Hello friend !! Aaj hum equilateral triangle ke area ka formula proof karna sikhenge. Triangle kya hai ? aur isske kitne types hote hai ? Iss par humne ek article likha hai aap yaha click karke read kar sakte hai.
Ab humare mind me ye question aata hai ki, equilateral triangle ka area kya hota hai ? Aayiye issko bhi jaan lete hai.
" Equilateral triangle ke 3 sides ke bheetar ka space (jagah) ko iss triangle ka area kahte hai."
Niche diye figure -1 ko dekhiye. Iss figure me humne ek sambaahu tribhuj ∆ABC diya hai. Iss triangle ke bheetar ka purple part , iss triangle ke area ko represent karta hai.
Triangle ke area ko unit² me naapa jata hai. Isliye iss triangle ke area ka maatrak bhi unit² hai.
Proof: Sabse pahle hum ek equilateral triangle ∆ABC lete hai. Jiske sabhi 3 sides ki length "L" hai. Iss triangle ke vertex A se base (BC) par humne ek perpendicular (AD) draw kiya. Perpendicular (AD) base BC ko 2 equal length me divide karti hai.
Means BD = 1/2 (BC) = L/2
Niche diya figure-2 dekiye.
Ab hum jaante hai ki, kisi bhi triangle ka area (A) ho to
Matalab ki agar hume figure-2 ka area maalum karna hai to isske liye hume base (BC) aur height (AD) ki length chahiye.
Base (BC) ki length (L) hai . Ye to hume maalum hai. But height (AD) ko hume calculate karna hoga.
Figure -2 me ∆ABC ka height maalum karne ke liye hum ∆ABD ko lete hai. Hum dekh rahe hai ki ∆ABD ek right angle triangle hai.
Isliye pythagoras theorem se,
Figure-2 me AB = L, BD = L/2 aur AD = h (suppose)
AB² = AD² + BD²
L² = h² + L²/4
h² = L² - L²/4
h² = (4L² - L²)/4
h² = 3L²/4
h = √3L/2
Ab hume base aur height dono ki length maalum hai. Isliye ab hum ∆ABC ka area nikal sakte hai.
Equilateral triangle (sambaahu tribhuj) ka area formula proof
Equilateral triangle jisko hum hindi me sambaahu tribhuj kahte hai, ye aisa triangle hota hai jiska every side ka length equal hota hai aur saath hi saath iss ke all angles ( 3 angle ) 60°ke hote hai.Ab humare mind me ye question aata hai ki, equilateral triangle ka area kya hota hai ? Aayiye issko bhi jaan lete hai.
" Equilateral triangle ke 3 sides ke bheetar ka space (jagah) ko iss triangle ka area kahte hai."
Niche diye figure -1 ko dekhiye. Iss figure me humne ek sambaahu tribhuj ∆ABC diya hai. Iss triangle ke bheetar ka purple part , iss triangle ke area ko represent karta hai.
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| Figure - 1 |
Equilateral triangle ke Area ka formula:
Iss triangle ke area ka formula ko hum iss prakar se likhate hai:
Iss formula me "L" equilateral triangle ke side ki length hai.
Formula to aapne dekh liya . Ab hum issi formula ko proof karna start karte.
Proof: Sabse pahle hum ek equilateral triangle ∆ABC lete hai. Jiske sabhi 3 sides ki length "L" hai. Iss triangle ke vertex A se base (BC) par humne ek perpendicular (AD) draw kiya. Perpendicular (AD) base BC ko 2 equal length me divide karti hai.
Means BD = 1/2 (BC) = L/2
Niche diya figure-2 dekiye.
 |
| Figure-2 |
Base (BC) ki length (L) hai . Ye to hume maalum hai. But height (AD) ko hume calculate karna hoga.
Figure -2 me ∆ABC ka height maalum karne ke liye hum ∆ABD ko lete hai. Hum dekh rahe hai ki ∆ABD ek right angle triangle hai.
Isliye pythagoras theorem se,
Figure-2 me AB = L, BD = L/2 aur AD = h (suppose)
AB² = AD² + BD²
L² = h² + L²/4
h² = L² - L²/4
h² = (4L² - L²)/4
h² = 3L²/4
h = √3L/2
Ab hume base aur height dono ki length maalum hai. Isliye ab hum ∆ABC ka area nikal sakte hai.
Some related questions:
#Q-1. Yadi ek sambaahu tribhuj ∆ABC ke bhuja ki lambai 7 cm ho to iss ka kshetraphal (Area) kya hoga ?
Solution-: Diya hai sambaahu tribhuj ∆ABC ke side ki length (L) = 7 cm
Isliye , hum jaante hai ki sambaahu tribhuj ka Area = √3/4 × L²
= √3/4 × 7²
= √3/4 × 49
= 21.21 cm² Ans.
#Q-2. Ek sambaahu tribhuj ka area 14 cm² hai to iss ke side (bhuja) ki lambai kya hogi?
Solution-: Diya hai A = 30 cm² aur hume maalum karna hai iss tribhuj ke side ki length L = ?
Area ka formula use karne par:
(√3/4)×L² = 30
L² = (30×4)/√3 = 120/√3
= 69.284
L = √69.284 = 8.32
Isliye , iss triangle ke side ki length (L) = 8.32 Ans.
I hope ki aapko aaj ka math topic
equilateral triangle (sambaahu tribhuj) ke area ka formula proof samjh me aaya hoga. Agar phir bhi kisi jagah par aapko nahi samjh aa raha hai ya koi other problem ho rahi ho to aap humare facebook page par massage karke puchh sakte hai. Aap nichey diye comment box ki help se bhi comment karke bhi apni problem hume send kar sakte ho.Hum aapki puri help karne ki koshish karenge.
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