Area-of-a-Sector (1)

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Area of a Sector

Calculating fractions of circular space in real-world design

The Pizza Slice Challenge

Consider a whole circular pizza cut evenly into eight identical slices:

• What fraction of the complete 360° rotation does one single slice represent?

• If the entire circular pizza has an area of 800 cm², what must be the area of one slice?

WHATS IS THE TOTAL OF A CIRCLE

A

360 DEGREE

B

180 DEGREE

C

90 DEGREE

D

40 DEGREE

Learning Objectives

Derive the formula for the area of a sector

Apply the formula for the area of a sector to solve mathematical and real-life problems.

Text

Recap: Circle Fundamentals

• Radius (r) and Diameter (d): The diameter passes across the centre, so d = 2r.

• Area of a full circle: Area = πr².

• Full revolution: A complete rotation around the centre measures exactly 360°.

Any sector is simply a fraction of this complete 360° angle.

WHATS YOUR CONCLUSION

Deriving the Sector Formula

A sector is enclosed by two radii and an arc. The central angle θ defines what portion of the circle is taken:

Rally Coach:- Calculate the area of shaded region.

Step 1: Identify Given Values

Radius r = 6 cm, Central angle θ = 60°.

Calculate the fractional multiplier: 60° / 360° = 1/6.

Tip

Simplify θ/360 before multiplying to keep calculations clean.

Step 2: Calculate Area

Full circle area = π × 6² = 36π cm².

Sector area = (1/6) × 36π = 6π ≈ 18.85 cm² (to 2 d.p.).

Rally Coach: Show Me & Reasoning

Discussion

Turn to your partner and explain your mathematical reasoning:

• Core Question: A garden sector has radius r = 7 cm and angle θ = 90°. What is its area? Express in terms of π .

.Stretch Reasoning: Explain geometrically and algebraically why a 180° sector always has exactly half the area of the full circle.

Consider a whole circular pizza cut evenly into eight identical slices:

• What fraction of the complete 360° rotation does one single slice represent?

• If the entire circular pizza has an area of 800 cm², what must be the area of one slice?

Urban Planning in the UAE

Roundabouts & Roads

Civil engineers calculate sector areas to estimate asphalt, kerbing, and safety clearances on junctions.

Fountains & Plazas

Plazas divide circular spaces into decorative sectors to allocate paving tiles and drainage accurately.

Irrigated Parks

Sprinklers sweep across defined sector angles to distribute precise water volumes across landscaping.

SDG 11: Cities

Accurate area calculation prevents material wastage, supporting sustainable development and resource efficiency.

Choice Board

Listening: Engineering Sustainable Plazas

Listen to the audio and answer the questions

1

What is the radius of the circular lawn sector?

2

What fraction of the complete circle does 72° represent?

3

What is the exact sector area in terms of pi?

4

Which UN Sustainable Development Goal does this planning support?

Calculating Sector Greenery in Dubai

▶ Play audio

Listening: Engineering Sustainable Plazas

Check your answers below

✅

1

20 metres

2

One fifth (1/5)

3

80π square metres

4

SDG 11: Sustainable Cities and Communities

Calculating Sector Greenery in Dubai

▶ Play audio

Exit Ticket: Sector Area Mastery

Question 1:

Calculate the area of a sector with radius 7 cm and central angle 90° (use π ≈ 22/7).

Question 2:

If a circular sector has central angle 120°, what simplified fraction of the total circle area does it occupy?

Question 3:

Why does halving the central angle while keeping the radius constant halve the sector area?

Exit Ticket: Sector Area Mastery

Answer 1:

38.5 cm² (calculated as 1/4 × π × 49)

Answer 2:

1/3 (since 120° / 360° = 1/3)

Answer 3:

Sector area is directly proportional to the central angle θ.

✅

Enter your question here

1

ITS CORRECT

2

ITS WRONG