Class 10 Mathematics – Areas Related to Circles | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: Think of the face of a watch — the hour hand sweeps a sector of the circular dial as time passes. Farmers water fields using circular sprinkler systems that cover sectors. Pizza is cut into sectors. Knowing the area of a sector or segment helps calculate land coverage, material cost, and even how much topping fits on a slice — all using the formulas in this chapter.

⚪ Topic 1: Perimeter of a Circle

The perimeter of a circle is called circumference.
Formula: C = 2πr (r = radius).
Also written as C = πd (d = diameter).
π is roughly equal to 3.14 or 22/7.
Circumference means the distance around the circle.

⚪ Topic 2: Area of a Circle

Area of a circle = πr².
This gives the space inside the circle.
Always use square units for area.
If diameter is given, find r = d/2 first.
Example: r = 7 cm, area = 22/7 × 49 = 154 cm².

⚪ Topic 3: What Is a Sector

A sector is a slice of a circle.
Think of it like a pizza slice shape.
It is bounded by two radii and an arc.
The angle at the centre is called θ (theta).
A semicircle is a sector with θ = 180°.

⚪ Topic 4: Arc Length

An arc is a part of the circumference.
Arc length = θ/360 × 2πr.
Here θ is the angle in degrees.
Bigger angle means longer arc.
Full circle arc (θ = 360°) = 2πr.

⚪ Topic 5: Area of a Sector

Area of sector = θ/360 × πr².
This is a fraction of the full circle area.
For θ = 90°, area = one-quarter of circle.
For θ = 180°, area = half of circle.
Always state the angle and radius clearly.

Area of Sector = θ/360 × πr²

⚪ Topic 6: What Is a Segment

A segment is the region between a chord and its arc.
A chord cuts the circle into two segments.
The smaller part is the minor segment.
The larger part is the major segment.
Segment area = sector area minus triangle area.

⚪ Topic 7: Area of a Segment

First find the area of the sector.
Then find the area of the triangle formed.
Minor segment = sector area − triangle area.
Major segment = circle area − minor segment.
The triangle has two sides equal to radius.

Segment Area = θ/360 × πr² − Area of Triangle

⚪ Topic 8: Area of a Ring

A ring is the space between two circles.
Both circles share the same centre.
Ring area = πR² − πr².
This simplifies to π(R² − r²).
R = outer radius, r = inner radius.

⚪ Topic 9: Combination Figures

Some shapes combine circles with other figures.
Break the shape into simpler parts.
Find each part's area separately.
Add or subtract areas as needed.
Draw a rough sketch to see the parts.

⚪ Topic 10: Key Formulas to Remember

Circumference = 2πr.
Area of circle = πr².
Arc length = θ/360 × 2πr.
Sector area = θ/360 × πr².
Segment area = sector area − triangle area.

🎨 Section 2 — Easy Illustrations

Picture 1: Circle with Radius and Diameter

A circle with its radius (r), diameter (d = 2r), and circumference labeled.

Picture 2: Sector and Arc

A sector is the shaded slice. The curved edge is the arc. The angle at centre is θ.

Picture 3: Segment of a Circle

A chord divides the circle into a minor segment (small) and a major segment (large).

Picture 4: Ring (Annulus)

A ring (annulus) is the shaded region between two concentric circles.

Picture 5: Combination Figure

A combination figure: add the areas of the rectangle and the semicircle together.

✏️ Section 3 — Worked Examples

Example 1: Area of a Sector

Question: Find the area of a sector with radius 14 cm and angle 90°. (Use π = 22/7)

1 Write the formula: Area = θ/360 × πr².

2 Put the values: θ = 90°, r = 14 cm.

3 Area = 90/360 × 22/7 × 14 × 14.

4 Simplify: 90/360 = 1/4.

5 Area = 1/4 × 22/7 × 196.

6 Area = 1/4 × 22 × 28 = 1/4 × 616.

7 Area = 154 cm².

Area of the Sector = 154 cm²

Example 2: Area of a Segment

Question: Find the area of the minor segment of a circle with radius 21 cm and central angle 120°. (Use π = 22/7, √3 = 1.73)

1 Sector area = 120/360 × 22/7 × 21 × 21.

2 = 1/3 × 22/7 × 441 = 1/3 × 1386.

3 Sector area = 462 cm².

4 Triangle area: the triangle has two sides = 21 cm, angle = 120°.

5 Triangle area = 1/2 × r² × sinθ = 1/2 × 441 × sin120°.

6 sin120° = √3/2 = 1.73/2 = 0.865.

7 Triangle area = 1/2 × 441 × 0.865 = 190.93 cm².

8 Segment area = 462 − 190.93 = 271.07 cm².

Area of Minor Segment = 271.07 cm²

Example 3: Combination Figure (Park Problem)

Question: A rectangular park is 50 m long and 30 m wide. Two semicircular flower beds are made on the shorter sides. Find the area of the remaining park. (Use π = 22/7)

1 Rectangle area = 50 × 30 = 1500 m².

2 Each semicircle has diameter = 30 m, so r = 15 m.

3 Area of one semicircle = πr²/2 = 22/7 × 225/2.

4 = 22 × 225 / 14 = 4950/14 = 353.57 m².

5 Two semicircles = 2 × 353.57 = 707.14 m².

6 Remaining area = 1500 − 707.14 = 792.86 m².

Remaining Park Area = 792.86 m²

Two semicircles = one full circle of radius 15 m

📊 Section 4 — Cheat Sheet

Term / Formula	What It Means	Key Fact
Circumference	Distance around the circle	C = 2πr
Area of Circle	Space inside the circle	A = πr²
Sector	Pizza-slice region of a circle	Bounded by 2 radii + arc
Arc Length	Part of the circumference	θ/360 × 2πr
Sector Area	Area of the pizza slice	θ/360 × πr²
Segment	Region between chord and arc	Minor and Major
Segment Area	Sector area minus triangle area	Sector − Triangle
Ring Area	Between two concentric circles	π(R² − r²)
Semicircle Area	Half of a circle	πr² / 2
Quadrant Area	Quarter of a circle	πr² / 4
Combination Figures	Add or subtract areas	Break into parts
π Values	Constant ratio	22/7 or 3.14

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. The perimeter of a circle is called _________.
2. Area of a sector = θ/360 × _________.
3. The region between a chord and its arc is a _________.
4. Area of a ring = π(R² − _________).
5. A semicircle has a central angle of _________ degrees.

✓ B. True or False

1. Arc length = θ/360 × πr². _______
2. A sector is bounded by two radii and an arc. _______
3. The area of a segment is always more than the sector area. _______
4. A quadrant is a sector with angle 90°. _______
5. Two semicircles of the same radius make one full circle. _______

✏️ C. Solve These

1. Find the area of a sector with radius 28 cm and angle 45°. (Use π = 22/7)

Show your work here

2. A circular garden has radius 35 m. A path 7 m wide runs around it. Find the area of the path.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Circumference	a. θ/360 × πr²
2. Area of sector	b. π(R² − r²)
3. Area of ring	c. 2πr
4. Segment area	d. Sector − Triangle

🎨 E. Draw and Label

Draw a circle. Shade a minor segment and label the chord, arc, and segment.

Draw your diagram here

✅ Detailed Solutions

Solution to Q1: Area of a Sector

Question: Find the area of a sector of a circle with radius 28 cm and angle 45°. (Use π = 22/7.)

1 Formula: Area of sector = (θ/360) × πr².

2 θ = 45°, r = 28 cm, π = 22/7.

3 Area = (45/360) × (22/7) × 28² = (1/8) × (22/7) × 784.

4 = (1/8) × (22 × 112) = (1/8) × 2464 = 308 cm².

∴ Area of sector = 308 cm²

Solution to Q2: Area of a Circular Path

Question: A circular garden has a radius of 35 m. A path 7 m wide is laid around it on the outside. Find the area of the path. (Use π = 22/7.)

1 Inner radius (garden) r = 35 m. Outer radius R = 35 + 7 = 42 m.

2 Area of path = π(R² − r²) = (22/7) × (42² − 35²).

3 42² = 1764, 35² = 1225. Difference = 1764 − 1225 = 539.

4 Area = (22/7) × 539 = 22 × 77 = 1694 m².

∴ Area of the circular path = 1694 m²

🃏 Section 6 — Revision Flashcards

👉

⚪

What is the formula for circumference?

C = 2πr

Also written as C = πd

🍕

What is a sector of a circle?

A sector is a pizza-slice shape bounded by two radii and an arc. Its area = θ/360 × πr².

📏

How do you find the arc length?

Arc = θ/360 × 2πr

It is a fraction of the full circumference.

🔬

How do you find the area of a segment?

Segment Area = Sector Area − Triangle Area

The triangle is formed by two radii and the chord.

🔘

What is a ring (annulus)?

The region between two concentric circles.

Area = π(R² − r²)

😐

What is a semicircle and its area?

Half of a circle (θ = 180°).

Area = πr² / 2

🧩

How do you solve combination figure problems?

Break the figure into simpler shapes. Find each area separately. Then add or subtract as required.

💡

What are the two common values of π?

π = 22/7 or 3.14

Use whichever value the question specifies.

Class 10 Mathematics — Areas Related to Circles

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⚪ Class 10 Mathematics — Areas Related to Circles Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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