GANITA MANJARI • CLASS 9 • CHAPTER 6
From athletics tracks to ancient Indian formulas
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Ganita Manjari • Class 9 • Chapter 6
Akash Srivastva | www.akashmaths.online
INTRODUCTION • ATHLETICS TRACK STAGGER
Outer lanes have larger radius on the curves. Without stagger, outer-lane athletes would run a longer distance.
Stagger = forward offset that makes the total distance equal for every lane.
Think & Reflect
If a school has a 200 m track instead of 400 m, does it need a smaller stagger for the same 4×100 m race?
Book p.118
Akash Srivastva | www.akashmaths.online
6.1 PERIMETER OF A SHAPE
Square
Perimeter = 4a
Ratio 4 : 1
Equilateral Triangle
Perimeter = 3a
Ratio 3 : 1
Rectangle
Perimeter = 2(a+b)
Think & Reflect
What is the connection between perimeter of a circle and the stagger calculation on a track?
Book p.119
Akash Srivastva | www.akashmaths.online
6.2 CIRCUMFERENCE & THE RATIO π
For every circle: C / D = π
C = 2πr = πD
D
C ≈ 3.14 × D
Circumference unrolled
Book p.120–121
Akash Srivastva | www.akashmaths.online
6.2 HISTORICAL JOURNEY OF π
Mesopotamia (~1900 BCE)
Archimedes (~250 BCE)
Zu Chongzhi (480 CE)
Mādhava (~1400 CE)
Book p.121–123
Akash Srivastva | www.akashmaths.online
6.2 MĀDHAVA’S INFINITE SERIES
π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − …
Mādhava of Saṅgamagrāma (Kerala School, ~1400 CE) calculated π correct to 11 decimal places using this series.
Book p.123
Akash Srivastva | www.akashmaths.online
6.3 CALCULATING TRACK STAGGERS
Extra distance on a half-turn = πw
For w = 1.22 m → Stagger ≈ 3.83 m
Inner radius does not affect the stagger.
Book p.124
Akash Srivastva | www.akashmaths.online
6.4 ARC LENGTH
Arc Length = 2πr × (θ / 360°)
Book p.125
Akash Srivastva | www.akashmaths.online
6.5 PERIMETER PARADOX
A large semicircle and several smaller semicircles drawn on the same diameter have the same total length.
Because the sum of the smaller diameters equals the large diameter, their semicircular paths add up to the same value.
Book p.126
Akash Srivastva | www.akashmaths.online
6.6–6.7 AREA OF RECTANGLE & PARALLELOGRAM
Rectangle
Area = length × breadth
Parallelogram
Area = base × height
A parallelogram can be cut and rearranged into a rectangle of the same area.
Book p.127–128
Akash Srivastva | www.akashmaths.online
6.8 AREA OF A TRIANGLE
Area = ½ × b × h
Any triangle occupies exactly half the area of the rectangle that has the same base and height.
Book p.129
Akash Srivastva | www.akashmaths.online
6.8.1 HERON’S FORMULA
Area = √[s(s−a)(s−b)(s−c)]
where s = (a+b+c)/2
No height is required. Discovered by Heron of Alexandria.
Book p.130
Akash Srivastva | www.akashmaths.online
6.8.2 BRAHMAGUPTA’S FORMULA
Area = √[(s−a)(s−b)(s−c)(s−d)]
where s = (a+b+c+d)/2
When one side becomes zero, Brahmagupta’s formula reduces to Heron’s formula.
Book p.131
Akash Srivastva | www.akashmaths.online
6.9 BAUDHĀYANA’S SQUARING
Baudhāyana (~800 BCE) gave a geometric construction to make a square equal in area to a given rectangle.
Side of the square = √(ab)
Using the identity: ab = [(a+b)/2]² − [(a−b)/2]²
Book p.132–133
Akash Srivastva | www.akashmaths.online
6.9 SQUARING A TRIANGLE
1. Convert the triangle into a rectangle of equal area (base × half-height).
Think & Reflect
Why is converting a triangle into a rectangle first a helpful step?
Book p.133
Akash Srivastva | www.akashmaths.online
6.10 AREA OF A CIRCLE
Archimedes: Area of circle = Area of a right triangle with base = circumference (2πr) and height = radius (r)
½ × (2πr) × r = πr²
Book p.134
Akash Srivastva | www.akashmaths.online
6.10 NĪLAKAṆṬHA’S VISUAL PROOF
Thin sectors of the circle are rearranged alternately to form a near-rectangle.
Base ≈ πr Height = r
Area = πr × r = πr²
≈ πr
r
Sectors rearranged
Book p.135
Akash Srivastva | www.akashmaths.online
KEY TAKEAWAYS & THANK YOU
From tracks to temples — measuring space connects mathematics with the real world.
“Geometry is the foundation of all painting.”
— Albrecht Dürer
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Ganita Manjari • Class 9
Akash Srivastva | www.akashmaths.online
