8th Class Mathematics Mensuration

 Mensuration

 

• Perimeter: The length of the boundary of a plane figure is called its perimeter.

 

• Area: The amount of surface enclosed by a plane figure is called its area.

 

• Rectangle: Given a rectangle of length T units and breadth 'b' units,

 

(i) Perimeter of the rectangle \[=2\left( l+b \right)\] units

(ii) Diagonal of the rectangle,\[~d=\sqrt{{{l}^{2}}+{{b}^{2}}}\] units

(iii) Area of the rectangle \[=\text{ }\left( \text{l}\times b \right)\]sq. units

(iv)\[\operatorname{Length}=\left( \frac{area}{breadth} \right)units\]

(v) \[\operatorname{Breadth}=\left( \frac{area}{length} \right)units\]

 

• Area of four walls of room: Let there be a room with length T units, breadth 'b' units and height 'h' units.

Then (i) Area of four walls \[=2\left( l+2 \right)\times h\]sq. units

• (ii) Diagonal of room \[=\sqrt{{{l}^{2}}+{{b}^{2}}+}{{h}^{2}}\]units

 

• Perimeter and area of a square: Let each side of a square be 'a' units. Then

(i) Perimeter of the square = (4a) units

(ii) Diagonal of the square

\[=\sqrt{{{a}^{2}}+{{a}^{2}}}=\sqrt{2a}=a\sqrt{2}\] Units.

(iii) Area of the square \[={{a}^{2}}\]sq. units

(iv) Area of the square \[=\frac{1}{2}\times {{\left( diagonal \right)}^{2}}\] sq. units

(v) Side of the square \[=\sqrt{Area}\]units.

 

• Perimeter and area of a triangle:

(i) Let 'a', 'b' and 'c' be the lengths of sides of a triangle. Then, perimeter of the triangle is given by (a + b + c) units.

\[s=\frac{1}{2}\left( a+b+c \right)\]is called semi-perimeter of the triangle.

 

(ii) Area of the triangle \[=\sqrt{s(s-a)(s-b)(s-c)}\] sq. units

(iii) Let the base of a triangle be 'b' units and its corresponding height (or altitude) be 'h' units.

 

• Then the area of the triangle \[=\left( \frac{1}{2}\times b\times h \right)\]sq. units

Note:  We may consider any side of the triangle as its base.

The the corresponding height would be the length perpendicular to this side from the opposite vertex.

• (iv) Area of an equilateral triangle with each side 'a' units \[=\left( \frac{\sqrt{3}}{4}\times {{a}^{2}} \right)\]sq. Units.

(v) Height of an equilateral triangle of side 'a' units   \[\left( \frac{\sqrt{3}a}{2} \right)\]Units.

• (vi) Area of a right triangle \[=\frac{1}{2}\times \](product of legs) sq. units

 

Note: The sides containing the right angle are known as legs of a right triangle.

 

• Area of a parallelogram: Let ABCD be a parallelogram with base 'b' units and height 'h' units.

 

 

Then area of parallelogram = (base x height) sq. units

 

• Area of a rhombus: Let ABCD be a rhombus in which diagonal \[AC={{d}_{1}}\]units and diagonal \[BD={{d}_{2}}\]units.

 

• Then area of rhombus \[ABCD=\left( \frac{1}{2}\times {{d}_{1}}\times {{d}_{2}} \right)\]sq. units.

 

• Area of a trapezium: Let ABCD be a trapezium in which \[AB\parallel DC.\]Let AB be 'a' units and DC be 'b' units.

 

 

Then area of trapezium ABCD

• \[=\frac{1}{2}\times \](sum of parallel sides)\[\times \] distance between them

\[=\frac{1}{2}\times \left( a+b \right)\times h\]sq. units

 

• Circle: A circle is a closed curve in a plane drawn in such a way that every point on this curve is at a constant distance (r units) from a fixed point 0 inside it.

The fixed point 0 is called the centre of the circle and the constant distance V is called the length of radius of the circle.

 

Note: Distance between the parallel sides = height

 

 

• Circumference of a circle: The perimeter of a circle is called its circumference. The length of the thread that goes around the circle exactly once gives the circumference of the circle.

 

• Circumference \[=2\pi r=\pi d,\] where r = radius and d = diameter

Here, \[\pi \left( Pi \right)\]is a constant.

Note: The approximate value as \[\frac{\mathbf{22}}{\mathbf{7}}\] or 3.14

However is not a rational number. It is an irrational number and is defined as the ratio of circumference of a circle to its diameter

 

• Area of a circle: Area of a circle with radius 'r' units is equal to\[\pi ~{{r}^{2}}\] sq. units.

 

Area of a ring: The region enclosed between two concentric circles of different radii is called a ring.                               
                           

Area of the ring \[=\text{ }\left( \pi {{R}^{2}}-\pi {{r}^{2}} \right)\]sq. units   \[=\pi \left( {{R}^{2}}-{{r}^{2}} \right)\] sq. units

\[=\pi \left( R+r \right)\left( R-r \right)\]sq. units

 

• Length of arc of a circle: Let A and B be any two points on a circle. The length of the thread that will wrap along this arc from A to B is the length of AB written as\[\overset\frown{AB}\].

 

• In a circle of radius 'r', we have

 

\[\frac{l\left( \overset\frown{AB} \right)}{Circumference}=\frac{{{x}^{o}}}{{{360}^{o}}}\]or \[l\left( \overline{AB} \right)=\frac{2\pi r{{x}^{o}}}{{{360}^{o}}}\]

 

• Area of a sector: A sector of a circle is the region enclosed by an are of a circle and two radii to its end points.

 

                         

• Area of sector \[=\frac{{{x}^{o}}}{{{360}^{o}}}\times \pi {{r}^{2}}\] where 'x' is sector angle and 'r' is the radius of circle.       

 

• Segment of a circle: A segment of a circle is the region enclosed by an arc of the circle and its chord.

                                   

Area of minor segment AXB = area of sector OAXB - area of \[\Delta \]OAB

Area of major segment AYB = area of circle - area of minor segment AXB   

 

• Volume: The space occupied by a solid body is called its volume.

The units of volume are cubic centimeters \[\left( c{{m}^{3}} \right)\]or cubic metres \[\left( {{m}^{3}} \right)\]etc.

 

• Cuboid: A solid bounded by six rectangular plane faces is called a cuboid.

For a cuboid of length \['l'\] units, breadth 'b' units and height 'h' units,

(i) Volume of the cuboid \[=\left( l\times b\times h \right)\]cubic sq. units

(ii) Diagonal of the cuboid \[\sqrt{{{l}^{2}}+{{b}^{2}}+{{h}^{2}}}\]units

(iii) Total surface area of the cuboid \[=2\left( lb+bh+lh \right)\] sq. units

(iv) Lateral surface area of the cuboid

 \[=\left[ 2\left( l+b \right) \right]\text{ }\times h\]sq. units

 

(vi) Area of 4 walls of a room \[=\left[ 2\left( l+b \right)\times h \right]\]units

 

• Cube: A cuboid whose length, breadth and height are all equal is called a cube.

 

For a cube of edge 'a' units,

(i) Volume of the cube \[={{a}^{3}}\]cubic units

(ii) Diagonal of the cube \[=a\sqrt{3}\]units

(iii) Total surface area of the cube \[=\left( 6{{a}^{2}} \right)\]sq. units

(iv) Lateral surface area of the cube = (4a2) sq. units

 

Relation between units
Units of length	Units of Volume
1cm =10 mm	1\[C{{m}^{3}}\]=1000\[m{{m}^{3}}\]
100 cm = 1 m	1\[{{m}^{3}}\]=1000000\[C{{m}^{3}}\]
	1 liter =10\[C{{m}^{3}}\]
	kiloliter = 1000 liters = 1\[{{m}^{3}}\]

 

• Cylinder: A solid having a curved surface and a uniform circular cross-section is known as a cylinder.

 

Note: If the axis of the Cylinder is perpendicular to each Cross – Section, the Cylinder is called a right  cylinder.

 

• Volume of a cylinder: For a cylinder whose height is 'h' units and the radius of base is \['r'\] units,

 

(i) Volume of cylinder \[\left( \pi {{r}^{2}}h \right)\]cubic units

= (base area) \[\times \] height

(ii) Area of curved surface \[==\left( 2\pi rh \right)\]sq. units

Total surface area\[=\left( 2\pi rh+2\pi {{r}^{2}} \right)\] sq. units

\[=\text{ }2\pi r\text{ }\left( h+r \right)\]sq. units

Volume and Capacity are similar words

(i) Volume is the amount of space occupied by an object

(ii) Capacity is the quantity that a container holds.