GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6

CIRCLE — Area, Sector and Applications

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a question.

What shape do alternating narrow circle sectors approach? How are a sector and a segment different?

Understand

Rearranged sectors approach a parallelogram of base πr and height r. Thus circle area is π $r^{2}$ . A sector uses two radii; a segment uses a chord and an arc.

Key Ideas

A = \frac{θ}{360} π r^{2}

Finite sectors do not make an exact rectangle. A segment is not a sector. Areas need square units.

Visualise

Number of equal sectors: 8

For finitely many sectors the edges remain curved. The limiting base approaches πr and height approaches r.

Minor central angle (degrees): 60

Sector: bounded by the arc and two radii.

Method

For a sector use the central-angle fraction of circle area. For a minor segment with angle at most 180°, subtract the triangle between the radii from the sector.

Worked Example

Given

Radius 7 cm, sector angle 60° (Exercise 6.3 Q1).

To Find

Find the area using the exercise approximation π ≈ $\frac{22}{7}$ .

Formula / Property / Theorem / Reasoning

Sector area = (θ/360)π $r^{2}$ .

Solution

Exact area = $\frac{49 π}{6}$ ${cm}^{2}$ .
Using π ≈ $\frac{22}{7}$ gives about $\frac{77}{3}$ ${cm}^{2}$ .

Result

About 25.67 ${cm}^{2}$ ; Q1 does not prescribe a final number of decimal places.

AIMS Method

A — Attempt independently. Make a prediction or start a sketch.
I — Investigate the idea. Identify the boundary, region and given conditions.
M — Make connections. Relate the figure to a known property or rearrangement.
S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.

Think first. Use AI purposefully. Verify always.

Focused Practice

A quadrant has radius 4 cm. Give its exact area and explain the fraction used.

Your reasoning

Check & Feedback

What must be subtracted from a 60° sector to obtain its minor segment?

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.10–6.10.1 · Figures 6.36–6.40 · printed pages 144–148.