Mensuration

Mensuration

It is branch of mathematics which is concerned about the measurement of length, area and Volume of plane and Solid figure

Recall from Previous Class

Perimeter

• The perimeter of plane figure is defined as the length of the boundary
• It units is same as that of length i.e. m ,cm,km  
  

1 Meter	10 Decimeter	100 centimeter
1 Decimeter	10 centimeter	100 millimeter
1 Km	10 Hectometer	100 Decameter
1 Decameter	10 meter	1000 centimeter

 
Area

• The area of the plane figure is the surface enclosed by its boundary
• It unit is square of length unit. i.e. m² ,  km²  
  

1 square Meter	100 square Decimeter	10000 square centimeter
1 square Decimeter	100 square  centimeter	10000 square  millimeter
1 Hectare	100 square  Decameter	10000 square meter
1 square myraimeter	100 square kilometer	108 square meter

 

Shapes where Area and Perimeter are known

 

Shapes	Perimeter	Area
Rectangle	P= 2(L+B) L and B are Length and Breadth of the   rectangle	A=L×B
Square	P=4a a is the side of the square	A=a2
Triangle	P=Sum of sides	A=(1/2)×(Base)×(Height/Altitude)
Parallelogram	P=2(Sum of Adjacent sides)	A=(Base) ×( Height)
Circle	P=2πr r is the radius of the circle	A=πr2

 

Area of Trapezium

Trapezium is a quadrilateral with a pair of parallel sides.
 

Isosceles trapezium
Trapezium when non-parallel sides of it are of equal length
Area of the Trapezium can be found by dividing the trapezium into two parts of three parts depending
On the shapes of the trapezium
 

(So two triangles and One rectangle)
Here h is the distance between the parallel sides
a   and b are the parallel sides 
A=  (1/2)hx + (1/2)hy + ah
=(1/2)h( x+y+ 2a)
Now b=x+y+a
So
=(1/2)h( a+b)
Similarly

(One triangle and One rectangle)
It can be proved easily here also
A=(1/2)h( a+b)
So to find the area of a trapezium we need to know the length of the parallel sides and the
perpendicular distance between these two parallel sides. Half the product of the sum of
the lengths of parallel sides and the perpendicular distance between them gives the area of
trapezium

Area of General Quadrilaterals

 
We can always divide a general quadrilateral into two triangles by drawing the diagonal. This split the quadrilateral into two triangles and Sum of areas of the triangles is equal to the area of the quadrilaterals
 
Here is the above figure, Diagonal AC divides the quadrilateral ABCD into two triangles.  We have drawn perpendicular to this diagonal from opposite vertices. They are of height h₁ and h₂ respectively
So Area of Quadrilateral ABCD will be given as
=Area of the Triangle ACD + Area of the Triangle ABC
=(1/2)×AC×h₁ + (1/2)×AC×h₂
=(1/2)AC(h₁+ h₂)
=(1/2)d(h₁+ h₂)
d is the length of the diagonal
 
We already know the formula for the area of special quadrilaterals (Rectangle, square and parallelogram). Those can also be derived from the above general quadrilaterals formula
We can similarly derive the formula for the Rhombus
Area of Rhombus is given by
A= (1/2)×d₁×d₂
Where d₁ and d₂ are the diagonals of the Rhombus.

Area of Polygons

We will encounter figures which are having more than 4 sides.  We can derive the area of the polygons by using the same approach i.e. dividing into triangles or quadrilaterals
 

Solid Shapes

Solid shapes are three dimensional figures.  We can identify common two dimensional figures on there faces
Example

It is a cube which has square shapes on the all the faces

It is cylinder and has circular figure at two bases
 

Volume,Surface area in case of Solid Figures

 

Surface Area	Surface area of a solid is the sum of the areas of its faces
Lateral Surface Are	The faces excluding the top and bottom) make the lateral surface area of the solid
Volume	Amount of space occupied by a three dimensional object (Solid figure) is called its volume. we use square units to find the area of a two dimensional region. In case of volume we will use cubic units to find the volume of a solid, as cube is the most convenient solid shape (just as square is the most convenient shape to measure area of a region) Volume is sometimes refer as capacity also

Volume Units

1 cm3	1mL	1000 mm3
1 Litre	1000ml	1000 cm3
1 m3	106 cm3	1000 L
1 dm3	1000 cm3	1 L

 

How to find the surface Area and Volume of the solid Figures

Surface Area is the area of the faces. So we need to find the area of the each of the faces and sum them to find the total surface area
In-case of Cuboid, all the faces are rectangles

Surface Area
=2×(Area of the rectangle having length and Breath  as h and b  )+2×(Area of the rectangle having length and Breath  as l and b  ) +2×(Area of the rectangle having length and Breath  as l and h  )
=2( LB + BH + LH ).
Volume is the space occupied by the solid figure. We can measure it by counting cubes in the space.
 if the base and top of the solid are congruent and parallel to each other and its edges are perpendicular to the base then volume can be deduced as
Volume of solid=area of base × height
 

Surface Area and Volume of Cube and Cubiod

 

Type	Measurement
Surface Area of Cuboid of Length L, Breadth B and Height H	2( LB + BH + LH ).
Lateral surface area of the cuboids	2( L + B ) H
Diagonal of the cuboids
Volume of a cuboids	LBH
Length of all 12 edges of the cuboids	4 (L+B+H).
Surface Area of Cube of side L	6L2
Lateral surface area of the cube	4L2
Diagonal of the cube
Volume of a cube	L3

 

Surface Area and Volume of Right circular cylinder

 

Radius	The radius (r) of the circular base is called the radius of the cylinder
Height	The length of the axis of the cylinder is called the height (h) of the cylinder
Lateral Surface	The curved surface joining the two base of a right circular cylinder is called Lateral Surface.

 
 

Type	Measurement
Curved or lateral Surface Area of cylinder	2πrh
Total surface area of cylinder	2πr (h+r)
Volume of Cylinder	π r2h

 

 

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Class 8 Maths Class 8 Science