Before you read

Answer first — see what you already know.

A pizza is cut into 4 equal slices (each with a $90°$ central angle). If the pizza has radius 10 inches, what is the area of one slice?

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Answer:

Before you read

Answer first — see what you already know.

A clock's minute hand is 10 inches long. How far does the tip of the minute hand travel in 20 minutes?

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Answer:

Theory

Arc and Sector Formulas

A sector is a 'pizza slice' of a circle.

Arc length (part of the circumference):
$\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

Sector area (part of the circle's area):
$\text{Sector area} = \frac{\theta}{360°} \times \pi r^2$

Arc length and sector area for a 60° angle

Where $\theta$ is the central angle in degrees.

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

Example 1

A circle with radius 12 cm has a sector with central angle $60°$ . Find the arc length and sector area.

Arc $= \frac{60}{360} \times 2\pi(12) = \frac{1}{6} \times 24\pi = 4\pi$ cm

Sector area $= \frac{60}{360} \times \pi(12)^2 = \frac{1}{6} \times 144\pi = 24\pi$ cm $^{2}$

Arc

= 4\pi

cm, Sector area

= 24\pi

^{2}

Strategy

SAT Pro Tip

Think of the fraction $\frac{\theta}{360°}$ as 'what fraction of the whole circle is this sector?' Then multiply by the full circumference or area.

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