Circle Sector

A Region Bounded by Two Radii and an Arc

A circle sector is the region of a disk bounded by two radii and the arc between them. A slice cut radially from a round cake or an orange wedge has roughly the same shape.

A sector consists of three main components:

• Two radii that meet at the center of the circle
• One arc that connects the ends of both radii

The two radii form a central angle at the center of the circle. For a fixed radius, this angle determines what fraction of the disk the sector occupies.

Relationship Between Central Angle and Sector Area

For a fixed radius, sector area is proportional to the central angle. Doubling the angle doubles the area of the corresponding sector.

The relationship between central angle and sector area can be expressed as a ratio. If the central angle is $\alpha$ degrees, then the sector area is $\frac{\alpha}{360^\circ}$ part of the total circle area.

For a fixed radius, sector area is proportional to the central angle. An angle of $90^\circ$ covers one-quarter of the circle, while $180^\circ$ covers one-half.

Calculating Sector Area

The area of a sector is determined by the circle radius and the central angle. The angle specifies what fraction of the full circle belongs to the sector.

Consider a circle with radius $14 \text{ cm}$ and a sector with central angle $60^\circ$. Substitution gives:

The exact sector area is $\frac{98\pi}{3} \text{ cm}^2$, approximately $102.63 \text{ cm}^2$.

Comparison of Central Angle with Sector Area

To understand the proportional relationship between central angle and sector area, look at various comparison examples:

Practice Problems

Each problem asks for the area of a sector, so express the central angle in radians and then apply the sector area formula.

1. A circle has a radius of $21 \text{ cm}$. Determine the sector area formed by a central angle of $120^\circ$.

2. A sector has area $49\pi \text{ cm}^2$ in a circle of radius $14 \text{ cm}$. What is its central angle?

3. A round flatbread with diameter $28 \text{ cm}$ is cut radially through its center into $8$ equal sectors. Determine the area of each piece.

Worked Solutions

1. Solution Steps:

   Given: $r = 21 \text{ cm}$, $\alpha = 120^\circ$

   $$\text{Sector Area} = \frac{\alpha}{360^\circ} \times \pi r^2$$

   $$= \frac{120^\circ}{360^\circ} \times \pi \times 21^2$$

   $$= \frac{1}{3} \times \pi \times 441$$

   $$= \frac{441\pi}{3}$$

   $$= 147\pi \text{ cm}^2$$

   $$\approx 461.81 \text{ cm}^2$$

2. Solution Steps:

   Given: sector area = $49\pi \text{ cm}^2$, $r = 14 \text{ cm}$

   $$49\pi = \frac{\alpha}{360^\circ} \times \pi \times 14^2$$

   $$49\pi = \frac{\alpha}{360^\circ} \times 196\pi$$

   $$49 = \frac{\alpha}{360^\circ} \times 196$$

   $$\frac{\alpha}{360^\circ} = \frac{49}{196}$$

   $$\frac{\alpha}{360^\circ} = \frac{1}{4}$$

   $$\alpha = \frac{1}{4} \times 360^\circ$$

   $$\alpha = 90^\circ$$

3. Solution Steps:

   Given: diameter = $28 \text{ cm}$, so $r = 14 \text{ cm}$ The flatbread is cut radially into $8$ equal sectors, so the central angle of each piece is $\frac{360^\circ}{8} = 45^\circ$

   $$\begin{aligned} &\text{Area of each piece} \\ &= \frac{45^\circ}{360^\circ} \times \pi \times 14^2 \end{aligned}$$

   $$= \frac{1}{8} \times \pi \times 196$$

   $$= \frac{196\pi}{8}$$

   $$= \frac{49\pi}{2} \text{ cm}^2$$

   $$\approx 76.97 \text{ cm}^2$$