find the perimeter.
Perimeter = 2[a + b] = 2.[2 + 3] = 2.[5] = 10.


That is example will clearly illustrates how to calculate the Area and Perimeter of a Kite Smart Math Formula.

Math Formula Basic | Polygon

Smart Math Formula, A polygon is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments.


Formula length of a side :
Area of Polygon = {[ side ]² . N] / [4.Tan(π / N]}
Perimeter of Polygon = N . [side]

Formula radius (circumradius) :
Area of Polygon = ½ . R² . Sin[2.π / N]

Formula apothem [inradius] :
Area of Polygon = A² . N . Tan(π / N)
where A = R . Cos(π / N)

Formula apothem and length of a side :
Area of Polygon = [A x P] / 2
where A = side / [2 x Tan(π / N]

Note :
N = Number of sides, A = Apothem, R = Radius, P = Perimeter



Example 

Question : Find the area and perimeter of a polygon with the length 2 and the number of sides is 4.

Smart Math Formula : Find the area.
Area = {[side]² . N] / [4.Tan[π / N]} = {[2]² . 4) / (4 . Tan[3.14 / 4]}
= (4 . 4) / 4 . Tan(0.785)
= 16 / 4 . 0.999
= 16 / 3.996
Area = 4.

Smart Math Formula : Find the perimeter.
Perimeter = [N .[side] = 4 . 2 = 8

Question : Find the area of a polygon with the given radius 2 and the number of sides is 5.
Smart Math Formula : Find the area.
Area = ½ . R² . Sin[2.π / N]
= [0.5] . 2² . Sin(2 . 3.14 / 5]
= 0.5 x 4 x Sin[6.28 / 5]
= 2 x Sin[1.26]
= 2 x 0.95
Area = 1.9.

Question : Find the area of a polygon with the given radius 2 and the number of sides is 5 using Apothem.

Smart Math Formula : Find the apothem.
Apothem = R . Cos[π / N]
= 2 . Cos[3.14 / 5]
= 2 . Cos[0.63]
= 2 . 0.81
Apothem = 1.62.

Smart Math Formula : Find the area.
Area = A² . N .Tan[π / N]
= 1.62² . 5 .Tan[3.14 / 5]
= 2.62 . 5 . Tan[0.63]
= 13.1 . 0.73
Area = 9.5.

Case 4: Find the area of a polygon with the length 2 and the number of sides is 4 using Apothem.

Smart Math Formula : Find the apothem.
Apothem = side / [2 .Tan[π / N] = 2 / [2 . Tan[π / 4]
= 2 / [2 .Tan(0.785]
= 2 / [2 . 0.999]
= 2 / 1.998
Apothem = 1.

Smart Math Formula : Find the perimeter.
Perimeter = [N x [side] = 4 x 2 = 8

Smart Math Formula : Find the area.
Area = [A x P] / 2
= [1 x 8] / 2
= 8 / 2
Area = 4.

That example be smart illustrates how to calculate the Area, Perimeter of a Polygon Smart Math Formula.

Math Formula Basic | Trapezium

Smart Math Formula TrapeziumSmart Math Formula Trapezium, a trapezium is a shape with four sides, that has one set of parallel sides.
 
Trapezium/Trapezoid :

Area of Trapezium = ½.[a + b].h
 

where  = a, b = sides, h = height 

Perimeter of Trapezium a+b+c+d
 

Note : sides = a, b, c, d  

Example

Trapezium using the sides length a= 6, b=3 and the height 4.
The Area is....
Question : Find the area. 
Area = ½ . [a + b] . h = ½ . [6 + 3] x 5 = ½ . 9 .4 = 18.
Trapezium using the sides length a= 6, b= 3, c= 4 and d= 6.
The Perimeter is.....
Question : Find the perimeter.
Perimeter = a + b + c + d = 6 + 3 + 4 + 6 = 19.

That example Smart Math Formula Trapezium how to calculate the Area and Perimeter of a Trapezium.

Math Formula Basic | Rhombus


Smart Math Formula Rhombus,  a Rhombus is a four sided polygon in which every side has the same length. 

Rhombus Formula : It is an equilateral quadrilateral.

Base Times Height Method : Area of Rhombus = b . h

Diagonal Method : Area of Rhombus = ½ . d1 . d2

Perimeter of Rhombus = 4 . [a]

Note :
a = side, b = breadth, h = height, d1, d2 [Are diagonals Rhombus]

Example

Question: Find the area of a rhombus with the given base 6 and height 8 using Base Times Height Method.

Question: Find the area.

Area = b . h = 6 . 8 = 48

Question: Find the perimeter of a rhombus with the given side 6.

Question: Find the perimeter.

Perimeter = 4 . [a] = 4 . 6 = 24.

That example will clearly illustrates how to calculate the Area, Perimeter of a Rhombus manually.

Math Formula Basic | Triangle

Smart Math Formula Triangle, A triangle has three corners or vertices and three sides or edges which are straight line segments.

A Triangle has sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent.


Area of Triangle. A = ½ . p . h

Perimeter of Triangle. P = a + b + c
 

Hypotenuse of Triangle. H²[bc²]= ab² + ac²
 

Note :
p = pedestal , h = high , sides of triangle is a,b,c and Hypotenuse = H

Example


Question:

What is the area and perimeter of a triangle, if the triangle pedestal 4, height triangle 3,hypotenuse 5

Smart Math Answer :
Area of Triangle = ½ . p . h = ½ . 4 . 3 = 6
Perimeter of Triangle = a + b + c = 4 + 3 + 5 = 12

Question :
Right triangle pedestal [ab.4, height [ac)] 3. the length hypotenuse is

Smart Math Answer :

H² = ab² + ac²
H² = 4² + 3² = 16 + 9 = 25
H = √25 = 5

Math Formula Basic | Square

Smart Math Formula SquareSmart Math Formula, A square is a regular polygon with four sides. It has four right angles and parallel sides.

 Area of Square = s²

Perimeter of Square = 4.s

s = √A       

Note : s = side

 

Example 


Question: Find the area, perimeter and of a square with the given side 8.
 
Question : Find the area.
Area = s² = 8² = 64.
 
Question : Find the perimeter.
Perimeter = 4 . s = 4 . 8 = 32.


Question : Find the side for Area 121
side = √A =  √121 = 11
 
As with the example above was counting on, square area and perimeter.

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Math Formula Basic | Rectangle

Smart Math Formula RectangleSmart Math Formula Rectangle, The rectangle is defined as a quadrilateral where all four angles are right angles. It has two pairs of parallel sides.

b = breadth, l = length
A = Area, P = Perimeter 

If that is the area declared and asked to find, length or breadth of the formula is as below:
Area of Rectangle       = l . b

length of Rectangle     = A : b
breadth of Rectangle   = A : l

If that is the perimeter declared and asked to find, length or breadth of the formula is as below: 
Perimeter of Rectangle = 2 [ l + b ]
length of Rectangle       = P : 2 - b
breadth of Rectangle     = P : 2 - l
     

Example


Question : Find the area and perimeter of a rectangle for the breadth 5 and the length 20.

  Question : Find the area.
            Area = l . b = 20 . 5 = 100

  Question : Find the perimeter.
            Perimeter = 2 [l + b] = 2 [20 + 5] = 50.



Rectangle the perimeter declared
Question : Find the length and breadth of a rectangle for the perimeter 32, breadth 6

Question : Find the length.
length = P : 2 - b = 32 : 2 - 6 = 10
 
 
The above example will clearly illustrates how to calculate the Area and Perimeter of a Rectangle manually, so you will be Smart Math Formula Rectangle.