Ngā Porowhita me ngā Āwhata
He ariā taketake ngā porowhita me ngā pewa i roto i te pāngarau, ā, he whānuitia ngā tono i roto i ngā momo mara, mai i te hoahoa mīhini ki te whakairoiro rorohiko. Ka matapakihia e tēnei tuhinga ngā whakamāramatanga, ngā āhuatanga, me ngā tono o ngā porowhita me ngā pewa.
Te Whakamāramatanga o te Porowhita
Ko te porowhita te kohinga o ngā pūwāhi katoa i roto i te paparangi e rite ana te tawhiti mai i tētahi pūwāhi e kiia nei ko te pokapū. E mōhiotia ana tēnei tawhiti ko te radius. Mā te pāngarau, ka taea te whakaatu i tētahi porowhita hei whārite:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
ko \((a, b)\) ngā taunga o te pokapū o te porowhita, ā, ko \(r\) te pūtoro o te porowhita.
I roto i te oranga o ia rā, he maha ngā wā ka tūtaki tātou ki ngā āhua porowhita, mai i ngā wira waka ki ngā karaka pakitara me ngā taputapu whare. Ehara i te mea he ātaahua noa iho tēnei āhua porowhita engari he whai hua anō hoki mō ngā mahi maha.
Ngā Wāhanga i roto i te Porowhita
Hei mārama ake i ngā porowhita, he mea nui kia mōhio ki ētahi o ngā wāhanga matua:
1. Pūwāhi Waenga:
Ko te pūwāhi pokapū ko te pūwāhi kei waenganui o te porowhita. Koinei te pūwāhi tohutoro matua hei whakatau i te pūtoro me te āhuahanga katoa o te porowhita.
2. Pūtoro (Maihao):
Ko te pūtoro te tawhiti mai i te pokapū o te porowhita ki tētahi pūwāhi i te porowhita. Ko ngā rārangi katoa e tuhia ana mai i te pokapū ki te porowhita he pūtoro, ā, he rite te roa.
3. Te whānui:
Ko te diameter he rārangi tika e hono ana i ngā pūwāhi e rua i runga i te porowhita, ā, e tika ana mā roto i tōna pokapū. Ko te diameter e rua te roa o te radius (D = 2R).
4. Porowhita:
Ko te porowhita te roa katoa o ngā taha katoa o te porowhita. Ka taea te tatau i te porowhita mā te whakamahi i te tātai:
\[ K = 2\pi r \]
ko \(r\) te pūtoro o te porowhita, ā, ko \(\pi\) he pūmau pāngarau e tata ana ki te 3.14159.
5. Rohe:
Ko te horahanga o te porowhita ko te horahanga o te rohe e herea ana e te porowhita, ā, ka kitea mā te whakamahi i te tātai:
\[ A = \pi r^2 \]
Porowhita Porowhita
Ko te pewa te wāhanga o te taha o te porowhita e tapahia ana e ngā pūwāhi e rua o te porowhita. E rua ngā momo matua o ngā pewa: ko ngā pewa matua me ngā pewa iti. Ki te tuhia he porowhita, ā, ka whiriwhiria e tātou ētahi pūwāhi e rua i te porowhita, ko te rārangi piko e hono ana i aua pūwāhi e rua he pewa. Ki te iti iho i te haurua o te porowhita te taupoki o te pewa, ka kiia he pewa iti; ki te nui atu i te haurua, ka kiia he pewa matua.
Te Tātai i te Roa o te Āwhata
Ko te roa o te pewa e whakawhirinaki ana ki te koki i waenganui i ngā radius e rua e whakawhiti ana i te porowhita i ngā pūwāhi e rua. Ka taea te tatau i te roa o te pewa mā te whakamahi i te tātai:
\[ s = r \theta \]
ko \(s\) te roa o te pewa, ko \(r\) te radius, ā, ko \(\theta\) te koki pokapū i roto i ngā radian. Mēnā ka hoatu te koki i roto i ngā nekehanga, ka taea te huri i te roa o te pewa mā te whakamahi i:
\[ s = \frac{\theta}{360} \times 2\pi r \]
Rohe o te Rāngai
Ko te rāngai he rohe i roto i tētahi porowhita e herea ana e ngā radius e rua me tētahi pewa. Ka kitea te horahanga o tētahi rāngai mā te whakamahi i te tātai:
\[ L = \frac{1}{2} r^2 \theta \]
ko \(L\) te horahanga o te rāngai, ko \(r\) te radius, ā, ko \(\theta\) te koki pokapū i roto i ngā radian. Mēnā ka hoatu te koki i roto i ngā nekehanga:
\[ L = \frac{\theta}{360} \times \pi r^2 \]
Ngā Whakamahinga o ngā Porowhita me ngā Āwhata
He mea nui ngā porowhita me ngā pewa i roto i ngā mara mahi maha, tae atu ki te pūtaiao me te hangarau.
I roto i te Hangarau me te Hoahoa Whare
He maha ngā whakamahinga o ngā porowhita i roto i ngā momo peka hangarau nā te mea he ōrite, he tino pai hoki te āhua. Hei tauira, he porowhita te āhua o ngā wira waka hei whakarite kia maeneene, kia whai hua hoki te haere. I roto i te hangarau hanganga, ka taea e ngā āwhata, ngā pewa rānei te tautoko i ngā kawenga me te tohatoha pēhanga ōrite ake, e kitea ana i roto i ngā piriti piko, i ngā āwhata hoahoanga rānei.
I roto i te Hoahoa Whakairoiro me te Pakiwaituhi
I roto i te ao hoahoa whakairoiro me te pakiwaituhi, he rite tonu te hiranga o ngā porowhita me ngā pewa porowhita. Ka whakamahia ngā porowhita hei huānga taketake mō ngā momo taonga me ngā hoahoa. Hei tauira, i te wā e hanga ana i ngā kiripuaki pakiwaituhi, i ngā waitohu kamupene rānei, he maha ngā wā ka whakamahia ngā porowhita hei āhua taketake mō ngā momo huānga.
I roto i te Tātai Ahurea
I roto i te ao whetū, he maha ngā wā ka kiia ngā porowhita aorangi he porowhita, he porowhita rānei. He mea nui te mārama ki ngā porowhita hei matapae i te nekehanga o ngā aorangi me ētahi atu tinana o te rangi. I roto i tana ture tuatoru mō te nekehanga o ngā aorangi, i whakamahia e Johannes Kepler ngā ariā o ngā porowhita me ngā porowhita hei whakamārama i ngā porowhita o ngā aorangi i roto i te pūnaha rā.
I roto i te Whakatere me te Matawhenua
I roto i te whakatere, inā koa te moana me te rererangi, he mea nui te mahi a ngā porowhita ki te whakamahere i ngā huarahi. Ko te ariā o te porowhita nui, he porowhita kei waenganui o te Ao, ā, e whakawhiti ana i te mata o te Ao, he mea nui ki te whakatere whakawhiti whenua.
Porowhitahia te Pāngarau i roto i te Mātauranga
He kaupapa matua ngā porowhita i roto i ngā marautanga pāngarau puta noa i te ao. I ngā wāhanga tōmua o te mātauranga, ka āwhina ngā porowhita i ngā ākonga ki te mārama me te whakaatu i ngā ariā taketake o te āhuahanga. I a ngā ākonga e ahu whakamua ana i roto i te tukanga mātauranga, ka whānui ake ēnei ariā ki ngā tātaritanga uaua ake, tae atu ki te ine whārite me te tātaitai.
Ko ngā wāhanga matua o te pāngarau, pērā i te ine-taurite, e pā tika ana ki te porowhita wae (he porowhita me te radius 1). Ko ngā ariā o te sine, te cosine, me te tangent e ahu mai ana i te whakaaturanga o ngā pūwāhi ki runga i te porowhita wae.
Whakamutunga
He ariā taketake ngā porowhita me ngā pewa i roto i te āhuahanga, ā, he whānuitia ngā whakamahinga i roto i ngā mara mai i te hangarau me te hoahoa whare ki te hoahoa whakairoiro me te whetū. Ehara i te mea he mea nui te māramatanga hōhonu ki ngā āhuatanga o ngā porowhita me ngā pewa i roto i ngā horopaki pāngarau anake, engari he mea whai pānga hoki ki te oranga o ia rā me ngā momo mahi. E whakaatu ana tēnei i te hiranga o ēnei ariā i roto i te whanaketanga o te mātauranga me te hangarau.
