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ملخص الفراغية 1431

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هانى الريس
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‫@‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫@@ @@‬ ‫ﻣﻠﺤﻮﻇﺎﺕ :‬ ‫‪@Šì“ä¾a‬‬ ‫١~ ‪ ’bä¾a@Ñä—m‬ﲝﺴﺐ ﺷﻜﻞ ﻗﺎﻋﺪ‪‬ﺎ ﻓﻴﺴﻤﻰ ﺍﳌﻨﺸﻮﺭ ﺛﻼﺛﻴﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ﻭﻫﻜﺬﺍ ﺇﻥ ﻛﺎﻧﺖ ﻗﺎﻋﺪﺗﻪ‬ ‫ﻣﺜﻠﺜﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ٠٠٠٠٠‬ ‫ﻭﻫﻮ ﺃﻱ ﺣﻴﺰ ﻣﻦ ﺍﻟﻔﺮﺍﻍ ﳏﺪﺩ ﺑﺴﻄﺢ ﺃﻭ ﻋﺪﺓ ﺳﻄــﻮﺡ ٠‬ ‫ﺍ‪‬ﺴــﻢ :‬ ‫۲~ ‪ Šì“ä¾a@óàí‬ﻣﺘﻮﺍﺯﻱ ﺍﻟﺴﻄﻮﺡ ﺇﺫﺍ ﻛﺎﻥ ﺭﺑﺎﻋﻴﺎ ﻭﻗﺎﻋﺪﺗﻪ ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ ﻭﻳﺴﻤﻰ ﻣﺘﻮﺍﺯﻱ ﺍﻟﺴﻄﻮﺡ ﻣﺘﻮﺍﺯﻱ‬ ‫ﹰ‬ ‫ﻣﺴﺘﻄﻴﻼﺕ ﺇﺫﺍ ﻛﺎﻥ ﻗﺎﺋﻤﺎ ﻭﻗﺎﻋﺪﺗﻪ ﻣﺴﺘﻄﻴﻞ ٠‬ ‫ﹰ‬ ‫، ﺃﻣﺎ ﺍﳌﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﱵ ﺗﺘﻘﺎﻃﻊ ﻓﻴﻬﺎ‬ ‫ﻛﺜﲑﺍﺕ ﺍﻟﻮﺟﻮﻩ : ﳎﺴﻢ ﲨﻴﻊ ﺃﺳﻄﺤﻪ ﻣﺴﺘﻮﻳﺔ ﻧﺴﻤﻰ ﻫﺬﻩ ﺍﻷﺳﻄﺢ و‬ ‫ً ، ﻭﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﺍﻷﺣﺮﻑ و ً ﻭﺃﻱ ﻣﺴﺘﻘﻴﻢ ﻳﺼﻞ ﺑﲔ ﺭﺃﺳﲔ ﰲ ﻭﺟﻬﲔ ﳐﺘﻠﻔﲔ ‪‬ﻳﺴﻤﻰ‬ ‫ﻧﺴﻤﻴﻬﺎ‬ ‫٣~ @‪ : kÈؾa‬ﻓﻬﻮ ﻣﺘﻮﺍﺯﻱ ﻣﺴﺘﻄﻴﻼﺕ ﲨﻴﻊ ﺣﺮﻭﻓﻪ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ ٠‬ ‫ﺍﳌﻨﺸﻮﺭ ‪ Z‬ﻋﺒﺎﺭﺓ ﻋﻦ ﻛﺜﲑﺓ ﻭﺟﻮﻩ ٠‬ ‫) ﺍﻟﻮﺟﻬﺎﻥ ﳘﺎ ﺍﻟﻘﺎﻋﺪﺗﺎﻥ (‬ ‫١~ ﻓﻴﻪ ﻭﺟﻬﺎﻥ ﻣﻀﻠﻌﺎﻥ ﻭ ﻳﻘﻌﺎﻥ ﰲ ﻣﺴﺘﻮﻳﲔ ﻣﺘﻮﺍﺯﻳﲔ ٠‬ ‫‪@Z@áÄnä¾a@ÉÜ›¾a@òybß‬‬ ‫) ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﻣﺘﻮﺍﺯﻳﺔ ﻭﻣﺘﻄﺎﺑﻘﺔ (‬ ‫۲~ ﻛﻞ ﻣﻦ ﺍﻷﻭﺟﻪ ﺍﻟﺒﺎﻗﻴﺔ ﻣﺘﻮﺍﺯﻳﺔ ﺍﻷﺿﻼﻉ ٠‬ ‫) ﻥ – ۲( × ٠٩‬ ‫ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻷﺿﻼﻉ ، ﻝ ﻃﻮﻝ ﺍﻟﻀﻠﻊ‬ ‫ـــــــ‬ ‫= ٤ ﻝ@ ﻇـﺎ‬ ‫ــ‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﻟﻘﺎﺋﻢ@‪ Z‬ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﲨﻴﻊ ﺃﺣﺮﻓﻪ ﺍﳉﺎﻧﺒﻴﺔ ﺗﻌﺎﻣﺪ ﺍﻟﻘﺎﻋﺪﺗﲔ )ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﻠﻰ ﻫﻴﺌﺔ ﻣﺴﺘﻄﻴﻞ(‬ ‫ﻥ‬ ‫ﻥ‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﳌﻨﺘﻈﻢ : ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﻗﺎﺋﻢ ﻗﺎﻋﺪﺗﻪ ﻣﻀﻠﻌﺎ ﻣﻨﺘﻈﻤﺎ ٠‬ ‫ﹰ‬ ‫ﹰ‬ ‫ﻧﻈﺮﻳﺔ:‬ ‫) ﺍﳌﻀﻠﻊ ﺍﳌﻨﺘﻈﻢ ﺃﺿﻼﻋﻪ ﻣﺘﺴﺎﻭﻳﺔ ﻭﺯﻭﺍﻳﺎﻩ ﻣﺘﺴﺎﻭﻳﺔ (‬ ‫‪ª‬‬ ‫א ‪ě‬‬ ‫ن א‬ ‫‪ ú‬א‪Ğ‬‬ ‫ א‬ ‫אز‬ ‫‪ª‬‬ ‫‪Ĩ‬‬ ‫ذא‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﳌﺎﺋﻞ@‪ Z‬ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﲨﻴﻊ ﺃﺣﺮﻓﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻻ ﺗﻌﺎﻣﺪ ﺍﻟﻘﺎﻋﺪﺗﲔ ) ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﻠﻰ ﻫﻴﺌﺔ ﻣﺘﻮﺍﺯﻱ‬ ‫ﻣﺴﺘﻄﻴﻞ ( ٠‬ ‫ﺗﻌﺮﻳﻒ :‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﻟﻨﺎﻗﺺ :ﺇﺫﺍ ﻗﻄﻊ ﲨﻴﻊ ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﳌﻨﺸﻮﺭ ﲟﺴﺘﻮ ﻻ ﻳﻮﺍﺯﻱ ﺍﻟﻘﺎﻋﺪﺓ ﻓﺈﻧﻨﺎ ﳓﺼﻞ ﻋﻠﻰ ﺟﺰﺃﻳﻦ‬ ‫ٍ‬ ‫= ﳎﻤﻮﻉ ﻣﺴﺎﺣﺎﺕ ﺃﻭﺟﻬﻪ ﺍﳉﺎﻧﺒﻴﺔ‬ ‫א‪Ğ‬‬ ‫א‬ ‫ﻧﺴﻤﻲ ﻛﻞ ﻣﻨﻬﺎ ﻣﻨﺸﻮﺭﹰﺍ ﻧﺎﻗﺼﺎ ٠‬ ‫ﹰ‬ ‫= ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ﳎﻤﻮﻉ ﻣﺴﺎﺣﱵ ﺍﻟﻘﺎﻋﺪﺗﲔ‬ ‫א‬ ‫א‬ ‫ﺍﳌﻘﻄﻊ ﺍﻟﻘﺎﺋﻢ : ﻫﻮ ﻣﺴﺘﻮﻯ ﻳﻘﻄﻊ ﲨﻴﻊ ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﻟﻠﻤﻨﺸﻮﺭ ﻭﻳﻜﻮﻥ ﻋﻤﻮﺩﻳﺎ ﻋﻠﻴﻬﺎ ٠‬ ‫ﹰ‬ ‫= ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ۲ ﻕ‬ ‫א‬ ‫ن : א‬ ‫ﻣﺴﻠﻤﺔ ﺍﳊﺠﻢ ﺍﻷﻭﱃ : ﺣﺠﻢ ﻣﺘﻮﺍﺯﻱ ﺍﳌﺴﺘﻄﻴﻼﺕ ﺫﻱ ﺍﻷﺑﻌﺎﺩ ﺱ ، ﺹ ، ﻉ ﻫﻮ ﺱ٠ ٠ﺹ٠ ٠ﻉ‬ ‫ﻣﺴﻠﻤﺔ ﺍﳊﺠﻢ ﺍﻟﺜﺎﻧﻴـﺔ : ﺍﻓﺮﺽ ﺃﻥ ﺍ ، ﺏ ﳎﺴﻤﺎﻥ ﻫﻨﺪﺳﻴﺎﻥ ، ﺻـﺺ ﻣﺴﺘﻮ ﻣﺎ ٠ ﺇﺫﺍ ﻭﺟﺪﻧﺎ ﺃﻥ ﻛﻞ ﻣﺴﺘﻮ ﻣـﻮﺍﺯ‬ ‫ﻟﻠﻤﺴﺘﻮﻯ ﺻﺺ ﺇﻣﺎ ﻳﻘﻄﻊ ﻛﻼ ﻣﻦ ﺍ‪‬ﺴﻤﲔ ﺃﻭ ﻻ ﻳﻘﻄﻊ ﺃﻳﺎ ﻣﻨﻬﻤﺎ ، ﻭﺇﺫﺍ ﻗﻄﻌﻬﻤﺎ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺘﺎ ﺍﳌﻘﻄﻌﲔ ﺍﻟﻨﺎﲡﲔ ﻣﺘﺴﺎﻭﻳﺘﲔ‬ ‫@‬ ‫@@@@@‪@ïŽa‡Ž@Šì“äß@@@@ïŽb½@Šì“äß@@@@ïqýq@Šì“äß@@@@ìЏÛa@ðŒaìnß@@@@@@@@@pýîĐn¾a@ðŒaìnß@@@@@@@@kÈØß‬‬ ‫ﻓﺈﻥ ﻟﻠﻤﺠﺴﻤﲔ ﺍ ، ﺏ ﺍﳊﺠﻢ ﻧﻔﺴﻪ ٠‬ ‫٦٥١‬ ‫٥٥١‬
‫@@@@@‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫‪W‬‬ ‫@‪N‬‬ ‫@‬ ‫@‪@L‬و ل‬ ‫א‪١٠@ Ğ‬‬ ‫ل‬ ‫א‬ ‫‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫‪@@Z@áöbÔÛa@Šì“ä¾a@À‬‬ ‫ﺀ~ ٠٨٤ﺳﻢ@‬ ‫ﺝ~ ٨٤ﺳﻢ@‬ ‫ﺏ~ ٠٨٤ﺳﻢ#‬ ‫ﺍ~ ٠٨ ﺳﻢ@‬ ‫١~ ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ = ﳏﻴﻂ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ‬ ‫@‬ ‫................................................................................................................................................................................................................................................................‬ ‫۲~ ﺍﳌﺴﺎﺣﺔ ﺍﻟﻜﻠﻴﺔ = ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ۲ × ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫٣~ ﺣﺠﻢ ﺍﳌﻨﺸﻮﺭ = ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ‬ ‫................................................................................................................................................................................................................................................................‬ ‫ﺣﻴﺚ ﻕ : ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ ، ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳌﻨﺸﻮﺭ‬ ‫ﺡ =ﻕ ×ﻉ‬ ‫ﺃﻱ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪@@Ýöb¾a@Šì“ä¾a@À‬‬ ‫ل א‪ ú ğ‬א‪ Ğ‬‬ ‫‪C‬‬ ‫א‬ ‫א‬ ‫=‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫א‬ ‫‪C۲H‬‬ ‫א‪Ğ‬‬ ‫= א‬ ‫א‬ ‫۲~ א‬ ‫..................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ ‫ل א‪ ú ğ‬א‪ Ğ‬‬ ‫‪C‬‬ ‫א‬ ‫א‬ ‫=‬ ‫א‬ ‫٣~‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪@Z@@áöbÔÛa@Šì“ä¾a@À‬‬ ‫€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€@‬ ‫٠٤ ـﻢ@ و‬ ‫ﺳـ‬ ‫٠٣٥ و‬ ‫אو‬ ‫א‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫۲~‬ ‫‪ LL‬א‬ ‫א‬ ‫١~ א‬ ‫‪@K‬‬ ‫א‬ ‫א‬ ‫ل א‪ ú ğ‬א‪ F Ğ‬ع ‪ Z‬ل ‪E‬‬ ‫ع‪Z‬‬ ‫۲~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪@Z@Ýöb¾a@Šì“ä¾a@À‬‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪ ú‬א‪Ğ‬‬ ‫⊥א‬ ‫א‬ ‫١~ א‬ ‫) ﺣﻴﺚ ﻫـ ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﳌﻨﺸﻮﺭ (‬ ‫ل א‪ ú ğ‬א‪ × Ğ‬ﺟﺎﻩ‬ ‫ع=‬ ‫۲~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫٣~ ﻣﺴﺎﺣﺔ ﺍﳌﻘﻄﻊ ﺍﻟﻘﺎﺋﻢ = ﻕ × ﺟﺎﻩ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪ C‬ﺟﺎﻩ‬ ‫א‬ ‫‪Z‬‬ ‫א‬ ‫א‬ ‫٤~ ‬ ‫................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫ﻣﻠﺤﻮﻇﺔ :‬ ‫@@@@@@@@‬ ‫אل א ول و‬ ‫‪ @Z@k튇m‬א‬ ‫א‬ ‫و‬ ‫א‪Ğ‬‬ ‫‪@N‬‬ ‫ﺏ~ א‪ğ‬‬ ‫א‬ ‫@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ﺍ~‬ ‫٨٥١‬ ‫@‬ ‫٧٥١‬
‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫و‬ ‫אو ٠٣٥ وא‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫‬ ‫٤~‬ ‫א‪ ٦] ١٠ Ğ‬ﺳﻢ @‪‡uëc‬‬ ‫٠١ ﺳﻢ و ل‬ ‫٥۲ ]٦ ﺳﻢ@ א‬ ‫٣~‬ ‫‪K‬‬ ‫א‬ ‫א‬ ‫ﺏ~‬ ‫ل א‪ ú ğ‬א‪K Ğ‬‬ ‫ﺍ~‬ ‫‪Z@kya‬‬ ‫٧٨١ﺳﻢ@‬ ‫‪K‬‬ ‫א‬ ‫א‬ ‫ﺏ~‬ ‫א‬ ‫ﺍ~‬ ‫:‬ ‫................................................................................................................................................................................................................................................................‬ 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‫..................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ ‫ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ@@@@@‬ ‫ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫%؛ ؛ ؛ ؛ ؛٢ ؛ﺵ#؛ ؛: @ ، و‪ī‬‬ ‫א‬ ‫‪: k튇m‬‬ ‫ذא‬ ‫١٨ ﺳﻢ@‬ ‫א‬ ‫ل א‪ ú ğ‬א‪ ١٠@ Ğ‬ﺳﻢ و‬ ‫‪@@@Z@k튇m‬‬ ‫ א א‬ ‫א‬ ‫א‬ ‫٠٦ ْ ، א‬ ‫אو‬ ‫א‬ ‫ﺏ~‬ ‫א‪Ğ‬‬ ‫ﺍ~‬ ‫‪Z‬‬ ‫٠٠٥ﺳﻢ@ ، و‬ ‫א‬ ‫٠٦١‬ ‫٩٥١‬
‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫א‪ Ğ‬‬ ‫٠٦% و ل‬ ‫אو‬ ‫א‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫٦~‬ ‫‪JW‬‬ ‫‪ K‬אو‬ ‫‪Ħ‬‬ ‫א‬ ‫٥~‬ ‫@‬ ‫‪W‬‬ ‫و‬ ‫ل‬ ‫א‬ ‫و‬ ‫٠۲‬ ‫‪K‬‬ ‫א‬ ‫ﺫ~‬ ‫א‬ ‫١~ א‬ ‫א‬ ‫٣~‬ ‫ع א‬ ‫۲~ א‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ 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‫................................................................................................................................................................................................................................................................‬ ‫...............................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ 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‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫.‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫א‬ ‫@٠٨۲ﺳﻢ# @‬ ‫‪:@k튇m‬‬ ‫..................................................................................................................................................................................................................................................................‬ ‫@‪N‬‬ ‫א‪ Ğ‬‬ ‫ل‬ ‫ن‬ ‫ل‬ ‫‪@L%٣٠@a‬א‬ ‫אو‬ ‫۲٦١‬ ‫١٦١‬
‫.‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫א‬ ‫٣‬ ‫٠٣ ﺳﻢ‬ ‫و‬ ‫۲١‬ ‫و‬ ‫א‪٥ Ğ‬‬ ‫ل‬ ‫٨~‬ ‫‪W‬‬ ‫٠٠۲ ﺳﻢ# ، و‬ ‫٨ ﺳﻢ ، و‬ ‫א‬ ‫٧~‬ ‫א‬ ‫ﺏ~‬ ‫ﺍ~‬ ‫‪K‬‬ ‫א‪Ğ‬‬ ‫ﺏ~‬ ‫‪K‬‬ ‫ل‬ ‫ﺍ~‬ ‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ 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‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫ل‬ ‫و‬ ‫٠۲ﺳـﻢ@ وא‬ ‫٠۲١ﺳـﻢ٣ و‬ ‫‪: @ k튇m‬‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪K‬‬ ‫א‬ ‫א‪ Ğ‬א‬ ‫..................................................................................................................................................................................................................................................................‬ ‫٤٦١‬ ‫٣٦١‬
‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫@‬ ‫‪ Z@òí‹Äã‬ﺇﺫﺍ ﻛﺎﻥ ﻉ ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﻭﻛﺎﻧﺖ ‪ ۲Q‬ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ ﻭﻗﻄﻊ ﺍﳍﺮﻡ ﲟﺴﺘﻮ ﻳﻮﺍﺯﻱ ﺍﻟﻘﺎﻋﺪﺓ ﻭﻳﺒﻌﺪ ﻙ ﻋﻦ‬ ‫ﻙ@‬ ‫‪@â‹a‬‬ ‫ﺍﻟﺮﺃﺱ ﻓﺈﻥ ﻣﺴﺎﺣﺔ ﺍﳌﻘﻄﻊ ﻕ ١= ــ × ﻕ ٢‬ ‫ﻉ@‬ ‫א م ‪W‬‬ ‫‬ ‫‪ª‬‬ ‫‪E‬‬ ‫‪ F‬א‪Ğ‬‬ ‫א‬ ‫و‬ ‫‪E‬‬ ‫‪F‬א‬ ‫א‬ ‫‪ Ù‬و‬ ‫ﻣﻼﺣﻈﺔ : ﻉ = ﻉَ + ﻙ‬ ‫ــ ١= ـﻙ@‬ ‫ﻕ‬ ‫ﺃﻱ ﺃﻥ :‬ ‫‪ F‬س א م ‪E‬‬ ‫وא‬ ‫ﻕ ٢ ﻉ@‬ ‫ﺣﻴﺚ ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ،، ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻨﺎﻗﺺ‬ ‫‪‬‬ ‫@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ﻙ : ﺍﳌﺴﺎﻓﺔ ﻣﻦ ﺭﺃﺱ ﺍﳍﺮﻡ ﺇﱃ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺼﻐﺮﻯ‬ ‫א‬ ‫س א م ‪ħ‬‬ ‫د‬ ‫א‬ ‫א‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ :‬ ‫ﻓﻴﺴﻤﻰ ﺍﳍﺮﻡ ﺛﻼﺛﻴﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ﻭﻫﻜﺬﺍ ﺇﻥ ﻛﺎﻧﺖ ﻗﺎﻋﺪﺗﻪ ﻣﺜﻠﺜﺎ ﺃﻭ‬ ‫א ‪ęª‬‬ ‫א‬ ‫‪@P@@@ñ‡ÇbÔÛa@òybß@À@ÊbÐmŠüa@l‹š@sÜq@ðëbí@â‹ç@ðc@ávy@Z@@òí‹Äã‬‬ ‫ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ٠٠٠٠٠‬ ‫ﺭﺃﺱ ﺍﳍﺮﻡ‬ ‫ﻭﻳﺪﻋﻰ ﺍﳍﺮﻡ ﺍﻟﺜﻼﺛﻲ ﺑـ ﺭﺑﺎﻋﻲ ﺍﻟﻮﺟﻮﻩ‬ ‫ = !؛٣ ‪ × Q‬ﻉ‬ ‫@‪: æc@ðc‬‬ ‫ﺍﳊﺮﻑ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺃﺣﺮﻓﻪ ﻣﺘﺴﺎﻭﻳﺔ ﲰﻲ ﺭﺑﺎﻋﻲ ﻭﺟﻮﻩ ﻣﻨﺘﻈﻢ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ‬ ‫ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺃﻭﺟﻬﻪ ﻣﺜﻠﺜﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﺍﻷﺿﻼﻉ‬ ‫ﻝ١‬ ‫‪@Z@˜ÓbäÛa@â‹a‬‬ ‫‪١Q‬‬ ‫ﺇﺫﺍ ﻗﻄﻊ ﺍﳍﺮﻡ ﲟﺴﺘﻮ // ﺍﻟﻘﺎﻋﺪﺓ ﻓﺈﻥ ﺍﳉﺰﺀ ﺍﶈﺼﻮﺭ ﺑﲔ ﺍﳌﻘﻄﻊ ﻭﺍﻟﻘﺎﻋﺪﺓ ﻳﺴﻤﻰ ﻫﺮﻣﺎ ﻧﺎﻗﺼﺎ ٠‬ ‫ﹰ‬ ‫ﹰ‬ ‫ﻗﺎﻋﺪﺓ ﺍﳍﺮﻡ‬ ‫‪W‬‬ ‫و‬ ‫א‪Ğ‬‬ ‫و‬ ‫م‬ ‫ﺍﳍﺮﻡ ﺍﻟﻘﺎﺋﻢ :‬ ‫) ﻫﻮ ﺍﳉﺰﺀ ﺍﶈﺼﻮﺭ ﺑﲔ ﻗﺎﻋﺪﺓ ﺍﳍﺮﻡ ﻭﺍﳌﻘﻄﻊ ﺍﳌﻮﺍﺯﻱ ﻟﻠﻘﺎﻋﺪﺓ ( ﻭﻓﻴﻪ ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ‬ ‫١~ ﻣﻮﻗﻊ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻣﻦ ﺍﻟﺮﺃﺱ ﻋﻠﻰ ﺍﻟﻘﺎﻋﺪﺓ ﻫﻮ ﻣﺮﻛﺰﻫﺎ‬ ‫ﻉ‬ ‫ﻣﺮﻛﺰ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫ﻋﻠﻰ ﺷﻜﻞ ﺷﺒﻪ ﻣﻨﺤﺮﻑ ٠‬ ‫ﻫﺮﻡ ﺛﻼﺛﻲ ﻗﺎﺋﻢ‬ ‫۲~ ﺃﻭﺟﻬﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﺒﺎﺭﺓ ﻋﻦ ﻣﺜﻠﺜﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﺍﻟﻀﻠﻌﲔ‬ ‫‪٢Q‬‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻨﺎﻗﺺ@@‪ Z‬ﻫﻮ ﺍﻟﺒﻌﺪ ﺍﻟﻌﻤﻮﺩﻱ ﺑﲔ ﺍﻟﻘﺎﻋﺪﺗﲔ ٠‬ ‫٣~ ﺍﺭﺗﻔﺎﻋﺎﺕ ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ‬ ‫@‬ ‫ﻝ۲‬ ‫‪@Z@@òí‹Äã‬‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻘﺎﺋﻢ : ﻫﻮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻣﻦ ﺭﺃﺱ ﺍﳍﺮﻡ ﺍﱃ ﻣﺮﻛﺰ ﺍﻟﻘﺎﻋﺪﺓ )ﺃﻱ ﺍﻟﻨﻘﻄﺔ ﺍﻟﱵ ﳍﺎ ﺍﻟﺒﻌﺪ ﻧﻔﺴﻪ ﻋﻦ ﲨﻴﻊ‬ ‫ﻫﺮﻡ ﻗﺎﺋﻢ ﻧﺎﻗﺺ‬ ‫‪= ˜ÓbäÛa@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬‬ ‫ﺭﺅﻭﺱ ﺍﻟﻘﺎﻋﺪﺓ (‬ ‫ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ ﻟﻠﻬﺮﻡ ﺍﻟﻨﺎﻗﺺ = !؛٢ ﳎﻤﻮﻉ ﳏﻴﻄﻲ ﺍﻟﻘﺎﻋﺪﺗﲔ × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫‪@@Z@òí‹Äã‬‬ ‫@@‪Z@æc@ðc‬‬ ‫‪! = áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬؛٢ ﳏﻴﻂ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ ﺍﳉﺎﻧﱯ‬ ‫ﺃﻱ ﺃﻥ :‬ ‫‪! = ˜ÓbäÛa@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬؛٢ ﻥ )ﻝ ١ + ﻝ ٢ ( × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫@@‪@@]@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬‬ ‫!؛٢ ﻥ ﺱ × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫@@‪ Z@sîy‬ﻥ ﻋﺪﺩ ﺃﺿﻼﻉ ﺍﻟﻘﺎﻋﺪﺓ ، ﻝ ١ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺼﻐﺮﻯ، ﻝ ٢ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﻜﱪﻯ‬ ‫ﻫﺮﻡ ﺭﺑﺎﻋﻲ‬ ‫ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺃﺿﻼﻉ ﺍﻟﻘﺎﻋﺪﺓ ، ﺱ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫‪ = ˜ÓbäÛa@â‹èÜÛ@òîÜØÛa@òyb¾a‬ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ‪۲Q + ١Q‬‬ ‫‪ = â‹èÜÛ@òîÜØÛa@òyb¾a‬ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫٦٦١‬ ‫٥٦١‬
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431
ملخص الفراغية 1431

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ملخص الفراغية 1431

  • 1. ‫@‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫@@ @@‬ ‫ﻣﻠﺤﻮﻇﺎﺕ :‬ ‫‪@Šì“ä¾a‬‬ ‫١~ ‪ ’bä¾a@Ñä—m‬ﲝﺴﺐ ﺷﻜﻞ ﻗﺎﻋﺪ‪‬ﺎ ﻓﻴﺴﻤﻰ ﺍﳌﻨﺸﻮﺭ ﺛﻼﺛﻴﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ﻭﻫﻜﺬﺍ ﺇﻥ ﻛﺎﻧﺖ ﻗﺎﻋﺪﺗﻪ‬ ‫ﻣﺜﻠﺜﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ٠٠٠٠٠‬ ‫ﻭﻫﻮ ﺃﻱ ﺣﻴﺰ ﻣﻦ ﺍﻟﻔﺮﺍﻍ ﳏﺪﺩ ﺑﺴﻄﺢ ﺃﻭ ﻋﺪﺓ ﺳﻄــﻮﺡ ٠‬ ‫ﺍ‪‬ﺴــﻢ :‬ ‫۲~ ‪ Šì“ä¾a@óàí‬ﻣﺘﻮﺍﺯﻱ ﺍﻟﺴﻄﻮﺡ ﺇﺫﺍ ﻛﺎﻥ ﺭﺑﺎﻋﻴﺎ ﻭﻗﺎﻋﺪﺗﻪ ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ ﻭﻳﺴﻤﻰ ﻣﺘﻮﺍﺯﻱ ﺍﻟﺴﻄﻮﺡ ﻣﺘﻮﺍﺯﻱ‬ ‫ﹰ‬ ‫ﻣﺴﺘﻄﻴﻼﺕ ﺇﺫﺍ ﻛﺎﻥ ﻗﺎﺋﻤﺎ ﻭﻗﺎﻋﺪﺗﻪ ﻣﺴﺘﻄﻴﻞ ٠‬ ‫ﹰ‬ ‫، ﺃﻣﺎ ﺍﳌﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﱵ ﺗﺘﻘﺎﻃﻊ ﻓﻴﻬﺎ‬ ‫ﻛﺜﲑﺍﺕ ﺍﻟﻮﺟﻮﻩ : ﳎﺴﻢ ﲨﻴﻊ ﺃﺳﻄﺤﻪ ﻣﺴﺘﻮﻳﺔ ﻧﺴﻤﻰ ﻫﺬﻩ ﺍﻷﺳﻄﺢ و‬ ‫ً ، ﻭﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﺍﻷﺣﺮﻑ و ً ﻭﺃﻱ ﻣﺴﺘﻘﻴﻢ ﻳﺼﻞ ﺑﲔ ﺭﺃﺳﲔ ﰲ ﻭﺟﻬﲔ ﳐﺘﻠﻔﲔ ‪‬ﻳﺴﻤﻰ‬ ‫ﻧﺴﻤﻴﻬﺎ‬ ‫٣~ @‪ : kÈؾa‬ﻓﻬﻮ ﻣﺘﻮﺍﺯﻱ ﻣﺴﺘﻄﻴﻼﺕ ﲨﻴﻊ ﺣﺮﻭﻓﻪ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ ٠‬ ‫ﺍﳌﻨﺸﻮﺭ ‪ Z‬ﻋﺒﺎﺭﺓ ﻋﻦ ﻛﺜﲑﺓ ﻭﺟﻮﻩ ٠‬ ‫) ﺍﻟﻮﺟﻬﺎﻥ ﳘﺎ ﺍﻟﻘﺎﻋﺪﺗﺎﻥ (‬ ‫١~ ﻓﻴﻪ ﻭﺟﻬﺎﻥ ﻣﻀﻠﻌﺎﻥ ﻭ ﻳﻘﻌﺎﻥ ﰲ ﻣﺴﺘﻮﻳﲔ ﻣﺘﻮﺍﺯﻳﲔ ٠‬ ‫‪@Z@áÄnä¾a@ÉÜ›¾a@òybß‬‬ ‫) ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﻣﺘﻮﺍﺯﻳﺔ ﻭﻣﺘﻄﺎﺑﻘﺔ (‬ ‫۲~ ﻛﻞ ﻣﻦ ﺍﻷﻭﺟﻪ ﺍﻟﺒﺎﻗﻴﺔ ﻣﺘﻮﺍﺯﻳﺔ ﺍﻷﺿﻼﻉ ٠‬ ‫) ﻥ – ۲( × ٠٩‬ ‫ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻷﺿﻼﻉ ، ﻝ ﻃﻮﻝ ﺍﻟﻀﻠﻊ‬ ‫ـــــــ‬ ‫= ٤ ﻝ@ ﻇـﺎ‬ ‫ــ‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﻟﻘﺎﺋﻢ@‪ Z‬ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﲨﻴﻊ ﺃﺣﺮﻓﻪ ﺍﳉﺎﻧﺒﻴﺔ ﺗﻌﺎﻣﺪ ﺍﻟﻘﺎﻋﺪﺗﲔ )ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﻠﻰ ﻫﻴﺌﺔ ﻣﺴﺘﻄﻴﻞ(‬ ‫ﻥ‬ ‫ﻥ‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﳌﻨﺘﻈﻢ : ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﻗﺎﺋﻢ ﻗﺎﻋﺪﺗﻪ ﻣﻀﻠﻌﺎ ﻣﻨﺘﻈﻤﺎ ٠‬ ‫ﹰ‬ ‫ﹰ‬ ‫ﻧﻈﺮﻳﺔ:‬ ‫) ﺍﳌﻀﻠﻊ ﺍﳌﻨﺘﻈﻢ ﺃﺿﻼﻋﻪ ﻣﺘﺴﺎﻭﻳﺔ ﻭﺯﻭﺍﻳﺎﻩ ﻣﺘﺴﺎﻭﻳﺔ (‬ ‫‪ª‬‬ ‫א ‪ě‬‬ ‫ن א‬ ‫‪ ú‬א‪Ğ‬‬ ‫ א‬ ‫אز‬ ‫‪ª‬‬ ‫‪Ĩ‬‬ ‫ذא‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﳌﺎﺋﻞ@‪ Z‬ﻭﻫﻮ ﻣﻨﺸﻮﺭ ﲨﻴﻊ ﺃﺣﺮﻓﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻻ ﺗﻌﺎﻣﺪ ﺍﻟﻘﺎﻋﺪﺗﲔ ) ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﻠﻰ ﻫﻴﺌﺔ ﻣﺘﻮﺍﺯﻱ‬ ‫ﻣﺴﺘﻄﻴﻞ ( ٠‬ ‫ﺗﻌﺮﻳﻒ :‬ ‫ﺍﳌﻨﺸﻮﺭ ﺍﻟﻨﺎﻗﺺ :ﺇﺫﺍ ﻗﻄﻊ ﲨﻴﻊ ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﳌﻨﺸﻮﺭ ﲟﺴﺘﻮ ﻻ ﻳﻮﺍﺯﻱ ﺍﻟﻘﺎﻋﺪﺓ ﻓﺈﻧﻨﺎ ﳓﺼﻞ ﻋﻠﻰ ﺟﺰﺃﻳﻦ‬ ‫ٍ‬ ‫= ﳎﻤﻮﻉ ﻣﺴﺎﺣﺎﺕ ﺃﻭﺟﻬﻪ ﺍﳉﺎﻧﺒﻴﺔ‬ ‫א‪Ğ‬‬ ‫א‬ ‫ﻧﺴﻤﻲ ﻛﻞ ﻣﻨﻬﺎ ﻣﻨﺸﻮﺭﹰﺍ ﻧﺎﻗﺼﺎ ٠‬ ‫ﹰ‬ ‫= ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ﳎﻤﻮﻉ ﻣﺴﺎﺣﱵ ﺍﻟﻘﺎﻋﺪﺗﲔ‬ ‫א‬ ‫א‬ ‫ﺍﳌﻘﻄﻊ ﺍﻟﻘﺎﺋﻢ : ﻫﻮ ﻣﺴﺘﻮﻯ ﻳﻘﻄﻊ ﲨﻴﻊ ﺍﻷﺣﺮﻑ ﺍﳉﺎﻧﺒﻴﺔ ﻟﻠﻤﻨﺸﻮﺭ ﻭﻳﻜﻮﻥ ﻋﻤﻮﺩﻳﺎ ﻋﻠﻴﻬﺎ ٠‬ ‫ﹰ‬ ‫= ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ۲ ﻕ‬ ‫א‬ ‫ن : א‬ ‫ﻣﺴﻠﻤﺔ ﺍﳊﺠﻢ ﺍﻷﻭﱃ : ﺣﺠﻢ ﻣﺘﻮﺍﺯﻱ ﺍﳌﺴﺘﻄﻴﻼﺕ ﺫﻱ ﺍﻷﺑﻌﺎﺩ ﺱ ، ﺹ ، ﻉ ﻫﻮ ﺱ٠ ٠ﺹ٠ ٠ﻉ‬ ‫ﻣﺴﻠﻤﺔ ﺍﳊﺠﻢ ﺍﻟﺜﺎﻧﻴـﺔ : ﺍﻓﺮﺽ ﺃﻥ ﺍ ، ﺏ ﳎﺴﻤﺎﻥ ﻫﻨﺪﺳﻴﺎﻥ ، ﺻـﺺ ﻣﺴﺘﻮ ﻣﺎ ٠ ﺇﺫﺍ ﻭﺟﺪﻧﺎ ﺃﻥ ﻛﻞ ﻣﺴﺘﻮ ﻣـﻮﺍﺯ‬ ‫ﻟﻠﻤﺴﺘﻮﻯ ﺻﺺ ﺇﻣﺎ ﻳﻘﻄﻊ ﻛﻼ ﻣﻦ ﺍ‪‬ﺴﻤﲔ ﺃﻭ ﻻ ﻳﻘﻄﻊ ﺃﻳﺎ ﻣﻨﻬﻤﺎ ، ﻭﺇﺫﺍ ﻗﻄﻌﻬﻤﺎ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺘﺎ ﺍﳌﻘﻄﻌﲔ ﺍﻟﻨﺎﲡﲔ ﻣﺘﺴﺎﻭﻳﺘﲔ‬ ‫@‬ ‫@@@@@‪@ïŽa‡Ž@Šì“äß@@@@ïŽb½@Šì“äß@@@@ïqýq@Šì“äß@@@@ìЏÛa@ðŒaìnß@@@@@@@@@pýîĐn¾a@ðŒaìnß@@@@@@@@kÈØß‬‬ ‫ﻓﺈﻥ ﻟﻠﻤﺠﺴﻤﲔ ﺍ ، ﺏ ﺍﳊﺠﻢ ﻧﻔﺴﻪ ٠‬ ‫٦٥١‬ ‫٥٥١‬
  • 2. ‫@@@@@‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫‪W‬‬ ‫@‪N‬‬ ‫@‬ ‫@‪@L‬و ل‬ ‫א‪١٠@ Ğ‬‬ ‫ل‬ ‫א‬ ‫‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫‪@@Z@áöbÔÛa@Šì“ä¾a@À‬‬ ‫ﺀ~ ٠٨٤ﺳﻢ@‬ ‫ﺝ~ ٨٤ﺳﻢ@‬ ‫ﺏ~ ٠٨٤ﺳﻢ#‬ ‫ﺍ~ ٠٨ ﺳﻢ@‬ ‫١~ ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ = ﳏﻴﻂ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ‬ ‫@‬ ‫................................................................................................................................................................................................................................................................‬ ‫۲~ ﺍﳌﺴﺎﺣﺔ ﺍﻟﻜﻠﻴﺔ = ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ۲ × ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫٣~ ﺣﺠﻢ ﺍﳌﻨﺸﻮﺭ = ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ‬ ‫................................................................................................................................................................................................................................................................‬ ‫ﺣﻴﺚ ﻕ : ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ ، ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳌﻨﺸﻮﺭ‬ ‫ﺡ =ﻕ ×ﻉ‬ ‫ﺃﻱ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪@@Ýöb¾a@Šì“ä¾a@À‬‬ ‫ل א‪ ú ğ‬א‪ Ğ‬‬ ‫‪C‬‬ ‫א‬ ‫א‬ ‫=‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫א‬ ‫‪C۲H‬‬ ‫א‪Ğ‬‬ ‫= א‬ ‫א‬ ‫۲~ א‬ ‫..................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ ‫ل א‪ ú ğ‬א‪ Ğ‬‬ ‫‪C‬‬ ‫א‬ ‫א‬ ‫=‬ ‫א‬ ‫٣~‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪@Z@@áöbÔÛa@Šì“ä¾a@À‬‬ ‫€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€@‬ ‫٠٤ ـﻢ@ و‬ ‫ﺳـ‬ ‫٠٣٥ و‬ ‫אو‬ ‫א‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫۲~‬ ‫‪ LL‬א‬ ‫א‬ ‫١~ א‬ ‫‪@K‬‬ ‫א‬ ‫א‬ ‫ل א‪ ú ğ‬א‪ F Ğ‬ع ‪ Z‬ل ‪E‬‬ ‫ع‪Z‬‬ ‫۲~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪@Z@Ýöb¾a@Šì“ä¾a@À‬‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪ ú‬א‪Ğ‬‬ ‫⊥א‬ ‫א‬ ‫١~ א‬ ‫) ﺣﻴﺚ ﻫـ ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﳌﻨﺸﻮﺭ (‬ ‫ل א‪ ú ğ‬א‪ × Ğ‬ﺟﺎﻩ‬ ‫ع=‬ ‫۲~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫٣~ ﻣﺴﺎﺣﺔ ﺍﳌﻘﻄﻊ ﺍﻟﻘﺎﺋﻢ = ﻕ × ﺟﺎﻩ‬ ‫..................................................................................................................................................................................................................................................................‬ ‫‪ C‬ﺟﺎﻩ‬ ‫א‬ ‫‪Z‬‬ ‫א‬ ‫א‬ ‫٤~ ‬ ‫................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫ﻣﻠﺤﻮﻇﺔ :‬ ‫@@@@@@@@‬ ‫אل א ول و‬ ‫‪ @Z@k튇m‬א‬ ‫א‬ ‫و‬ ‫א‪Ğ‬‬ ‫‪@N‬‬ ‫ﺏ~ א‪ğ‬‬ ‫א‬ ‫@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ﺍ~‬ ‫٨٥١‬ ‫@‬ ‫٧٥١‬
  • 3. ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫و‬ ‫אو ٠٣٥ وא‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫‬ ‫٤~‬ ‫א‪ ٦] ١٠ Ğ‬ﺳﻢ @‪‡uëc‬‬ ‫٠١ ﺳﻢ و ل‬ ‫٥۲ ]٦ ﺳﻢ@ א‬ ‫٣~‬ ‫‪K‬‬ ‫א‬ ‫א‬ ‫ﺏ~‬ ‫ل א‪ ú ğ‬א‪K Ğ‬‬ ‫ﺍ~‬ ‫‪Z@kya‬‬ ‫٧٨١ﺳﻢ@‬ ‫‪K‬‬ ‫א‬ ‫א‬ ‫ﺏ~‬ ‫א‬ ‫ﺍ~‬ ‫:‬ ‫................................................................................................................................................................................................................................................................‬ 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Ğ‬ﺳﻢ و‬ ‫‪@@@Z@k튇m‬‬ ‫ א א‬ ‫א‬ ‫א‬ ‫٠٦ ْ ، א‬ ‫אو‬ ‫א‬ ‫ﺏ~‬ ‫א‪Ğ‬‬ ‫ﺍ~‬ ‫‪Z‬‬ ‫٠٠٥ﺳﻢ@ ، و‬ ‫א‬ ‫٠٦١‬ ‫٩٥١‬
  • 4. ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫א‪ Ğ‬‬ ‫٠٦% و ل‬ ‫אو‬ ‫א‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫٦~‬ ‫‪JW‬‬ ‫‪ K‬אو‬ ‫‪Ħ‬‬ ‫א‬ ‫٥~‬ ‫@‬ ‫‪W‬‬ ‫و‬ ‫ل‬ ‫א‬ ‫و‬ ‫٠۲‬ ‫‪K‬‬ ‫א‬ ‫ﺫ~‬ ‫א‬ ‫١~ א‬ ‫א‬ ‫٣~‬ ‫ع א‬ ‫۲~ א‬ ‫א‪Ğ‬‬ ‫١~ א‬ ‫................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ 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‫................................................................................................................................................................................................................................................................‬ ‫...............................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ 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‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫.‬ ‫א‪Ğ‬‬ ‫‪ĩ‬‬ ‫א‬ ‫@٠٨۲ﺳﻢ# @‬ ‫‪:@k튇m‬‬ ‫..................................................................................................................................................................................................................................................................‬ ‫@‪N‬‬ ‫א‪ Ğ‬‬ ‫ل‬ ‫ن‬ ‫ل‬ ‫‪@L%٣٠@a‬א‬ ‫אو‬ ‫۲٦١‬ ‫١٦١‬
  • 5. ‫.‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫א‬ ‫٣‬ ‫٠٣ ﺳﻢ‬ ‫و‬ ‫۲١‬ ‫و‬ ‫א‪٥ Ğ‬‬ ‫ل‬ ‫٨~‬ ‫‪W‬‬ ‫٠٠۲ ﺳﻢ# ، و‬ ‫٨ ﺳﻢ ، و‬ ‫א‬ ‫٧~‬ ‫א‬ ‫ﺏ~‬ ‫ﺍ~‬ ‫‪K‬‬ ‫א‪Ğ‬‬ ‫ﺏ~‬ ‫‪K‬‬ ‫ل‬ ‫ﺍ~‬ ‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................................‬ 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‫................................................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................................................‬ ‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬ ‫ل‬ ‫و‬ ‫٠۲ﺳـﻢ@ وא‬ ‫٠۲١ﺳـﻢ٣ و‬ ‫‪: @ k튇m‬‬ ‫................................................................................................................................................................................................................................................................‬ ‫‪K‬‬ ‫א‬ ‫א‪ Ğ‬א‬ ‫..................................................................................................................................................................................................................................................................‬ ‫٤٦١‬ ‫٣٦١‬
  • 6. ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫‪ ª‬א‬ ‫א‪ğ‬‬ ‫א‬ ‫א م‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫.......................................................................................‬ ‫................................‬ ‫.....................................................‬ ‫................................‬ ‫@‬ ‫‪ Z@òí‹Äã‬ﺇﺫﺍ ﻛﺎﻥ ﻉ ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﻭﻛﺎﻧﺖ ‪ ۲Q‬ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ ﻭﻗﻄﻊ ﺍﳍﺮﻡ ﲟﺴﺘﻮ ﻳﻮﺍﺯﻱ ﺍﻟﻘﺎﻋﺪﺓ ﻭﻳﺒﻌﺪ ﻙ ﻋﻦ‬ ‫ﻙ@‬ ‫‪@â‹a‬‬ ‫ﺍﻟﺮﺃﺱ ﻓﺈﻥ ﻣﺴﺎﺣﺔ ﺍﳌﻘﻄﻊ ﻕ ١= ــ × ﻕ ٢‬ ‫ﻉ@‬ ‫א م ‪W‬‬ ‫‬ ‫‪ª‬‬ ‫‪E‬‬ ‫‪ F‬א‪Ğ‬‬ ‫א‬ ‫و‬ ‫‪E‬‬ ‫‪F‬א‬ ‫א‬ ‫‪ Ù‬و‬ ‫ﻣﻼﺣﻈﺔ : ﻉ = ﻉَ + ﻙ‬ ‫ــ ١= ـﻙ@‬ ‫ﻕ‬ ‫ﺃﻱ ﺃﻥ :‬ ‫‪ F‬س א م ‪E‬‬ ‫وא‬ ‫ﻕ ٢ ﻉ@‬ ‫ﺣﻴﺚ ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ،، ﻉ : ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻨﺎﻗﺺ‬ ‫‪‬‬ ‫@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ﻙ : ﺍﳌﺴﺎﻓﺔ ﻣﻦ ﺭﺃﺱ ﺍﳍﺮﻡ ﺇﱃ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺼﻐﺮﻯ‬ ‫א‬ ‫س א م ‪ħ‬‬ ‫د‬ ‫א‬ ‫א‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ :‬ ‫ﻓﻴﺴﻤﻰ ﺍﳍﺮﻡ ﺛﻼﺛﻴﺎ ﺃﻭ ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ﻭﻫﻜﺬﺍ ﺇﻥ ﻛﺎﻧﺖ ﻗﺎﻋﺪﺗﻪ ﻣﺜﻠﺜﺎ ﺃﻭ‬ ‫א ‪ęª‬‬ ‫א‬ ‫‪@P@@@ñ‡ÇbÔÛa@òybß@À@ÊbÐmŠüa@l‹š@sÜq@ðëbí@â‹ç@ðc@ávy@Z@@òí‹Äã‬‬ ‫ﺭﺑﺎﻋﻴﺎ ﺃﻭ ﲬﺎﺳﻴﺎ ٠٠٠٠٠‬ ‫ﺭﺃﺱ ﺍﳍﺮﻡ‬ ‫ﻭﻳﺪﻋﻰ ﺍﳍﺮﻡ ﺍﻟﺜﻼﺛﻲ ﺑـ ﺭﺑﺎﻋﻲ ﺍﻟﻮﺟﻮﻩ‬ ‫ = !؛٣ ‪ × Q‬ﻉ‬ ‫@‪: æc@ðc‬‬ ‫ﺍﳊﺮﻑ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺃﺣﺮﻓﻪ ﻣﺘﺴﺎﻭﻳﺔ ﲰﻲ ﺭﺑﺎﻋﻲ ﻭﺟﻮﻩ ﻣﻨﺘﻈﻢ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ‬ ‫ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺃﻭﺟﻬﻪ ﻣﺜﻠﺜﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﺍﻷﺿﻼﻉ‬ ‫ﻝ١‬ ‫‪@Z@˜ÓbäÛa@â‹a‬‬ ‫‪١Q‬‬ ‫ﺇﺫﺍ ﻗﻄﻊ ﺍﳍﺮﻡ ﲟﺴﺘﻮ // ﺍﻟﻘﺎﻋﺪﺓ ﻓﺈﻥ ﺍﳉﺰﺀ ﺍﶈﺼﻮﺭ ﺑﲔ ﺍﳌﻘﻄﻊ ﻭﺍﻟﻘﺎﻋﺪﺓ ﻳﺴﻤﻰ ﻫﺮﻣﺎ ﻧﺎﻗﺼﺎ ٠‬ ‫ﹰ‬ ‫ﹰ‬ ‫ﻗﺎﻋﺪﺓ ﺍﳍﺮﻡ‬ ‫‪W‬‬ ‫و‬ ‫א‪Ğ‬‬ ‫و‬ ‫م‬ ‫ﺍﳍﺮﻡ ﺍﻟﻘﺎﺋﻢ :‬ ‫) ﻫﻮ ﺍﳉﺰﺀ ﺍﶈﺼﻮﺭ ﺑﲔ ﻗﺎﻋﺪﺓ ﺍﳍﺮﻡ ﻭﺍﳌﻘﻄﻊ ﺍﳌﻮﺍﺯﻱ ﻟﻠﻘﺎﻋﺪﺓ ( ﻭﻓﻴﻪ ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ‬ ‫١~ ﻣﻮﻗﻊ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻣﻦ ﺍﻟﺮﺃﺱ ﻋﻠﻰ ﺍﻟﻘﺎﻋﺪﺓ ﻫﻮ ﻣﺮﻛﺰﻫﺎ‬ ‫ﻉ‬ ‫ﻣﺮﻛﺰ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫ﻋﻠﻰ ﺷﻜﻞ ﺷﺒﻪ ﻣﻨﺤﺮﻑ ٠‬ ‫ﻫﺮﻡ ﺛﻼﺛﻲ ﻗﺎﺋﻢ‬ ‫۲~ ﺃﻭﺟﻬﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻋﺒﺎﺭﺓ ﻋﻦ ﻣﺜﻠﺜﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﺍﻟﻀﻠﻌﲔ‬ ‫‪٢Q‬‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻨﺎﻗﺺ@@‪ Z‬ﻫﻮ ﺍﻟﺒﻌﺪ ﺍﻟﻌﻤﻮﺩﻱ ﺑﲔ ﺍﻟﻘﺎﻋﺪﺗﲔ ٠‬ ‫٣~ ﺍﺭﺗﻔﺎﻋﺎﺕ ﺍﻷﻭﺟﻪ ﺍﳉﺎﻧﺒﻴﺔ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ‬ ‫@‬ ‫ﻝ۲‬ ‫‪@Z@@òí‹Äã‬‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ ﺍﻟﻘﺎﺋﻢ : ﻫﻮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻣﻦ ﺭﺃﺱ ﺍﳍﺮﻡ ﺍﱃ ﻣﺮﻛﺰ ﺍﻟﻘﺎﻋﺪﺓ )ﺃﻱ ﺍﻟﻨﻘﻄﺔ ﺍﻟﱵ ﳍﺎ ﺍﻟﺒﻌﺪ ﻧﻔﺴﻪ ﻋﻦ ﲨﻴﻊ‬ ‫ﻫﺮﻡ ﻗﺎﺋﻢ ﻧﺎﻗﺺ‬ ‫‪= ˜ÓbäÛa@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬‬ ‫ﺭﺅﻭﺱ ﺍﻟﻘﺎﻋﺪﺓ (‬ ‫ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ ﻟﻠﻬﺮﻡ ﺍﻟﻨﺎﻗﺺ = !؛٢ ﳎﻤﻮﻉ ﳏﻴﻄﻲ ﺍﻟﻘﺎﻋﺪﺗﲔ × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫‪@@Z@òí‹Äã‬‬ ‫@@‪Z@æc@ðc‬‬ ‫‪! = áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬؛٢ ﳏﻴﻂ ﺍﻟﻘﺎﻋﺪﺓ × ﺍﻻﺭﺗﻔﺎﻉ ﺍﳉﺎﻧﱯ‬ ‫ﺃﻱ ﺃﻥ :‬ ‫‪! = ˜ÓbäÛa@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬؛٢ ﻥ )ﻝ ١ + ﻝ ٢ ( × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﳍﺮﻡ‬ ‫ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫@@‪@@]@áöbÔÛa@â‹èÜÛ@òîjãb§a@òyb¾a‬‬ ‫!؛٢ ﻥ ﺱ × ﺍﺭﺗﻔﺎﻉ ﺍﻟﻮﺟﻪ ﺍﳉﺎﻧﱯ‬ ‫@@‪ Z@sîy‬ﻥ ﻋﺪﺩ ﺃﺿﻼﻉ ﺍﻟﻘﺎﻋﺪﺓ ، ﻝ ١ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺼﻐﺮﻯ، ﻝ ٢ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﻜﱪﻯ‬ ‫ﻫﺮﻡ ﺭﺑﺎﻋﻲ‬ ‫ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺃﺿﻼﻉ ﺍﻟﻘﺎﻋﺪﺓ ، ﺱ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫‪ = ˜ÓbäÛa@â‹èÜÛ@òîÜØÛa@òyb¾a‬ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ‪۲Q + ١Q‬‬ ‫‪ = â‹èÜÛ@òîÜØÛa@òyb¾a‬ﺍﳌﺴﺎﺣﺔ ﺍﳉﺎﻧﺒﻴﺔ + ﻣﺴﺎﺣﺔ ﺍﻟﻘﺎﻋﺪﺓ‬ ‫٦٦١‬ ‫٥٦١‬