Heron's formula (derivation using trig) — Hinglish notes

3.1.22 · Maths › Advanced Trigonometry

WHAT is Heron's formula?

WHY it's beautiful: isme sirf lengths chahiye — na height, na angle, na coordinates. Perfect jab tumhare paas sirf ruler ho.

Figure — Heron's formula (derivation using trig)

HOW we derive it (from first principles)

Hum shuru karte hain us ek area formula se jo sides aur angle ko mix karta hai, phir angle ko khatam karte hain.

Step 1 — Trig area se shuru karo. $Area = \frac{1}{2} ab sin C$ Ye step kyun? Do sides $a, b$ aur unka included angle $C$ triangle ko pin kar dete hain, isliye ye area hona hi chahiye. Isme angle $C$ hai — wahi dushman hai jise hum eliminate karenge.

Step 2 — Law of Cosines laao taaki $cos C$ ko sirf sides mein describe kar sako. $c^{2} = a^{2} + b^{2} - 2 ab cos C \Rightarrow cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab} .$ Ye step kyun? Hum $sin C$ ko $sin^{2} C = 1 - cos^{2} C$ se express kar sakte hain, aur $cos C$ ab hum purely sides mein jaante hain.

Step 3 — area $^{2}$ likho (squaring se $sin$ ka square root hat jaata hai). $Area^{2} = \frac{1}{4} a^{2} b^{2} sin^{2} C = \frac{1}{4} a^{2} b^{2} (1 - cos^{2} C) .$ Ye step kyun? $sin^{2} = 1 - cos^{2}$ sab kuch $cos C$ mein badal deta hai, jo pure sides hai.

Step 4 — $1 - cos^{2} C = (1 - cos C) (1 + cos C)$ factor karo aur substitute karo. $1 + cos C = \frac{2 ab + a ^{2} + b ^{2} - c ^{2}}{2 ab} = \frac{( a + b ) ^{2} - c ^{2}}{2 ab} = \frac{( a + b + c ) ( a + b - c )}{2 ab} .$ $1 - cos C = \frac{2 ab - a ^{2} - b ^{2} + c ^{2}}{2 ab} = \frac{c ^{2} - ( a - b ) ^{2}}{2 ab} = \frac{( c + a - b ) ( c - a + b )}{2 ab} .$ Ye step kyun? Har numerator ek difference of squares hai, jo linear factors mein toot jaata hai jo "semi-perimeter" chillate hain.

Step 5 — Wapis plug karo. $Area^{2} = \frac{a ^{2} b ^{2}}{4} \cdot \frac{( a + b + c ) ( a + b - c )}{2 ab} \cdot \frac{( c + a - b ) ( c - a + b )}{2 ab} .$ $a^{2} b^{2}$ do $2 ab$ 's ko cancel kar deta hai: $Area^{2} = \frac{( a + b + c ) ( a + b - c ) ( b + c - a ) ( c + a - b )}{16} .$

Step 6 — Semi-perimeter mein convert karo. $s = \frac{a + b + c}{2}$ ke saath: $a + b + c = 2 s, a + b - c = 2 (s - c), b + c - a = 2 (s - a), c + a - b = 2 (s - b) .$ To $Area^{2} = \frac{( 2 s ) 2 ( s - a ) 2 ( s - b ) 2 ( s - c )}{16} = s (s - a) (s - b) (s - c) .$

Worked examples

Steel-manned mistakes

Active recall

Recall Kya tum bina dekhe derivation rebuild kar sakte ho?

Shuru: Area $= \frac{1}{2} ab sin C$ .
Square karo: Area $^{2} = \frac{1}{4} a^{2} b^{2} (1 - cos^{2} C)$ .
Law of cosines: $cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}$ .
$1 \pm cos C$ ko difference of squares se factor karo.
$a^{2} b^{2}$ cancel karo, $\frac{( a + b + c ) ( a + b - c ) ( b + c - a ) ( c + a - b )}{16}$ milega.
$s$ se rewrite karo → $s (s - a) (s - b) (s - c)$ .

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tumhare paas teen sticks hain aur tum triangle banate ho. Ek baar sticks choose ho gayi, triangle sirf ek hi shape ki ho sakti hai — tum use stretch nahi kar sakte. To uske "andar ki jagah" pehle se sides ki lengths se tay ho jaati hai. Heron ka rule ek magic recipe hai: teeno lengths jodo, aadha karo (use $s$ kaho), phir $s$ ko ( $s$ minus har ek stick) se multiply karo. Sab ka square root lo — boom, area! Na height measure karni, na protractor. Humne ye ek aise formula se shuru karke nikala jo ek angle use karta hai, phir ek aur rule se us angle ko stick lengths se swap kiya, jab tak angle poori tarah gayab nahi ho gaya.

Flashcards

Semi-perimeter

s

kya hai?

s = \frac{a + b + c}{2}

, perimeter ka aadha.

Heron's formula batao.

Area = s (s - a) (s - b) (s - c)

jahan

s = \frac{a + b + c}{2}

Derivation shuru karne wala trig area formula kaun sa hai?

Area = \frac{1}{2} ab sin C

Kaun si identity

sin C

cos C

ke favor mein khatam karti hai?

sin^{2} C = 1 - cos^{2} C

cos C

ko sides mein kaise express karte hain?

cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}

(Law of Cosines).

1 - cos^{2} C

ko factor karne ki algebraic trick kaun si hai?

Difference of squares:

(1 - cos C) (1 + cos C)

, har ek sides mein difference of squares hai.

s - a, s - b, s - c

positive kyun hone chahiye?

Triangle inequality; warna koi real triangle nahi (root negative ho jaata hai).

Side

a

ke equilateral triangle ka area Heron se?

\frac{3}{4} a^{2}

13-14-15 triangle ka area?

84

(

s = 21

ke saath).

s

mein switch karne se pehle denominator mein kya aata hai?

16

, product

(2 s) (2 (s - a)) (2 (s - b)) (2 (s - c))

se.

Connections

Law of Cosines — wo engine jo $cos C$ ko sides mein badalta hai.
Area of a triangle = ½ab sin C — shuru karne ki jagah.
Triangle Inequality — guarantee deta hai ki root real hai.
Semi-perimeter and Incircle (r = Area/s) — $s$ inradius mein wapas aata hai.
Difference of Squares Factoring — key algebra move.
Heronian Triangles — integer sides, integer area wale triangles.