To find the area of a circle circumscribing a triangle, we need to use some known geometric relationships involving the triangle and the circle. Here's a detailed, step-by-step solution:
### Step 1: Identify the Given Information
- The side [tex]\( BC \)[/tex] of the triangle [tex]\( ABC \)[/tex] is [tex]\( 60 \)[/tex] cm.
- The angle [tex]\( \angle BAC \)[/tex] is [tex]\( 20^\circ \)[/tex].
### Step 2: Recall the Formula for the Circumradius
The circumradius [tex]\( R \)[/tex] of a triangle can be found using the following formula for a given side and its opposite angle:
[tex]\[ R = \frac{a}{2 \sin(A)} \][/tex]
where [tex]\( a \)[/tex] is the length of side [tex]\( BC \)[/tex] and [tex]\( A \)[/tex] is the angle opposite to it, [tex]\( \angle BAC \)[/tex].
### Step 3: Compute the Circumradius
- The side [tex]\( BC = 60 \)[/tex] cm.
- The angle [tex]\( \angle BAC = 20^\circ \)[/tex].
To find the circumradius [tex]\( R \)[/tex], convert the angle to radians:
[tex]\[ \text{Angle in radians} = 20^\circ \times \frac{\pi}{180} = \frac{\pi}{9} \][/tex]
Now, use the trigonometric function [tex]\(\sin\)[/tex]:
[tex]\[ \sin\left(20^\circ\right) \][/tex]
Then the formula for the circumradius is:
[tex]\[ R = \frac{60}{2 \sin(20^\circ)} \][/tex]
[tex]\[ R \approx 87.71413200489262 \text{ cm} \][/tex]
### Step 4: Area of the Circumcircle
The area of the circle is given by the formula:
[tex]\[ \text{Area} = \pi R^2 \][/tex]
Substitute the value of [tex]\( R \)[/tex] into the area formula:
[tex]\[ \text{Area} = \pi \times (87.71413200489262)^2 \][/tex]
[tex]\[ \text{Area} \approx 24170.68802232985 \text{ cm}^2 \][/tex]
### Step 5: Select the Correct Answer
Based on the calculations, the area of the circle is close to [tex]\( 24170.69 \text{ cm}^2 \)[/tex]. Hence, the correct answer is:
[tex]\[ \boxed{24170.69 \text{ cm}^2} \][/tex]
