Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Express a central angle as a fraction of a turn.
- Calculate a sector's arc and area.
- Include straight edges in its perimeter.
Before you start
Fractions, circle circumference and area formulas, and multiplication involving π.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A paper fan is an ideal 120° sector of a circle with radius 9 cm. Find its curved-edge length, surface area, and full perimeter.
Why this math matters
The same angle fraction can select a portion of circumference or a portion of area. Those calculations have different units and answer different questions. A sector's outside boundary also includes two straight radii, which a curved-edge calculation alone omits.
Set up the model
A useful answer starts with clear assumptions:
- The fan is a flat circular sector with its vertex exactly at the centre.
- Its radius is 9 cm along either straight edge.
- Paper thickness and any handle are excluded; π forms are exact.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Measure a circular fan's edge and surface
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A paper fan is an ideal 120° sector of a circle with radius 9 cm. Find its curved-edge length, surface area, and full perimeter.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Find the circle fraction
120°/360° = 1/3
The two radii enclose one third of a complete turn. Both the chosen arc and sector area use this fraction.
Measure the curved edge
s = (1/3)(2π × 9) = 6π cm
Circumference measures length. Its one-third share is approximately 18.85 cm, with no square units.
Measure surface and full boundary
A = (1/3)π × 9² = 27π cm²; P = 6π + 18 cm
The surface uses radius squared. The full perimeter adds both 9 cm radii to the curved edge, giving approximately 36.85 cm.
The result
Arc length is 6π cm, area is 27π cm², and perimeter is 18 + 6π cm.
An arc and a chord connecting the same endpoints are different paths. The arc follows the curved edge; substituting a straight connecting segment would change the shape whose perimeter you measure.
Common mistakes to catch
- The sector perimeter includes two radii in addition to the arc.
- Using 18 cm as the radius confuses the diameter with the given radius.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find the arc length of a 90° sector of radius 8 cm.
Show a hint
Take one quarter of the circumference.
Reveal answer and explanation
4π cm
(90/360)(2π × 8) = 4π cm.
Practice 2
A radius-6 cm sector has area 12π cm². Find its central angle.
Show a hint
Compare with the full-circle area 36π.
Reveal answer and explanation
120°
The sector is one third of the circle, so its angle is 360°/3.
Take the idea with you
For a circular slice, name the requested quantity before choosing circumference, area, or a combined boundary formula.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Separate a box's covering from its capacity
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