Question

Each side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm². The area of the triangle is

(a) $\frac{7\sqrt{3}}{4}{\mathrm{cm}}^2$
(b) $\frac{49\sqrt{3}}{4}{\mathrm{cm}}^2$
(c) $35{\mathrm{cm}}^2$
(d) $49{\mathrm{cm}}^2$

Answer 1

(b) $\frac{49\sqrt{3}}{4}{\mathrm{cm}}^2$

Let the radius of the circle be r cm.
Then, its area = ${\mathrm{\pi r}}^2$ cm²
∴ ${\mathrm{\pi r}}^2$ = 154
⇒ $\frac{22}{7}\times r\times r=154$
⇒ r² = $\left(\frac{154\times 7}{22}\right)$ = 49
⇒ r = $\sqrt{49}$cm = 7 cm

Side of the equilateral triangle = Radius of the circle
= 7 cm
∴ Area of the equilateral triangle = $\frac{\sqrt{3}}{4}{\left(\mathrm{side}\right)}^2$ sq. units

= $\frac{\sqrt{3}}{4}{\left(7\right)}^2$ cm²

= $\frac{49\sqrt{3}}{4}$ cm²