📏 Maths Class 9 NCERT Lesson 6
Measuring Space: Perimeter and Area
Special Edition for Easy Learning
What is inside?
Section 1 — Classroom Notes
Section 2 — Illustrations
Section 3 — Worked Examples
Section 4 — Cheat Sheet
Section 5 — Handout
Section 6 — Revision Flashcards
Section 7 — Mind Map
📝 Section 1 — Classroom Notes
🌎 Real-World Hook: Imagine you want to put a fence around your garden. To know how much fencing wire you need, you calculate the perimeter. If you want to tile the garden floor, you need the area. A farmer fencing a triangular field uses Heron's formula. A builder laying pipes in a circular tank uses circumference. Perimeter and area are not just classroom topics — they are skills used every day by architects, farmers, tailors, and engineers!
Topic 1: What is Perimeter?
Perimeter = total length around a shape.
Walk along the edges. That total distance = perimeter.
Rectangle: P = 2 × (length + width).
In words: add length and width. Multiply by 2.
Square: P = 4 × side.
Triangle: P = side 1 + side 2 + side 3.
Unit is always cm, m, km, etc.
Topic 2: Circumference of a Circle
Circumference = perimeter of a circle.
It is the distance around the circle.
Formula: C = 2 × π × r
In words: two times pi times radius.
Also: C = π × d (pi times diameter).
r = radius. d = diameter = 2 × r.
C = 2 × π × r = π × d
Topic 3: What is Pi (π)?
π (pi) is a special number.
π = circumference ÷ diameter.
In words: divide circumference by diameter.
This ratio is the same for every circle!
π is about 3.14.
We often use 22/7 for easy calculation.
π is irrational. Its digits never end or repeat.
π ≈ 3.14159265...
Topic 4: Length of an Arc
An arc is a part of the circle's edge.
The arc makes an angle at the centre.
We call this angle θ (theta).
Arc length = 2πr × θ/360
In words: circumference times angle over 360.
Full circle = 360°. Half circle = 180°.
Half circle arc = π × r.
Arc length = 2πr × θ° ÷ 360°
Topic 5: Area of a Rectangle
Area = space inside a flat shape.
We measure area in square units.
Like cm², m², km².
Rectangle area = length × width.
In words: multiply length by width.
Square area = side × side = side².
Rectangle area = length × width
Topic 6: Area of a Parallelogram
A parallelogram has 2 pairs of parallel sides.
The bottom side is the base (b).
The height (h) is the perpendicular distance.
Height goes straight up from base to top.
Area = base × height = b × h.
In words: multiply base by height.
Height is NOT the slant side!
Parallelogram area = base × height
Topic 7: Area of a Triangle
A triangle is half of a parallelogram.
Area = ½ × base × height.
In words: half times base times height.
The base is any side you pick.
The height is perpendicular to that base.
Height goes from the base to the opposite corner.
Triangle area = ½ × base × height
Topic 8: Heron's Formula
Heron's formula uses all three sides.
No need to know the height!
First find s (semi-perimeter).
s = (a + b + c) ÷ 2
In words: add all sides, then divide by 2.
Area = √[s × (s−a) × (s−b) × (s−c)]
In words: take the square root of the product.
Area = √[s(s−a)(s−b)(s−c)]
Topic 9: Area of a Circle
Area of a circle = π × r².
In words: pi times radius squared.
r² means r × r.
Archimedes proved this formula long ago.
Cut the circle into thin slices.
Rearrange them into a rectangle-like shape.
Width = r, Length = πr. So area = πr².
Circle area = π × r²
Topic 10: Area of a Sector
A sector is a "pie slice" of a circle.
It is made by two radii and an arc.
The angle at the centre = θ.
Area = πr² × θ/360
In words: circle area times angle over 360.
Half circle (θ = 180°): area = ½ × πr².
Quarter circle (θ = 90°): area = ¼ × πr².
Sector area = πr² × θ° ÷ 360°
🎨 Section 2 — Illustrations
Picture 1: Circumference of a Circle
The circumference is the total distance around a circle.
Picture 2: Area of a Parallelogram
Area of parallelogram = base times height. Height is perpendicular!
Picture 3: Area of a Triangle
Triangle is half the rectangle. So area = half base times height.
Picture 4: Heron's Formula
Heron's formula lets you find area using only the three side lengths.
Picture 5: Area of a Sector
A sector is a pie slice of the circle. Its area depends on the angle.
📐 Section 3 — Worked Examples
Example 1: Circumference and Area of a Circle
A circle has radius 7 cm.
Find the circumference and area. Use π = 22/7.
1 Write the circumference formula: C = 2 × π × r
2 Put in the values: C = 2 × 22/7 × 7
3 The 7s cancel: C = 2 × 22 = 44 cm
Circumference = 44 cm
4 Write the area formula: A = π × r²
5 Put in the values: A = 22/7 × 7²
6 7² = 49. So A = 22/7 × 49
7 49 ÷ 7 = 7. So A = 22 × 7 = 154 cm²
Area = 154 cm²
C = 2πr A = πr²
Example 2: Heron's Formula
A triangle has sides 7 cm, 15 cm, 20 cm.
Find its area using Heron's formula.
1 Write down: a = 7, b = 15, c = 20.
2 Find s = (a + b + c) ÷ 2
3 s = (7 + 15 + 20) ÷ 2 = 42 ÷ 2 = 21
4 s − a = 21 − 7 = 14
5 s − b = 21 − 15 = 6
6 s − c = 21 − 20 = 1
7 Multiply: 21 × 14 × 6 × 1 = 1764
8 Area = √1764 = 42 cm²
Area = 42 cm²
s = 21 Area = 42 cm²
Example 3: Area of a Sector
A circle has radius 14 cm.
Find the area of a 90° sector. Use π = 22/7.
1 Write the formula: Area = πr² × θ/360
2 Put in values: A = 22/7 × 14² × 90/360
3 14² = 196. And 90/360 = 1/4.
4 A = 22/7 × 196 × 1/4
5 196 ÷ 7 = 28. So A = 22 × 28 × 1/4
6 22 × 28 = 616. Then 616 ÷ 4 = 154
Sector area = 154 cm²
Formula: πr² × θ/360
📋 Section 4 — Cheat Sheet
Word / Rule
What It Means
Perimeter
Total length around a shape
Circumference
Perimeter of a circle. C = 2πr
Pi (π)
About 3.14 or 22/7. Same for all circles
Arc length
Part of circumference. l = 2πr × θ/360
Rectangle area
length × width
Parallelogram area
base × height (h is perpendicular!)
Triangle area
½ × base × height
Heron's formula
√[s(s−a)(s−b)(s−c)], s = (a+b+c)/2
Circle area
π × r²
Sector area
πr² × θ/360 (pie slice of circle)
Semi-perimeter
Half the perimeter. s = (a+b+c)/2
Irrational number
Decimals never end, never repeat. π is one
✏️ Section 5 — Handout
A. Fill in the Blanks
1. The distance around a shape is called ________.
2. Circumference of a circle = 2 × ________ × r.
3. π is approximately ________ or 22/7.
4. Area of a triangle = ½ × ________ × height.
5. In Heron's formula, s = (a + b + c) ÷ ________.
B. True or False
1. π is a rational number. ________
2. Area of parallelogram = base × slant side. ________
3. Area of a circle = π × r². ________
4. A sector is a pie slice of a circle. ________
C. Solve
1. Find the circumference of a circle with radius 14 cm.
Use π = 22/7.
Show your work here
2. A triangle has sides 5 cm, 12 cm, 13 cm.
Find its area using Heron's formula.
Show your work here
D. Match the Columns
1. Circumference — (a) ½ × base × height
2. Triangle area — (b) πr²
3. Circle area — (c) base × height
4. Parallelogram area — (d) 2πr
E. Draw and Label
Draw a circle. Shade a 90° sector.
Label: radius, arc, angle, sector.
Draw your sector here
✅ Detailed Solutions — Handout Practice
✓ A. Fill in the Blanks — Answers
1 The distance around a shape is called perimeter.
2 Circumference of a circle = 2 × π × r.
3 π is approximately 3.14 (or 22/7).
4 Area of a triangle = ½ × base × height.
5 In Heron's formula, s = (a + b + c) ÷ 2. (s = semi-perimeter)