In this article, we'll walk you through the process using formulas, explanations, and real number examples to make it easy to follow and understand.
Step 1: Show the Formula
To calculate the area of a circular sector, we use the following formula:
\[ A = \frac{1}{2} r^2 \theta \]
where:
- \( A \) is the area of the sector.
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in radians.
Step 2: Explain the Formula
The formula \( A = \frac{1}{2} r^2 \theta \) is derived from the fact that the area of a sector is a fraction of the area of the entire circle. The fraction is determined by the ratio of the angle \( \theta \) to the full angle of a circle (2\(\pi\) radians).
Step 3: Use Actual Numbers as an Example
Let's assume we have a circle with a radius (\( r \)) of 5 units and a central angle (\( \theta \)) of \( 60^\circ \).
Step 4: Convert the Angle to Radians
Since the formula requires the angle in radians, we first convert \( 60^\circ \) to radians:
\[ \theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \text{ radians} \]
Step 5: Calculate the Area
Now, we substitute the values into the formula:
\[ A = \frac{1}{2} \times 5^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 25 \times \frac{\pi}{3} \]
\[ A = \frac{25\pi}{6} \]
For a numerical value, we can use \(\pi \approx 3.14159\):
\[ A \approx \frac{25 \times 3.14159}{6} \]
\[ A \approx \frac{78.53975}{6} \]
\[ A \approx 13.09 \, \text{square units} \]
Final Value
The area of the circular sector with a radius of 5 units and a central angle of \( 60^\circ \) is approximately \( 13.09 \, \text{square units} \).
By following these steps, you can easily determine the area of a circular sector using the given radius and angle. Understanding these geometric principles will enhance your problem-solving skills and mathematical knowledge.
