Question

Prove that the area of a sector of sectorial angle \( \theta \) and radius \( r \) is \( \dfrac{\pi r^2 \theta}{360} \).

Hint:

Remember: A sector is a fraction of a full circle, so its area is found by multiplying the total area \( \pi r^2 \) by \( \dfrac{\theta}{360} \).

Answer 1

Step 1: Write the area of a complete circle.}
We know that the area of a circle of radius \( r \) is \[ \pi r^2 \]
Step 2: Understand the meaning of a sector.}
A sector is a part of a circle enclosed by two radii and the corresponding arc. If the angle of the sector is \( \theta^\circ \), then it is only a fraction of the whole circle.
Step 3: Find the fraction of the whole circle.}
Since the angle of the complete circle is \( 360^\circ \), the sector with angle \( \theta^\circ \) represents \[ \frac{\theta}{360} \] part of the whole circle.
Step 4: Multiply this fraction by the total area.}
Therefore, the area of the sector is equal to \( \dfrac{\theta}{360} \) times the total area of the circle. So, \[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
Step 5: Simplify the expression.}
Thus, \[ \text{Area of sector} = \frac{\pi r^2 \theta}{360} \]
Step 6: Write the final result.}
Hence, the area of a sector of angle \( \theta \) and radius \( r \) is \[ \boxed{\frac{\pi r^2 \theta}{360}} \]