Understanding Sector Area
A sector is a portion of a circle enclosed by two radii and an arc. It resembles a slice of pie or pizza. The area of a sector depends on the radius of the circle and the central angle that defines the sector.
Formula
The formula for calculating the area of a sector is:
$$A = \pi \times r^2 \times \frac{\theta}{360^\circ}$$
Where:
- r is the radius of the circle
- θ (theta) is the central angle in degrees
If the angle is given in radians, convert it to degrees first by multiplying by 180/π.
Example Calculation
Let's calculate the area of a sector with:
- Radius = 5 meters
- Central angle = 120 degrees
Step 1: Identify the values
- r = 5 m
- $\theta = 120^\circ$
Step 2: Apply the formula
$$A = \pi \times 5^2 \times \frac{120^\circ}{360^\circ}$$
Step 3: Calculate
$$A = \pi \times 25 \times \frac{1}{3}$$
$$A = \frac{25\pi}{3} \approx 26.18$$ square meters
Practical Applications
Sector area calculations are useful in:
- Engineering design for circular components
- Architecture for curved structures
- Land surveying for circular plots
- Manufacturing of circular parts
- Mathematical problem solving involving circles
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