Chapter 06 • Grade 9

Measuring Space: Perimeter and Area

From the staggers of an athletics track to the infinite series of Mādhava. Discover the laws governing the boundaries and contents of shapes.

6.1 The Stagger Logic

Why do Athletes start differently?

In a 4x100m relay, runners in outer lanes start ahead of those in inner lanes. This distance is called the stagger. Without it, the outer runner would travel a much longer distance to reach the same finish line. To calculate this, we must master the geometry of circles.

Basic Perimeters
  • Square: \( 4a \)
  • Equilateral Triangle: \( 3a \)
  • Rectangle: \( 2(a + b) \)
Circumference

The total length around a circle. It is directly proportional to its diameter.

6.2 The Ratio

The Adventurous Journey of \(\pi\)

The ratio of Circumference (\(C\)) to Diameter (\(D\)) is a universal constant. It connects the straight-edged world with the curves of nature.

1900 BCE • Mesopotamia

Used \(\pi \approx 3 + 1/8 = 3.125\). Realised it must be larger than 3.

250 BCE • Archimedes

"Trapped" \(\pi\) between 96-sided polygons: \( 3\frac{10}{71} < \pi < 3\frac{1}{7} \).

480 CE • Zu Chongzhi

Found the "Close Ratio": \( 355/113 \). Accurate for over 800 years!

c. 1400 • Mādhava

Discovered the first exact infinite series for \(\pi/4\). Birthed Calculus.

\( \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \)
6.4 Arc Logic

Length of an Arc

Using Rotational Symmetry, we can find parts of the circumference easily:

\( \text{Arc Length} = 2\pi r \times \frac{\theta^\circ}{360^\circ} \)
The 400m Track Formula

Stagger needed = \( 2\pi(r + w) - 2\pi r = 2\pi w \), where \( w \) is the lane width.

6.7 & 6.8 Foundations

Areas of Polygons

Parallelogram

Area = Base \(\times\) Height

Transformation: Cut a right triangle from one side and slide it to the other to form a rectangle.

Triangle

Area = \(\frac{1}{2} \times \text{Base} \times \text{Height}\)

Two congruent copies of a triangle fit together perfectly to form a parallelogram. Thus, triangle area is half of parallelogram area.
The Median Theorem

A median of a triangle divides it into two triangles of equal area (even if they have different shapes!).

6.8.1 Advanced

Heron’s & Brahmagupta’s Formulas

Can we find the area if we only know the side lengths? Yes, for triangles and cyclic 4-gons.

\( \text{Area}_{\triangle} = \sqrt{s(s-a)(s-b)(s-c)} \)
where \( s = \frac{a+b+c}{2} \) (Semi-perimeter)
Brahmagupta's Generalisation

For a cyclic 4-gon with sides \( a, b, c, d \):
\( \text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)} \)
Note: If \( d=0 \), this becomes Heron's formula!

6.9 Ancient Construction

Baudhāyana’s Sulbasūtra

In 800 BCE, Indian mathematicians knew how to "square" a rectangle—construct a square with the exact same area as a rectangle of sides \( a \) and \( b \).

Geometric Identity

\( ab = \left(\frac{a+b}{2}\right)^2 - \left(\frac{a-b}{2}\right)^2 \)

By constructing a right-angled triangle with hypotenuse \( \frac{a+b}{2} \) and one side \( \frac{a-b}{2} \), the third side automatically becomes \( \sqrt{ab} \)!

6.10 The Limit

Area of a Circle

Nīlakaṇṭha Somayājī (c. 1500) provided a beautiful visual proof: slice a circle into tiny sectors and rearrange them. They form a parallelogram with base \( \pi r \) and height \( r \).

\( \text{Area} = \pi r^2 \)
Area of a Sector

Subtending angle \( \theta^\circ \):
\( \text{Area} = \pi r^2 \times \frac{\theta^\circ}{360^\circ} \)