7th Class Mathematics Mensuration Mesuration

Mensuration

Standard Units of Area

The inter relationship among various units of measurement of area are listed below.

\[1\,{{m}^{2}}\]                      =         \[(100\times 100)\,c{{m}^{2}}={{10}^{4}}\,c{{m}^{2}}\]

\[1\,{{m}^{2}}\]                      =         \[(10\times 10)\,d{{m}^{2}}=100\,d{{m}^{2}}\]

\[1\,d{{m}^{2}}\]        =         \[(10\times 10)\,c{{m}^{2}}=100\,c{{m}^{2}}\]

\[1\,da{{m}^{2}}\]       =         \[(10\times 10)\,{{m}^{2}}=100\,{{m}^{2}}\]

\[1\,h{{m}^{2}}\]                    =         \[(100\times 100)\,{{m}^{2}}={{10}^{4}}{{m}^{2}}\]

\[1\,k{{m}^{2}}\]                     =         \[(1000\times 1000)\,{{m}^{2}}={{10}^{6}}\,{{m}^{2}}\]

\[1\,hectare\]      =         \[10000\,{{m}^{2}}\]

\[1\,k{{m}^{2}}\]                     =         \[100\,hectare\]

Formula Related to Perimetre and Area

• Area of a triangle \[=\frac{1}{2}\times b\times h\]
• Area of an equilateral triangle \[=\frac{\sqrt{3}}{4}\times {{a}^{2}}\]
• Perimetre of a rectangle \[=2(Length+breadth)\]
• Area of a rectangle \[=Length\times breadth\]
• Diagonal of a rectangle \[=\sqrt{{{(length)}^{2}}+{{(breadth)}^{2}}}\]
• Perimetre of a square \[=4\times side\]
• Area of a square \[=sid{{e}^{2}}=\frac{1}{2}\times {{(diagonal)}^{2}}\]
• Side of a square \[=\sqrt{area}\]
• Diagonal of a square \[=side\times \sqrt{2}\]
• Perimetre of a parallelogram \[=2\times sum\,of\,length\,of\,adjacent\,sides.\]
• Area of a parallelogram \[=base\times corresponding\,height.\]
• Perimetre of a rhombus \[=4\times side\]
• Area of a rhombus \[=base\times vertical\,height.\]
• Area of a rhombus \[=\left( \frac{1}{2} \right)\times product\,of\,diagoanls\]
• Circumference of a circle \[=2\pi r\]
• Area of a circle \[=\pi \,{{r}^{2}}\]
• The volume of a cuboid \[=length\times breadth\times height\]
• The volume of a cube \[={{(length)}^{3}}\]

Example:

Find the area of a right-angled triangle whose sides are 15 cm, 9 cm and 2 cm.

(a) \[48\,c{{m}^{2}}\]              (b) \[80\,c{{m}^{2}}\]

(c)\[54\,c{{m}^{2}}\]               (d) \[78\,c{{m}^{2}}\]

(e) None of these

Answer (c)

Explanation: Here, a = 15 cm, b = 9 cm and c = 12 cm

Also, \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\Rightarrow \]The given triangle is a right triangle.

\[\therefore \]Area of the right triangle \[=\frac{1}{2}\times 9\times 12=54\,c{{m}^{2}}\]

Example:

The dimensions of the floor of a room are 15 m and 20 m. How many square tiles each of length 20 cm are required to furnish the floor?

(a) 2,400                       (b) 5,200        

(c) 7,500                        (d) 5,250

(e) None of these

Answer (c)

Explanation: Area of the room \[=15\,m\times 20\,m\]

\[=1500\,cm\times 2000\,cm=3\times {{10}^{6}}\,c{{m}^{2}}\]

Area of a tile \[=20\,cm\times 20\,cm=400\,c{{m}^{2}}\]

Total number of tiles required \[=\frac{3\times {{10}^{6}}}{400}=\frac{30000}{4}=7,500\]