Te whanaungatanga i waenga i te roa o te pewa me te horahanga o te rāngai
I roto i te pāngarau, inā koa i roto i te āhuahanga papatahi, he ariā nui kei roto e pā ana ki ngā porowhita. E rua ngā ariā matua e kōrerohia pinepine ana ko te roa o te pewa me te horahanga ā-wāhanga. Mā te mārama pai ki ēnei ariā e rua ka taea e tātou te tatau i ngā āhuatanga rerekē o ngā porowhita, ahakoa i roto i ngā akoranga pāngarau whānui, i ngā tono hangarau, i te oranga o ia rā rānei.
Te Whakamāramatanga o te Porowhita
I mua i tā tātou haere tonu, he mea āwhina kia mārama tātou he aha te porowhita. Ko te porowhita te huinga o ngā pūwāhi katoa i roto i te papa he tawhiti pumau mai i tētahi pūwāhi kua hoatuhia ko te pokapū. Ko tēnei tawhiti pumau e mōhiotia ana ko te radius o te porowhita. He maha ngā āhuatanga nui o te porowhita, tae atu ki:
1. Pūwāhi Pokapū (O): Ko te pūwāhi pumau e inehia ai ngā tawhiti o ngā pūwāhi katoa o te porowhita.
2. Pūtoro (r): Te tawhiti mai i te pūwāhi pokapū ki tētahi pūwāhi i runga i te porowhita.
3. Te whānui (d): Ko te tawhiti roa rawa atu i tētahi pūwāhi ki tētahi atu i runga i tētahi porowhita e tika ana mā te pūwāhi pokapū. Ko te whānui e rua te roa o te pūtoro.
4. Porowhita (C): Ko te roa o te rārangi e karapoti ana i te porowhita, i tatauhia mā te whakamahi i te tātai \( C = 2 \pi r \).
Te Mārama ki te Roa o te Āwhata
Ko te roa o te pewa ko te roa o tētahi wāhanga motuhake o te porowhita. Me whakaaro tātou he porowhita nui e tapahia ana e ngā rārangi porowhita e rua. Mā ēnei rārangi porowhita ka wehea te porowhita kia rua ngā pewa, ka kiia ēnei ko ngā pewa matua me ngā pewa iti, i runga i ō rāua roa.
Hei tatau i te roa o te pewa, me mōhio tātou ki te radius o te porowhita me te rahi o te koki pokapū i hangaia e ngā rārangi radius e rua. Ka taea te tatau i te roa o te pewa mā te whakamahi i te tātai:
\[ L = \theta \times r \]
kāore i te mana:
– Ko te roa o te pewa ko \( L \),
– Ko te koki pokapū i roto i ngā rātiana ko \( \theta \),
– Ko te \( r \) te pūtoro o te porowhita.
Mena kei roto i ngā nekehanga te koki pokapū, ka huri te tātai ki:
\[ L = \left( \frac{\theta}{360} \right) \times 2 \pi r \]
Hei tauira, ki te mea he porowhita tāu me te radius o te 10 waeine me te koki pokapū o te 60 nekehanga, ka taea te tatau i te roa o te pewa penei:
\[ L = \left( \frac{60}{360} \right) \times 2 \pi \times 10 = \left( \frac{1}{6} \right) \times 20 \pi = \frac{20 \pi}{6} \approx 10.47 \, \text{unit} \]
Te Whakamāramatanga o te Horahanga o tētahi Rāngai
Ko te horahanga o tētahi wāhanga ko te horahanga o tētahi wāhanga motuhake o tētahi porowhita i hangaia e ngā rārangi porowhita e rua me te pewa e hono ana i a rāua. He maha ngā wā ka whakaritea ngā wāhanga ki ngā poro o tētahi pai, piza rānei. Hei tatau i te horahanga o tētahi wāhanga, me whakamahi tātou i te porowhita o te porowhita me te koki pokapū i hangaia e ngā rārangi porowhita e rua.
Ko te tātai mō te tatau i te horahanga o tētahi rāngai ko:
\[ A = \frac{1}{2} r^2 \theta \]
kāore i te mana:
– Ko te horahanga o te rāngai ko \( A \),
– Ko te koki pokapū i roto i ngā rātiana ko \( \theta \),
– Ko te \( r \) te pūtoro o te porowhita.
Mena kei roto i ngā nekehanga te koki pokapū, ka hurihia te tātai ki:
\[ A = \left( \frac{\theta}{360} \right) \times \pi r^2 \]
Hei tauira, me kī he porowhita tā tātou he 10 ngā waeine te whānui, ā, he 60 ngā nekehanga te koki pokapū. Kātahi ka taea te tatau i te horahanga o te rāngai penei:
\[ A = \left( \frac{60}{360} \right) \times \pi \times 10^2 = \left( \frac{1}{6} \right) \times 100 \pi = \frac{100 \pi}{6} \approx 52.36 \, \text{unit}^2 \]
Te Hononga i waenganui i te Roa o te Āwhata me te Horahanga o te Wāhanga
He hononga tata ngā ariā o te roa o te pewa me te horahanga ā-wāhanga nā te mea e whakawhirinaki ana ēnei ki te pūtoro o te porowhita me te koki pokapū. Mā te mōhio ki tētahi o ēnei e rua me ētahi atu mōhiohio pēnei i te pūtoro, te koki pokapū rānei, ka taea e tātou te tatau i tētahi atu.
Ka taea te whakatakoto i te whanaungatanga pāngarau i waenga i te roa o te pewa me te horahanga o te rāngai penei. Mai i te tātai kua mōhiotia kētia e tātou:
1. Te roa o te pewa: \[ L = \left( \frac{\theta}{360} \right) \times 2 \pi r \]
2. Te horahanga o te kupenga: \[ A = \left( \frac{\theta}{360} \right) \times \pi r^2 \]
Ka kitea e tātou i ngā tātai e rua i runga ake nei he rite tonu te hautanga koki \(\left( \frac{\theta}{360} \right)\) e tohu ana i te ōwehenga o te porowhita katoa i hangaia.
Ki te hiahia tātou ki te hono atu i ngā mea e rua, kia mōhio koe ko te roa o te pewa \(L \) he ōrau o te porowhita o te porowhita, ā, ko te horahanga ā-rohe \(A \) he ōrau o te horahanga o te porowhita. Arā,
\[ L = \left( \frac{\theta}{360} \right) \times 2 \pi r \]
\[ A = \left( \frac{L}{2 \pi r} \right) \times \pi r^2 \]
Whakangāwaritia ngā hautau,
\[ A = \left( \frac{L}{2} \right) \times r \]
Nō reira, ka taea e tātou te kī tika, ka taea te hono i te horahanga o tētahi rāngai ki te roa o te pewa i roto i te roa o te pewa me te radius:
\[ A = \frac{1}{2} R r \]
Ngā Taupānga i roto i te Oranga o Ia Rā
Ehara i te mea he mea mātauranga noa te mārama ki te whanaungatanga i waenga i te roa o te pewa me te horahanga o te rāngai, engari he tono mahi anō hoki i roto i ngā āhuatanga o te oranga o ia rā. Ko ētahi o ēnei tono ko:
1. Hoahoa Whare: I te hoahoa i ngā whare porowhita, i ngā hanganga rānei, pērā i ngā tuanui, i ngā māra, i ngā whare porowhita rānei, he mea tino nui te tatau i te roa o te pewa me te horahanga o te wāhanga.
2. Hangarau Mīhini: I te hoahoa i ngā wāhanga mīhini e whakamahi ana i te nekehanga porowhita, porotaka rānei, ka āwhina tēnei mōhiotanga ki te tatau i ngā ara me ngā wāhi e hiahiatia ana.
3. Tātai whetū: I te whakatauira i ngā porowhita o ngā aorangi, o ngā amiorangi tūturu rānei, he porowhita, he porowhita rānei.
4. Ahuwhenua: Āwhina i te whakamahere i te whakamakuku hurihuri o te pokapū kia ōrite ai te tohatoha o te wai.
Whakamutunga
Mā te mārama ki te whanaungatanga i waenga i te roa o te pewa me te horahanga o te wāhanga ka taea e tātou te mārama ake ki te taunekeneke a ngā koki, ngā radius, me ētahi atu wāhanga o te porowhita. Mā te whakamahi i ēnei tātai taketake, ka taea e tātou te aro atu ki ngā wero maha o te pāngarau āhuahanga me ngā tono mahi i roto i ngā momo mara, tae atu ki te hoahoanga, te miihini, te ahuwhenua, me te whetū. Ahakoa te āhua māmā o ēnei ariā e rua, he whānuitia ā rātou tono puta noa i te oranga o ia rā, ā, he mea nui kia akohia, kia matatauhia hoki.
