Question

The area of an equilateral triangle is 49 square root 3 cm sq. Taking each angular point as centre, a circle is described with radius equal to half the length of the side triangle as shown in the figure. Find the area of the portion in the triangle not included in the circles.​

Answer 1

Answer:

The area of the portion in the triangle not included in the circles is $7.77\;cm^{2}.$

Step-by-step explanation:

Given: The area of an equilateral triangle is $49\sqrt{3} \;cm^{2}$.

To Find: The area of the shaded region.

Solution: From the figure, assume

$\theta=60^\circ$

$\pi = \frac{22}{7}$

Area of sector $=\frac{\theta}{360^\circ} \times \pi r^{2}$

Area of equilateral triangle $\frac{\sqrt{3}}{4} a^{2} =49\sqrt{3}$

                                                 $a^{2} =49\times4$

                                                   $a=7\times2$

                                                      $=14\;cm$

∴ The value of $a=14\;cm$.

Now, radius $r=\frac{a}{2}$

                       $=\frac{14}{2}$

                       $=7\;cm$

∴ The value of $r=7\;cm.$

To find the area of shaded region use the below formula,

Area of shaded region $=$ Area of $\triangle ABC$  $- \;3$ (Area of Sector)

                                      $= 49\sqrt{3} -3(\frac{\theta}{360^\circ} \times\pi r^{2})$

                                      $=49\sqrt{3} -3(\frac{60^\circ}{360^\circ} \times \frac{22}{7} \times 7 \times 7)$

                                      $=49\sqrt{3} -77$

                                      $=49\times1.76-77$

                                      $=7.77\;cm^{2}$

Hence, The area of the portion in the triangle not included in the circles is $7.77\;cm^{2}.$