SAT — mathMedium3 min read

Arc Length and Sector Area

Calculate arc length and sector area for the Digital SAT using the fraction-of-a-circle method.

Tutor AI Team2026-02-25T01:35:34.096Z

An arc is a portion of a circle's circumference, and a sector is a "pizza slice" portion of a circle's area. Both are calculated as a fraction of the full circle. The Digital SAT tests these calculations in both degree and radian contexts.

Core Concepts

Fraction of the Circle

A central angle $\theta$ out of $360°$ gives the fraction:

$\text{fraction} = \frac{\theta}{360°}$

Arc Length

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

In radians: $\text{Arc length} = r\theta$

Sector Area

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

In radians: $\text{Sector area} = \frac{1}{2}r^2\theta$

Relationship

Arc length and sector area use the same fraction of the circle, applied to circumference and area respectively.

Strategy Tips

Tip 1: Find the Fraction First

Always start by computing $\frac{\theta}{360}$ (or $\frac{\theta}{2\pi}$ in radians).

Tip 2: Radian Formulas Are Simpler

In radians, arc length $= r\theta$ and sector area $= \frac{1}{2}r^2\theta$ .

Tip 3: Check Degree vs. Radian

Make sure your angle is in the same units as your formula expects.

Worked Example: Example 1

Problem

Circle radius 10, central angle 72°. Find arc length.

Fraction: $\frac{72}{360} = \frac{1}{5}$

Arc = $\frac{1}{5} \times 2\pi(10) = 4\pi \approx 12.57$

Solution

Worked Example: Example 2

Problem

Circle radius 6, central angle $\frac{\pi}{3}$ . Find sector area.

$A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 6\pi \approx 18.85$

Solution

Worked Example: SAT-Style

Problem

A pizza with diameter 16 inches is cut into 8 equal slices. What is the area of one slice?

$r = 8$ . Angle per slice = $\frac{360}{8} = 45°$ .

$A = \frac{45}{360} \times \pi(8)^2 = \frac{1}{8} \times 64\pi = 8\pi \approx 25.1$ sq in.

Solution

Worked Example: Example 4

Problem

Arc length is 5π and radius is 10. Find the central angle in degrees.

$5\pi = \frac{\theta}{360} \times 2\pi(10)$

$5\pi = \frac{20\pi\theta}{360}$

$\theta = \frac{5 \times 360}{20} = 90°$

Solution

Practice Problems

Problem 1

Radius 12, angle 120°. Find arc length.

Problem 2

Radius 5, angle $\frac{\pi}{4}$ . Find sector area.

Problem 3

A sector has area $24\pi$ and radius $8$ . Find the central angle.

Problem 4

A circular track has radius 50 m. How far do you travel through a 60° turn?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Using diameter instead of radius.
Mixing up degrees and radians in the formula.
Forgetting to use the fraction. The full circumference is $2\pi r$ ; the arc is a fraction of it.
Computing arc length when sector area is asked (or vice versa).

Key Takeaways

✓
Fraction of circle: $\frac{\theta}{360°}$ (degrees) or $\frac{\theta}{2\pi}$ (radians).
✓
Arc length = fraction × circumference. In radians: $r\theta$ .
✓
Sector area = fraction × area. In radians: $\frac{1}{2}r^2\theta$ .
✓
Always check degree vs. radian mode.

Tags:

#sat#math#geometry#circles#arc-length#sector

Ready to Ace Your SAT math?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

Core Concepts

Fraction of the Circle

Arc Length

Sector Area

Relationship

Strategy Tips

Tip 1: Find the Fraction First

Tip 2: Radian Formulas Are Simpler

Tip 3: Check Degree vs. Radian

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?