Circumference of a sector of angle `p^(@)` of a circle with radius R is

Options

A. `(p)/(180) xx 2 pi ` R
B. `(p)/(180) xx pi R^(2)`
C. `(p)/(360) xx 2 pi ` R
D. `(p)/(720) xx 2pi R^(2)`

Answer

To find the circumference of a sector of a circle with a given angle \( p^\circ \) and radius \( R \), we can follow these steps:

### Step 1: Understand the full circle's circumference
The circumference \( C \) of a full circle is given by the formula:
\[ C = 2\pi R \]
where \( R \) is the radius of the circle.

**Hint:** Remember that the circumference of a full circle is based on the radius and involves the constant \( \pi \).

### Step 2: Determine the fraction of the circle represented by the sector
The sector we are interested in has an angle of \( p^\circ \). The fraction of the full circle that this sector represents can be calculated as:
\[ \text{Fraction of the circle} = \frac{p}{360} \]
This is because a full circle is \( 360^\circ \).

**Hint:** The angle of the sector compared to the total angle of a circle helps us find what portion of the circumference we need.

### Step 3: Calculate the circumference of the sector
To find the circumference of the sector, we multiply the full circumference of the circle by the fraction of the circle that the sector represents:
\[ \text{Circumference of the sector} = \frac{p}{360} \times (2\pi R) \]
This simplifies to:
\[ \text{Circumference of the sector} = \frac{p \cdot 2\pi R}{360} \]

**Hint:** Multiplying the full circumference by the fraction gives you the length of the arc of the sector.

### Step 4: Simplify the expression
We can simplify the expression further:
\[ \text{Circumference of the sector} = \frac{p \pi R}{180} \]

**Hint:** Simplifying the fraction helps in making the final answer clearer.

### Final Answer
Thus, the circumference of the sector of angle \( p^\circ \) of a circle with radius \( R \) is:
\[ \text{Circumference of the sector} = \frac{p \pi R}{180} \]