Question

A logo was designed with four orange petals made from semicircles inscribed in a circle of diameter $14\sqrt{2}$ units. The orange part is the total area of the four petals. Find its area. (Take $\pi = \tfrac{22}{7}$). 

Hint:

Petal problems are usually intersections of semicircles. Use the lens formula (two circle overlap) for exact area, then multiply by number of petals.

Answer 1

Step 1: Understand the figure.
- A circle of diameter $14\sqrt{2}$ is drawn. 
- Radius $R = \dfrac{14\sqrt{2}}{2} = 7\sqrt{2}$. 
- Inside, four semicircles are drawn with diameters along the sides of a square inscribed in the circle. 
- The petals are the overlaps of these semicircles at the center. 
Step 2: Side of the square.
If the circle radius = $R$, the side of an inscribed square = $R\sqrt{2} = 7\sqrt{2}\times \sqrt{2} = 14$. So, each semicircle has diameter $14$, radius $7$. 
Step 3: Area of one petal (lens formed by overlap of two semicircles).
Each petal is the intersection of two semicircles of radius 7. 
Formula for area of lens from two equal circles: \[ A_{\text{lens}} = 2r^2\cos^{-1}\left(\frac{d}{2r}\right) - \frac{d}{2}\sqrt{4r^2 - d^2} \] where $r = 7$, $d = 7$. 
Step 4: Substitute values.
\[ A_{\text{lens}} = 2(49)\cos^{-1}\left(\frac{7}{14}\right) - \frac{7}{2}\sqrt{196 - 49} \] \[ = 98\cos^{-1}\left(\tfrac{1}{2}\right) - \tfrac{7}{2}\sqrt{147} \] \[ = 98. \frac{\pi}{3} - \tfrac{7}{2}. 7\sqrt{3} \] \[ = \frac{98\pi}{3} - \frac{49\sqrt{3}}{2} \] Step 5: Total orange area.
There are 4 petals: \[ A_{\text{orange}} = 4\left(\frac{98\pi}{3} - \frac{49\sqrt{3}}{2}\right) \] \[ = \frac{392\pi}{3} - 98\sqrt{3} \] Substitute $\pi = \tfrac{22}{7}$: \[ A_{\text{orange}} = \frac{392}{3}. \frac{22}{7} - 98. 1.732 \] \[ = \frac{392\times 22}{21} - 169.736 \approx 411.81 \] Final Answer: \[ \boxed{412 \, \text{units}^2 \, \text{(approx)}} \]