Lissafin Yankin Sama na Sphere
Sphere yana ɗaya daga cikin siffofi na geometric da muka fi fuskanta a rayuwarmu ta yau da kullun. Siffarsa tana bayyana a cikin abubuwa daban-daban, daga ƙwallon ƙafa da marmara zuwa ƙwallon kwando da taurari da taurari. A cikin lissafi, sphere yana da halaye na musamman saboda duk wuraren da ke samansa suna da nisa daidai daga wuri ɗaya na tsakiya. Wani muhimmin batu da ake nazari akai-akai game da spheres shine yankin saman. Fahimtar yadda ake ƙididdige shi ba wai kawai yana da amfani ga dalilai na makaranta ba har ma yana da alaƙa da fannoni daban-daban kamar injiniyanci, kimiyyar lissafi, gine-gine, har ma da ƙirar samfura.
Fahimtar siffar da kuma samanta
Kafin mu shiga cikin dabarar, muna buƙatar fahimtar menene yankin saman wani yanki. Yankin saman wani yanki shine harsashin waje wanda ke kewaye da dukkan yanki. Idan za a iya "yanke" da "miƙa" wani yanki (kodayake wannan yana da wahala), to shine harsashin waje wanda muke son ƙididdige shi.
Lambun yana da wasu muhimman abubuwa, ciki har da:
1. Tsakiyar wurin: wurin da ke tsakiyar ƙwallon.
2. Radius (r): nisan daga tsakiyar wurin zuwa saman ƙwallon.
3. Diamita (d): mafi tsayin nisan da ke ratsa tsakiyar ƙwallon, ƙimarsa ta ninka radius sau biyu (d = 2r).
Lokacin da ake ƙididdige yankin saman wani yanki, ƙimar da aka fi amfani da ita ita ce radius.
Tsarin yankin saman fili na wani yanki
Tsarin da ake amfani da shi don yankin saman wani yanki shine:
L = 4πr²
Bayani:
– L = yankin saman ƙwallon
– π (pi) = daidaiton kusan 3,14159 (sau da yawa ana zagaye shi zuwa 3,14 ko 22/7)
– r = radius na ƙwallon
Wannan dabarar ta nuna cewa faɗin saman wani yanki ya dogara ne da murabba'in radius ɗin. Wannan yana nufin cewa idan aka ninka radius na wani yanki, faɗin saman zai ninka sau huɗu. Wannan ya faru ne saboda sinadarin r² da ke cikin dabarar.
Me yasa dabarar ta zama 4πr²?
A makaranta, wannan dabara yawanci ana bayar da ita ne a matsayin dabarar da aka riga aka shirya. Duk da haka, yana da sauƙin tunawa da amfani idan muka fahimci ainihin ra'ayin. A ra'ayinmu, akwai dangantaka tsakanin ƙwallo da da'ira.
Faɗin da'ira shine πr². Abin sha'awa, faɗin saman wani yanki ya ninka girman da'ira mafi girma sau huɗu (da'ira mai radius iri ɗaya da siffar). Saboda haka:
Faɗin saman wani yanki = 4 × (faɗin da'ira)
= 4 × πr²
= 4πr²
Ana iya fahimtar wannan dangantakar "sau huɗu" ta hanyar hanyoyin lissafi da gwaji (misali kwatanta yankin saman wani yanki da yankin saman wani silinda), kodayake yawanci ana nazarin shaidarsa ta asali a wani mataki mafi girma.
Matakai don ƙididdige yankin saman wani yanki
Domin ya zama mafi tsari, ga matakai na gaba ɗaya don ƙididdige yankin saman wani yanki:
1. Ƙayyade radius na ƙwallon (r)
Idan an san diamita, to r = d/2.
2. Yi amfani da dabarar L = 4πr²
Sanya ƙimar r a cikin dabarar.
3. Zaɓi ƙimar da ta dace ta π
– Yi amfani da π = 3,14 don lissafin aiki.
– Yi amfani da π = 22/7 idan r shine ninki na 7 don samun sakamako mai kyau.
4. Lissafa r², sannan a ninka da 4π
Tabbatar an gudanar da ayyukan lissafi a hankali.
5. Rubuta raka'o'in yanki
Idan r yana cikin cm, to yankin yana cikin cm²; idan r yana cikin mita, yanki yana cikin m², da sauransu.
Misali tambaya ta 1: radius da aka sani
Tambaya: Layin ƙasa yana da radius na 7 cm. Lissafa yankin samansa.
Mafita:
– r = 7 cm
– L = 4πr²
– L = 4 × (22/7) × 7²
– L = 4 × (22/7) × 49
– L = 4 × 22 × 7
– L = 616 cm²
Don haka, yankin saman ƙwallon shine 616 cm².
Misali tambaya ta 2: idan aka yi la'akari da diamita
Tambaya: Diamita na ƙwallon ƙafa shine santimita 20. Menene faɗin samansa?
Mafita:
– d = 20 cm
– r = d/2 = 10 cm
– L = 4πr²
– L = 4 × 3,14 × 10²
– L = 4 × 3,14 × 100
– L = 1256 cm²
Don haka, faɗin saman da'irar shine 1256 cm².
Kurakuran da aka saba yi don gujewa
Lokacin da ake ƙididdige yankin saman wani yanki, wasu kurakurai da ke faruwa galibi sun haɗa da:
1. Rashin fahimtar radius da diamita
Mutane da yawa suna haɗa diamita cikin dabarar kamar dai ita ce radius. Duk da haka, dabarar tana amfani da r, ba d ba.
2. Na manta da yin murabba'i mai siffar radius
Tsarin shine r², ba r ba. Don haka idan r = 10, to r² = 100, ba 10 ba.
3. Raka'o'in da ba su daidaita ba
Misali, idan radius ɗin yana cikin cm, amma sakamakon an rubuta shi a matsayin m², tabbatar da cewa raka'o'in sun yi daidai tun daga farko.
4. Zaɓin π mara kyau
Yana da kyau a yi amfani da 3,14 ko 22/7, amma a yi amfani da su yadda ya kamata kamar yadda ake buƙata. Idan r ba ninki na 7 ba ne, amfani da 22/7 na iya haifar da ƙarin rarrabuwa masu rikitarwa.
Amfani da yankin saman wani yanki a rayuwa ta ainihi
Lissafin saman wani yanki ba wai kawai aikin lissafi ba ne. Ana amfani da wannan ra'ayi a yanayi da yawa, misali:
– Lissafa buƙatun fenti ko shafa fenti don abubuwa masu siffar ƙwallo, kamar manyan kayan ado masu siffar ƙwallo ko tankunan ƙwallo masu siffar ƙwallo.
- Lissafi a fannin kimiyyar lissafi, kamar fannin saman duniya don kimanta hasken rana da aka karɓa ko aka fitar.
– Tsarin samfura, misali ƙayyade yankin kayan da za a yi amfani da su wajen yin ƙwallon filastik ko ƙwallon roba.
– Ilimin Taurari, misali lissafin yankin saman taurari ko duniyoyi don nazarin zafin jiki da makamashi.
Mafi girman siffar ƙwallo, haka nan girman girman samanta, kuma wannan alaƙar tana bin dabarar 4πr². Saboda haka, ƙaramin canji a radius na iya haifar da babban canji a yankin.
Penutup
Faɗin saman wani yanki shine jimlar faɗin waje na ɓangaren. Don ƙididdige shi, muna amfani da dabarar L = 4πr², inda r shine radius na ɓangaren. Mabuɗin yin lissafi mai kyau shine tabbatar da cewa radius ɗin daidai ne (musamman idan an ba da diamita), yin taka tsantsan wajen yin murabba'i r, zaɓar ƙimar π da ta dace, da kuma rubuta raka'o'in yanki daidai.
Ta hanyar fahimtar matakai da misalai, ƙididdige yankin saman wani yanki zai ji daɗi sosai. A ƙarshe, wannan ƙwarewar ba wai kawai za a iya amfani da ita ga matsalolin lissafi ba har ma da matsalolin duniya daban-daban da suka shafi abubuwan da ke kewaye da mu.
