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. m^{2} , km^{2}
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

10^{8} 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=a^{2}

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=πr^{2}

Watch this tutorial for example around Perimeter
Area of Trapezium
Trapezium is a quadrilateral with a pair of parallel sides.
Isosceles trapezium
Trapezium when nonparallel 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
Example
A trapezoid's two bases are 10 m and 6m, and it is 4m high. What is its Area?
Solution:
A=(1/2)h( a+b)
So
A= (1/2)4(10+6)
=32 m^{2}
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 h
_{1} and h
_{2} respectively
So Area of Quadrilateral ABCD will be given as
=Area of the Triangle ACD + Area of the Triangle ABC
=(1/2)×AC×h
_{1} + (1/2)×AC×h
_{2}
=(1/2)AC(h
_{1}+ h
_{2})
=(1/2)d(h
_{1}+ h
_{2})
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)×d
_{1}×d
_{2}
Where d
_{1} and d
_{2} are the diagonals of the Rhombus.
Examples
1) The area of a rhombus is 240 cm^{2} and one of the diagonals is 10 cm. Find the other diagonal.
Solution:
Area of Rhombus is given by
A= (1/2)×d_{1}×d_{2}
240=(1/2)10×d
d=45 cm
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 cm^{3}

1mL

1000 mm^{3}

1 Litre

1000ml

1000 cm^{3}

1 m^{3}

10^{6 }cm^{3}

1000 L

1 dm^{3}

1000 cm^{3}

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
Incase 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

6L^{2}

Lateral surface area of the cube

4L^{2}

Diagonal of the cube


Volume of a cube

L^{3}

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 r^{2}h

Example:
Find the height of a cylinder whose radius is 14 cm and the total surface area is 1936 cm^{2}
Assume π= 22/7
Solution
Let height of the cylinder = h, radius = r = 14cm
Total surface area = 2πr (h + r)
2× (22/7)× 14 × (h + 14) = 1936
h = 8 cm
Example
Given a cuboid tank, in which situation will you find surface area and in
which situation volume.
(a) To find how much water it can hold.
(b) Number of cement bags required to plaster it.
(c) To find the number of smaller tanks that can be filled with water from it.
Solution
a) Volume as we are talking about capacity
b) Surface area
c) Again Volume
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