In Grade 12 Geometry, it's really important to know the formulas for arc length and sector area. These help us solve problems about circles. Here are the main formulas to remember:

1. Arc Length

To find the arc length of a circle, you can use this formula:

$$L = r \theta$$

Where:

• L = arc length
• r = radius of the circle
• θ = angle in radians

Example of Arc Length Calculation

Let’s say we have a circle with a radius of 6 cm and an angle of 60 degrees (which is the same as $\frac{\pi}{3}$ radians). To find the arc length, we would do:

$$L = 6 \cdot \frac{\pi}{3} = 2\pi \text{ cm} \approx 6.28 \text{ cm}$$

2. Sector Area

To calculate the area of a sector (a slice of the circle), we use this formula:

$$A = \frac{1}{2} r^2 \theta$$

Where:

• A = area of the sector
• r = radius of the circle
• θ = angle in radians

Example of Sector Area Calculation

Using the same circle (6 cm radius) with an angle of 60 degrees (or $\frac{\pi}{3}$ radians), we find the sector area like this:

$$A = \frac{1}{2} \cdot 6^2 \cdot \frac{\pi}{3} = 6\pi \text{ cm}^2 \approx 18.85 \text{ cm}^2$$

Summary

• Arc Length Formula: $L = r \theta$
• Sector Area Formula: $A = \frac{1}{2} r^2 \theta$

Remember to convert degrees to radians when needed. You can do that by using this conversion: $\frac{\pi}{180}$.