Arc Length and Sector Area

Arc length and sector area are two of the most common circle calculation problems on standardized tests and in real-world applications. They answer two practical questions: “How long is a piece of the circumference?” and “How much area does a pie slice cover?” If you can find the circumference and area of a full circle, you already have everything you need — both formulas just take a fraction of the full measurement.

Arc Length Formula

The arc length is the distance along the curved edge of a circle between two points. If you know the central angle $\theta$ in degrees and the radius $r$, the arc length is:

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

Here is what each piece means:

• $\theta$ is the central angle in degrees — the angle at the center of the circle that “opens up” to the arc.
• $2\pi r$ is the full circumference of the circle.
• The fraction $\frac{\theta}{360}$ tells you what portion of the full circumference the arc covers. A $90\degree$ angle is $\frac{90}{360} = \frac{1}{4}$ of the circle, so the arc is one-quarter of the circumference.

Radian shortcut: If the angle is given in radians instead of degrees, the formula simplifies to $s = r\theta$, where $s$ is arc length. This works because the radian measure already encodes the fraction of the circle. For a full treatment of radians, see the Radians and Degrees topic in trigonometry.

Sector Area Formula

A sector is the “pie slice” region enclosed by two radii and the arc between them. Its area is:

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

The same logic applies:

• $\pi r^2$ is the full circle’s area.
• $\frac{\theta}{360}$ is the fraction of the circle the sector covers.

The Key Idea

Both formulas follow the same pattern:

$\text{Partial measurement} = \frac{\theta}{360} \times \text{Full measurement}$

If you know what fraction of the circle you are dealing with, you can find any partial measurement — arc length, sector area, or even the arc’s degree measure when working backward. This single idea is the foundation of every problem on this page.

Sector and Arc Diagram

In this diagram, the green-shaded region is the sector area, the blue curve along the edge is the arc length, the green lines are the two radii, and the amber arc at the center marks the central angle $\theta$. Every arc length and sector area problem comes down to identifying these four elements.

Worked Examples

Example 1: Arc length for a 90-degree sector

Find the arc length of a $90\degree$ sector with radius $10$ cm.

Step 1: Identify the values: $\theta = 90\degree$, $r = 10$ cm.

Step 2: Apply the arc length formula.

$\text{Arc Length} = \frac{90}{360} \times 2\pi(10) = \frac{1}{4} \times 20\pi = 5\pi \approx 15.71 \text{ cm}$

Answer: The arc length is $5\pi \approx$ 15.71 cm. This makes sense — a $90\degree$ arc is one-quarter of the full circumference, and the full circumference is $20\pi \approx 62.83$ cm. One-quarter of that is $15.71$ cm.

Example 2: Sector area for a 60-degree sector

Find the area of a sector with a central angle of $60\degree$ and radius $12$ in.

Step 1: Identify the values: $\theta = 60\degree$, $r = 12$ in.

Step 2: Apply the sector area formula.

$\text{Sector Area} = \frac{60}{360} \times \pi(12)^2 = \frac{1}{6} \times 144\pi = 24\pi \approx 75.40 \text{ in}^2$

Answer: The sector area is $24\pi \approx$ 75.40 in². A $60\degree$ sector is one-sixth of the circle, and the full area is $144\pi \approx 452.39$ in². One-sixth of that is $75.40$ in².

Example 3: Find the central angle from arc length

An arc has a length of $8\pi$ and the circle has a radius of $16$. Find the central angle.

Step 1: Write the arc length formula and substitute what you know.

$8\pi = \frac{\theta}{360} \times 2\pi(16)$

Step 2: Simplify the right side.

$8\pi = \frac{\theta}{360} \times 32\pi$

Step 3: Divide both sides by $32\pi$.

$\frac{8\pi}{32\pi} = \frac{\theta}{360}$

$\frac{1}{4} = \frac{\theta}{360}$

Step 4: Solve for $\theta$.

$\theta = \frac{360}{4} = 90\degree$

Answer: The central angle is 90 degrees. You can verify: $\frac{90}{360} \times 2\pi(16) = \frac{1}{4} \times 32\pi = 8\pi$. Confirmed.

Example 4: Sprinkler coverage area

A lawn sprinkler rotates through an angle of $120\degree$ and has a reach (radius) of $15$ ft. Find the area of lawn that gets watered.

Step 1: This is a sector area problem. The sprinkler sweeps out a sector with $\theta = 120\degree$ and $r = 15$ ft.

Step 2: Apply the sector area formula.

$\text{Sector Area} = \frac{120}{360} \times \pi(15)^2 = \frac{1}{3} \times 225\pi = 75\pi \approx 235.62 \text{ ft}^2$

Answer: The sprinkler waters approximately 235.62 square feet of lawn. Since $120\degree$ is one-third of a full rotation, the watered area is one-third of the area a full-circle sprinkler would cover ($225\pi \approx 706.86$ ft²).

Example 5: Pizza slice — area and crust length

A pizza has a 14-inch diameter and is cut into 8 equal slices. Find the area and the crust length (arc length) of one slice.

Step 1: Find the radius and the central angle per slice.

$r = \frac{14}{2} = 7 \text{ in}$

$\theta = \frac{360}{8} = 45\degree$

Step 2: Find the area of one slice (sector area).

$\text{Sector Area} = \frac{45}{360} \times \pi(7)^2 = \frac{1}{8} \times 49\pi = \frac{49\pi}{8} \approx 19.24 \text{ in}^2$

Step 3: Find the crust length (arc length along the outer edge).

$\text{Arc Length} = \frac{45}{360} \times 2\pi(7) = \frac{1}{8} \times 14\pi = \frac{14\pi}{8} = \frac{7\pi}{4} \approx 5.50 \text{ in}$

Answer: Each pizza slice has an area of approximately 19.24 in² and a crust length of approximately 5.50 inches.

Real-World Application: Windshield Wiper Sweep Area

A windshield wiper sweeps back and forth across the windshield in an arc. The wiper arm acts as the radius, and the sweep angle determines how much of the windshield gets cleaned. This is a sector area problem.

Problem: A wiper blade is 18 inches long and sweeps through an angle of $110\degree$. What area of the windshield does it clean?

Step 1: Identify the values: $r = 18$ in, $\theta = 110\degree$.

Step 2: Apply the sector area formula.

$\text{Sector Area} = \frac{110}{360} \times \pi(18)^2 = \frac{11}{36} \times 324\pi = \frac{3564\pi}{36} = 99\pi \approx 311.02 \text{ in}^2$

Answer: The wiper cleans approximately 311.02 square inches (about 2.16 square feet) of windshield per sweep.

Why this matters in carpentry and auto work: Carpenters and tradespeople frequently encounter arc-shaped cuts — arched doorways, curved trim, fan-shaped deck sections. The same formula applies: if you know the radius (the arm or span) and the sweep angle, you can calculate the material needed (area) or the curved edge length (arc length) for cutting.

Common Mistakes to Avoid

1. Forgetting to use the fraction. The most common error is computing the full circumference ($2\pi r$) or full area ($\pi r^2$) and forgetting to multiply by $\frac{\theta}{360}$. Always ask yourself: “Am I finding a part of the circle?”

2. Mixing up arc length and sector area. Arc length uses $2\pi r$ (circumference). Sector area uses $\pi r^2$ (area). The fraction $\frac{\theta}{360}$ is the same for both — only the base measurement changes.

3. Using diameter instead of radius. Both formulas require the radius $r$. If you are given a diameter, divide by 2 before substituting.

4. Forgetting that area is in square units. Arc length is in linear units (cm, in, ft). Sector area is in square units (cm², in², ft²).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Arc length is a fraction of the circumference: $\frac{\theta}{360} \times 2\pi r$
• Sector area is a fraction of the full area: $\frac{\theta}{360} \times \pi r^2$
• Both formulas use the same pattern: $(\text{fraction of circle}) \times (\text{full measurement})$
• To find the central angle, set up the formula with the known values and solve for $\theta$
• Always use the radius (not diameter) and check that your units match — linear units for arc length, square units for sector area
• The radian shortcut $s = r\theta$ works when the angle is in radians, but the degree formulas on this page work for all standard problems

Return to Geometry for more topics in this section.