AB and CD are diameters of a circle with centre O and radius 7 cm. If \(\angle BOD = 30^\circ\), then find the area and perimeter of the shaded region.
Given:
- Radius of the circle, \(r = 7\) cm.
- \(AB\) and \(CD\) are diameters.
- \(\angle BOD = 30^\circ\).
- Since \(AB\) and \(CD\) are diameters intersecting at \(O\), \(\angle AOC = \angle BOD = 30^\circ\) (vertically opposite angles).
- The shaded regions are sector \(BOD\) and sector \(AOC\).
Step 1: Calculate area of one sector
Area of one sector is given by:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]
Substitute values:
\[
\text{Area of sector } BOD = \frac{30^\circ}{360^\circ} \times \pi \times (7)^2 = \frac{1}{12} \times \frac{22}{7} \times 49 = \frac{1}{12} \times 22 \times 7 = \frac{154}{12} = \frac{77}{6} \text{ cm}^2
\]
Step 2: Calculate total shaded area
Since sector \(AOC\) is identical to sector \(BOD\), its area is also \(\frac{77}{6}\) cm\(^2\).
Total shaded area = Area of sector \(BOD\) + Area of sector \(AOC\):
\[
2 \times \frac{77}{6} = \frac{77}{3} \text{ cm}^2
\]
Numerical value:
\[
\frac{77}{3} \approx 25.67 \text{ cm}^2
\]
Step 3: Calculate arc length of one sector
Arc length of sector is:
\[
l = \frac{\theta}{360^\circ} \times 2 \pi r
\]
Substitute values:
\[
\text{Arc length } BD = \frac{30^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 7 = \frac{1}{12} \times 2 \times 22 = \frac{44}{12} = \frac{11}{3} \text{ cm}
\]
Step 4: Calculate total arc length
Arc length \(AC\) is also \(\frac{11}{3}\) cm.
Total arc length = \(\frac{11}{3} + \frac{11}{3} = \frac{22}{3}\) cm.
Step 5: Sum of radii
Radii involved are \(OA = OB = OC = OD = 7\) cm each.
Sum of radii = \(7 + 7 + 7 + 7 = 28\) cm.
