ACT — mathMedium1 min read

Circles, Arcs, and Sectors

Calculate circumference, area, arc length, and sector area for the ACT.

Tutor AI Team2026-02-25T02:03:17.402Z

Circle questions are common on the ACT. You need formulas for area, circumference, arc length, sector area, and central/inscribed angle relationships.

Key Formulas

Circumference: $C = 2\pi r = \pi d$
Area: $A = \pi r^2$
Arc length: $L = \frac{\theta}{360} \times 2\pi r$
Sector area: $A = \frac{\theta}{360} \times \pi r^2$

Angle Relationships

Central angle = intercepted arc.
Inscribed angle = $\frac{1}{2}$ intercepted arc.
Angle in a semicircle = 90°.

Circle Equation

$(x-h)^2 + (y-k)^2 = r^2$ . Centre $(h,k)$ , radius $r$ .

Worked Example: Example 1

Problem

Circle radius 10, central angle 72°. Arc length = $\frac{72}{360} \times 20\pi = 4\pi$ .

Solution

Worked Example: Example 2

Problem

Sector area: $r = 6$ , $\theta = 120°$ . $A = \frac{120}{360} \times 36\pi = 12\pi$ .

Solution

Practice Problems

1. Find the area of a circle with circumference $10\pi$ .
1. Arc length: $r = 8$ , $\theta = 45°$ .
1. An inscribed angle measures 35°. What is the central angle for the same arc?

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Key Takeaways

✓
Arc length and sector area: use the fraction $\frac{\theta}{360}$ .
✓
Inscribed angle = half the central angle.
✓
Circle equation: $(x-h)^2 + (y-k)^2 = r^2$ .

Tags:

#act#math#geometry#circles#arcs#sectors

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Key Formulas

Angle Relationships

Circle Equation

Worked Example: Example 1

Worked Example: Example 2

Practice Problems

Key Takeaways

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