Arcs and Sectors

Step 2: Simplify the calculation (leaving it in terms of $\pi$ mid-step to avoid premature rounding).

• $\text{Arc Length} = \frac{5}{12} \times 16\pi = \frac{20\pi}{3} \approx 20.9439...$

Step 3: State the final answer with appropriate units.

• $\text{Arc Length} = 20.9\text{ cm}$

Calculating Areas of Sectors

If an arc is the pizza crust, a sector is the actual slice you get to eat. A minor sector is less than half the circle ($\theta < 180^\circ$), while a major sector is more than half ($\theta > 180^\circ$). To find the sector area, we calculate the fractional area of the whole circle.

$$\text{Area of Sector} = \frac{\theta}{360} \times \pi r^2$$

Find the area of a sector with a radius of $9\text{ cm}$ and a central angle of $60^\circ$. Leave your answer in terms of $\pi$.

Step 1: Substitute the values into the formula, making sure to apply the radius squared.

• $\text{Area} = \frac{60}{360} \times \pi \times 9^2$

Step 2: Simplify the fraction and the squared term.

• $\text{Area} = \frac{1}{6} \times 81\pi$

Step 3: Provide the exact final answer with square units.

• $\text{Area} = \frac{27\pi}{2}\text{ cm}^2$ (or $13.5\pi\text{ cm}^2$)

Calculating Angles of Sectors

Sometimes you know the size of the slice but need to work backward to find out exactly how wide the angle was cut. You can use an inverse operation to rearrange either the arc length or sector area formula to isolate the central angle ($\theta$).

$$\theta = \frac{\text{Arc Length} \times 360}{2\pi r}$$

A sector has an arc length of $15\text{ cm}$ and a radius of $10\text{ cm}$. Calculate the central angle of the sector to 1 decimal place.

Step 1: Substitute the known values into the arc length formula.

• $15 = \frac{\theta}{360} \times 2 \times \pi \times 10$

Step 2: Rearrange to isolate $\theta$ by multiplying by 360 and dividing by the remaining terms.

• $15 \times 360 = \theta \times 20\pi$
• $5400 = 20\pi\theta$

Step 3: Calculate the final answer in degrees.

• $\theta = \frac{5400}{20\pi} \approx 85.943...$
• $\theta = 85.9^\circ$