Question

In the adjoining figure, $\triangle OAB$ is an equilateral triangle and the area of the shaded region is $750\pi$ cm$^2$. Find the perimeter of the shaded region.

Hint:

In area-ratio problems, if the radii and central angles are identical, the areas will always be equal regardless of the orientation of the figure.

Answer 1

Step 1: Understanding the Concept:
The shaded region in such figures usually represents a sector or a combination of a circle and a triangle. For an equilateral triangle, the central angle is $60^\circ$.
Step 2: Key Formula or Approach:
1. Area of Major Sector = $\frac{360 - \theta}{360} \times \pi r^2$.
2. Length of Arc = $\frac{\theta}{360} \times 2\pi r$.
Step 3: Detailed Explanation:
Assuming the shaded region is the major sector of the circle:
1. $\theta = 60^\circ$. Major sector angle $= 360 - 60 = 300^\circ$.
2. Area $= \frac{300}{360} \pi r^2 = \frac{5}{6} \pi r^2$.
3. Given Area $= 750\pi \implies \frac{5}{6} r^2 = 750 \implies r^2 = 900 \implies r = 30$ cm.
4. Perimeter of shaded region = Arc length + $2 \times$ radii (if it's just the sector):
- Arc length $= \frac{300}{360} \times 2 \pi (30) = 50\pi$ cm.
- Total Perimeter $= 50\pi + 60$ cm. (If the triangle side is included, the calculation adjusts accordingly).
Step 4: Final Answer:
The radius is 30 cm; perimeter is $(50\pi + 60)$ cm.