Unit 6 · GeometryG3 · Circles · Aerodynamics & Navigation

Circles, Arcs & Sectors

Connect radius, circumference, central angles, inscribed angles, arcs, and sectors.

30 min
Unlocks: Fairing Curvature
Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

inscribed angle: an angle with vertex on the circle

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Use the same fraction $\theta/360$ for arc length and sector area. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Use the same fraction

\theta/360

for arc length and sector area. Keep exact values through the setup; round only when the stem asks.

Core relationship

L=\frac{\theta}{360}(2\pi r)

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Keep $\pi$ exact when the choices contain $\pi$ . Desmos is useful when a decimal arc length or sector area is requested, after the central-angle fraction has been written over $360^\circ$ .

Explore in Desmos

Evaluate an arc length and sector area

Type the fraction of circumference or area exactly, keeping pi symbolic.

The calculator is live: change anything and see what happens.

Evaluate an arc length and sector area

Degree mode

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Trap

The main trap

An inscribed angle is half its intercepted arc, while a central angle equals it.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1

Problem

A circle has radius 9 and central angle

80^\circ

. Find the arc length.

Use

\frac{80}{360}

of the circumference.

Which fraction of the full circumference corresponds to $80^\circ$ ?

The arc length is

4\pi

Why does the setup use circumference rather than area?

Worked example

Example 2: ACT twist

Problem

An inscribed angle measures

37^\circ

. What is the measure of its intercepted arc?

Its intercepted arc is twice the angle.

Which relationship converts the inscribed angle to its intercepted arc?

The arc measures

74^\circ

Why is the arc twice the inscribed angle?

Faded example · you finish it

Example 3: You finish it

Problem

120^\circ

sector lies in a circle of radius 6. Find its area.

\frac{120}{360}\pi(6)^2

Complete the calculation.

Desmos way

Keep $\pi$ exact in circle calculations

Type the fraction of circumference or area exactly, keeping pi symbolic.

Keep \pi exact in circle calculations

Degree mode

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Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.

The decision that matters

Structure before arithmetic

Algebra or Desmos?

Evaluate an arc length and sector area

The main trap

Keep π\piπ exact in circle calculations

Keep $\pi$ exact in circle calculations