In a circle of diameter 42 cm, if an arc subtends an angle of `60^(@)` at the centre where `pi = (22)/( 7)` then what will be the length of arc?

Answer

To find the length of the arc in a circle with a diameter of 42 cm that subtends an angle of 60 degrees at the center, we can follow these steps:

### Step 1: Calculate the radius of the circle
The radius \( r \) is half of the diameter. Given that the diameter is 42 cm, we can calculate the radius as follows:
\[ r = \frac{\text{Diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm} \]

### Step 2: Use the formula for the length of the arc
The length of an arc \( L \) that subtends an angle \( \theta \) at the center of a circle can be calculated using the formula:
\[ L = \frac{\theta}{360} \times 2 \pi r \]
where \( \theta \) is the angle in degrees, \( r \) is the radius, and \( \pi \) is given as \( \frac{22}{7} \).

### Step 3: Substitute the known values into the formula
In this case, \( \theta = 60^\circ \) and \( r = 21 \text{ cm} \). Substituting these values into the formula gives:
\[ L = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21 \]

### Step 4: Simplify the expression
First, simplify \( \frac{60}{360} \):
\[ \frac{60}{360} = \frac{1}{6} \]
Now substitute this back into the equation:
\[ L = \frac{1}{6} \times 2 \times \frac{22}{7} \times 21 \]

### Step 5: Calculate the length of the arc
Now, calculate the length:
\[ L = \frac{1}{6} \times 2 \times \frac{22 \times 21}{7} \]
Calculating \( 2 \times 22 = 44 \):
\[ L = \frac{1}{6} \times \frac{44 \times 21}{7} \]
Calculating \( 44 \div 7 \approx 6.2857 \) (but we will keep it as a fraction for exactness):
\[ L = \frac{1}{6} \times \frac{924}{7} = \frac{924}{42} = 22 \text{ cm} \]

### Final Answer
The length of the arc is \( 22 \text{ cm} \).

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