Mathematics: Class 9 — Chapter 6: Measuring Space: Perimeter and Area

6.1 Perimeter of a Shape

perimeter is the total length around the border of a closed two-dimensional shape.
Square: $P = 4a$ (where $a$ is the side length). The ratio of perimeter to side is always $4:1$.
Equilateral Triangle: $P = 3a$ (where $a$ is the side length). The ratio of perimeter to side is always $3:1$.
Rectangle: $P = 2(a + b)$ (where $a$ is length and $b$ is width).

6.2 Perimeter of a Circle & The C/D Ratio

The ratio of the circumference ($C$) to the diameter ($D$) of any circle is constant and is denoted by the Greek letter π (pi).
irrational number: $\pi$ is irrational, meaning its decimal expansion goes on forever without repeating.
Historical Approximations:
- Mesopotamia (c. 1900 BCE): $\pi \to 3.125$
- Archimedes (c. 250 BCE): $3\frac{10}{71} < \pi < 3\frac{1}{7}$
- Zu Chongzhi (480 CE): $\pi \to \frac{355}{113} \to 3.1415929$
- Aryabhata (499 CE): $\pi \to \frac{62832}{20000} = 3.1416$
- Madhava of Sangamagrama: First exact infinite series: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \text{...}$

6.4 Length of an Arc of a Circle

Circumference: $C = 2\pi r$ (where $r$ is the radius).
Semicircle Arc: $l = \pi r$
Quarter Circle Arc: $l = \frac{\pi r}{2}$
General Arc Length: For an arc subtending an angle $\theta^\theta$ at the center: $$l = 2\pi r \times \frac{\theta^\theta}{360^\theta}$$

6.6 & 6.7 Area of Rectangles and Parallelograms

Rectangle: $A = ab$ (where $a$ and $b$ are adjacent sides).
Parallelogram: $A = \text{base} \times \text{height} = bh$. Any parallelogram can be geometrically transformed into a rectangle of equal base and height.

6.8 Area of a Triangle

Standard Formula: $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}bh$.
Median Property: A median of a triangle divides it into two triangles of equal area.
Heron's Formula: For a triangle with sides $a$, $b$, and $c$: $$A = \text{s(s-a)(s-b)(s-c)}$$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter.
Circumcircle Connection: $A = \frac{abc}{4R}$ (where $R$ is the circumradius).
Incircle Connection: $A = r \times s$ (where $r$ is the incircle radius and $s$ is the semi-perimeter).

6.9 Brahmagupta's Formula for Cyclic 4-gons

A cyclic 4-gon is a quadrilateral whose vertices all lie on a single circle.
Brahmagupta's Formula: $$A = \text{(s-a)(s-b)(s-c)(s-d)}$$ where $s = \frac{a+b+c+d}{2}$ is the semi-perimeter.
Generalisation: Heron's formula is a special case of Brahmagupta's formula where one side length $d = 0$.

6.10 Area of a Circle & Sectors

Area of a Circle: $A = \pi r^2$.
Sector of a Circle: The region bounded by an arc and two radii. Area of a sector with central angle $\theta^\theta$: $$A = \pi r^2 \times \frac{\theta^\theta}{360^\theta}$$
Segment of a Circle: The region bounded by an arc and a chord.