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  • In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. Find the area of the design.

6. In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. Find the area of the design.

                                                              

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Assume the centre of the circle to be point C and AD as the median of the equilateral triangle.

Then we can write:-                   

                                                         AO\ =\ \frac{2}{3}AD

or                                                      32\ =\ \frac{2}{3}AD

Thus                                                 AD\ =\ 48\ cm

Consider \Delta ABD,

                                                    AB^2\ =\ AD^2\ +\ BD^2

or                                                AB^2\ =\ 48^2\ +\ \left ( \frac{AB}{2} \right )^2

or                                                AB\ =\ 32\sqrt{3}\ cm

Thus the area of an equilateral triangle is:- 

                                                               =\ \frac{\sqrt{3}}{4}\times \left ( 32\sqrt{3} \right )^2

or                                                           =\ 768\sqrt{3}\ cm^2

And the area of the circle is :                       =\ \pi r^2\ =\ \pi\times 32^2

or                                                            =\ \frac{22528}{7} cm^2

Hence the area of the design is:- 

                                                            =\ \left ( \frac{22528}{7}\ -\ 768 \sqrt{3} \right )\ cm^2

Posted by

Devendra Khairwa

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