Find the length of an arc of circle of radius 6cm subtending an angle of `15^(@)` at the centre.

Answer

To find the length of an arc of a circle with a radius of 6 cm that subtends an angle of 15 degrees at the center, we can follow these steps:

### Step 1: Understand the relationship between the angle and the arc length
The length of an arc (L) is directly proportional to the angle (θ) it subtends at the center of the circle. The full circumference of the circle corresponds to an angle of 360 degrees.

### Step 2: Calculate the circumference of the circle
The formula for the circumference (C) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Here, the radius \( r = 6 \) cm.

Substituting the value of the radius:
\[ C = 2\pi \times 6 = 12\pi \text{ cm} \]

### Step 3: Set up the proportion for the arc length
The length of the arc (L) that subtends an angle of 15 degrees can be found using the proportion:
\[ \frac{L}{C} = \frac{\theta}{360} \]
where \( \theta \) is the angle in degrees.

### Step 4: Substitute the known values into the proportion
We know:
- \( C = 12\pi \) cm
- \( \theta = 15 \) degrees

Substituting these values into the proportion:
\[ \frac{L}{12\pi} = \frac{15}{360} \]

### Step 5: Solve for the arc length (L)
Cross-multiplying gives:
\[ L = 12\pi \times \frac{15}{360} \]

Now, simplify the fraction:
\[ \frac{15}{360} = \frac{1}{24} \]
Thus,
\[ L = 12\pi \times \frac{1}{24} = \frac{12\pi}{24} = \frac{\pi}{2} \text{ cm} \]

### Final Answer
The length of the arc is:
\[ L = \frac{\pi}{2} \text{ cm} \]

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