Question

on a circular wooden table of radius 70 cm a design is formed leaving the space of an equilateral triangle ABC in the middle as shown in the figure. find the total area to be designed

Answer 1

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Given:

• Radius of the circle = 70 cm
• An equilateral triangle ABC

To Find:

• The total area to be designed

Solution:

Using the formula,

$\boxed{\pink{\tt{Area\:of\:segment=\bigg(\dfrac{\pi r^{2} \theta}{360}\bigg)-\bigg(r^{2}sin\dfrac{\theta}{2}cos\dfrac{\theta}{2}\bigg)}}}$

We find here that theta is 120°, as 360÷3 =120°.

$\leadsto{\tt{\bigg(\dfrac{\dfrac{22}{7}\times {(70)}^{2} \theta}{360}\bigg)-\bigg({(70)}^{2}\times sin\dfrac{120}{2}\times cos\dfrac{120}{2}\bigg)}}$

$\leadsto{\tt{\bigg(\dfrac{22\times 10 \times 70}{10}\bigg)-(2,121.76)}}$

$\leadsto{\tt{5,133.33-2,121.76}}$

$\leadsto{\purple{\tt{3,011.57\:cm^{2}}}}$

The segment is to be multiplied by 3 to find the area of the design.

$\leadsto{\tt{3\times 3,011.57\:cm^{2}}}$

$\leadsto{\purple{\tt{9,034.71\:cm^{2}}}}$

$\rule{200}4$

Answer 2

Answer:

• Area of shaded region = 9042.25 cm² (Approx)

Step-by-step explanation:

Note:

• Diagram attached in Attachment.

Given:

• Radius of circular wooden table = 70 cm

To Find:

• Total area to be designed.

Construction:

• Draw a perpendicular bisector D i.e, BD = CD.

⇒ OCD = 30°

$\implies \sf \cos 30^{\circ}=\dfrac{DC}{OC}\\ \\ \\ \implies \sf \dfrac{\sqrt{3}}{2}=\dfrac{DC}{70}\\ \\ \\ \implies \sf DC=35\sqrt{3}\\ \\ \\ \implies \sf BC=2DC\\ \\ \\ \implies \sf BC = 2\times 35\sqrt{3}\\ \\ \\ \implies \sf BC = 70\sqrt{3}\\ \\ \\ \sf We\;know\;that,\;\sqrt{3}=1.73\\ \\ \\ \implies BC=70\times 1.73\\ \\ \\ \implies \sf BC=121.1\;cm\\ \\ \rule{100}{1}$

$\implies \sf \sin 30^{\circ}=\dfrac{OD}{OC}\\ \\ \\ \implies \sf \dfrac{1}{2}=\dfrac{OD}{70}\\ \\ \\ \implies \sf OD=35\;cm\\ \\ \\ \sf Now,\;AD=AO+OD\\ \\ \\ \implies \sf AD=70+35\\ \\ \\ \implies \sf AD = 105\;cm\\ \\ \rule{100}{1}$

$\sf Now,\;area\;of\;shaded\;region=Area\;of\;circle-Area\;of\;triangle\\ \\ \\ \implies \sf Area\;of\;shaded\;region=\pi r^{2}-\dfrac{1}{2}\times AD\times AB\\ \\ \\ \implies \sf Area\;of\;shaded\;region=\dfrac{22}{7} \times 70\times 70-\dfrac{1}{2}\times 105\times 121.1\\ \\ \\ \implies \sf Area\;of\;shaded\;region=15400-6357.75\\ \\ \\ \implies \sf Area\;of\;shaded\;region=9042.25\;cm^{2}$

Hence, Area of shaded region = 9042.25 cm² (Approx)