จากรูปที่ ให้นักเรียนน...

1

กลมสาระการเรยนรคณตศาสตร พนทผวและปรมาตร ครบญน า เกษ

ในชวตประจ าวนเราพบรปเรขาคณตสามมตอยในชวตประจ าวน

อยเสมอ ดงรป

1 2 3

4 5 6

7 8 9

10 11 12

13 14 15

16 17 18

จากรปท 1-18 ใหนกเรยนน ารปภาพทมลกษณะเดยวกนมาจด

ไวในกลมเดยวกน

กลมท 1 รปท ..........................................................................

รปมฐานเปนรป........................... ผวขางเปนรป..............................

มชอเรยกวาโดยทวไป .............................................................

กลมท 2 รปท ..........................................................................



กลมท 3 รปท ..........................................................................



กลมท 4 รปท ..........................................................................



กลมท 5 รปท ..........................................................................



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1. ใหบอกชอ และสวนตาง ๆ ของรปเรขาคณตสามมตใหถกตอง




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6. ใหนกเรยนเลอกรปเรขาคณตสามมตทนกเรยนสนใจ แลววาด

ภาพของรปเรขาคณตสามมตนนพรอมทงวาดรปคลของรป

เรขาคณตสามมตนนดวย

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พนทของรปเรขาคณตสองมตใด ๆ หมายถง บรเวณทอยภายใน

รปเรขาคณตสองมตนน ๆ สรปสตรการหาพนทรปเรขาคณตชนด

ตาง ๆ ไดดงน

สเหลยมจตรส = ดาน ดาน

สเหลยมผนผา = กวาง ยาว

สเหลยมดานขนานและสเหลยมขนมเปยกปน

= สวนสง ความยาวของฐาน

สเหลยมรปวาว รปสเหลยมจตรส และรปสเหลยมขนมเปยกปน

เมอก าหนดความยาวของเสนทแยงมม

=

ผลคณของเสนทแยงมม

สเหลยมคางหม =

สวนสง ผลบวกของดานคขนาน

สามเหลยม =

ฐาน สง

สามเหลยมในกรณททราบความยาวดานทงสามดาน แตไม

ทราบสวนสง

พนทรปสามเหลยม = )cs)(bs)(as(s

เมอ s = 2cba

a, b, c เปนความยาวดานของรปสามเหลยมทก าหนดให

พนทรปสามเหลยมดานเทา = 2)ดาน(4

3

พนทรปหกเหลยมดานเทา = 6)ดาน(43 2

พนทวงกลม = 2r

พนทรปสามเหลยมฐานโคง = 2r360

2. ใหนกเรยนหาพนทของรปเรขาคณตสองมตตอไปน

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หนวยความยาวพนฐานทควรจ าดงน

1 เซนตเมตร เทากบ 10 มลลเมตร

1 เมตร เทากบ 100 เซนตเมตร

1 กโลเมตร เทากบ 1,000 เมตร

1 ฟต เทากบ 12 นว

1 หลา เทากบ 3 ฟต

1 ไมล เทากบ 1,760 หลา

1 คบ เทากบ 12 นว

1 ศอก เทากบ 2 คบ

1 วา เทากบ 4 ศอก

1. จงเตมค าตอบในชองวางใหถกตอง

1) 1 ตารางเมตร เทากบ ......................... ตารางเซนตเมตร

4 ตารางเมตร เทากบ .......................... ตารางเซนตเมตร

2) 1 ตารางหลา เทากบ .......................... ตารางฟต

3 ตารางหลา เทากบ ............................ตารางฟต

3) …..........…ตารางฟต เทากบ 900 ตารางเซนตเมตร

4 ตารางฟต เทากบ ................ตารางเซนตเมตร

4) …......... ตารางหลา เทากบ 9 ตารางฟต

10 ตารางหลา เทากบ ................ตารางฟต

5) 1 ตารางวา เทากบ ................ตารางเมตร

…… ตารางวา เทากบ 80 ตารางเมตร

6) 1 ตารางฟต เทากบ ................ตารางนว

4 ตารางฟต เทากบ ................ตารางนว

7) …… ตารางวา เทากบ 40,000 ตารางเซนตเมตร

…… ตารางเมตร เทากบ 80,000 ตารางเซนตเมตร

8) …… ตารางหลา เทากบ 1,296 ตารางนว

…… ตารางหลา เทากบ 64,800 ตารางนว

9) 4 ตารางเมตร เทากบ ................ตารางวา

20 ตารางเมตร เทากบ ................ตารางวา

10) …… ตารางนว เทากบ 1 ตารางฟต

432 ตารางนว เทากบ..................ตารางฟต

11) 900 ตารางเซนตเมตร เทากบ.....................ตารางฟต

…… ตารางเซนตเมตร เทากบ 5 ตารางฟต

12) …… ตารางฟต เทากบ 1 ตารางหลา

…… ตารางฟต เทากบ 10 ตารางหลา

หนวยปรมาตรหรอหนวยการตวงทนยมใช

1 ลกบาศกเซนตเมตร เทากบ 1 มลลลตร

1 ลตร เทากบ 1,000 ลกบาศกเซนตเมตร

1 กโลลตร เทากบ 1,000 ลตร

1 ลกบาศกเมตร เทากบ 1,000 ลตร

หนวยการตวงในมาตราไทย

1 ถง เทากบ 20 ลตร

1 เกวยน เทากบ 100 ถง

1 เกวยน เทากบ 2 ลกบาศกเมตร

1 เกวยน เทากบ 2,000 ลตร

1. จงเตมค าตอบในชองวางใหถกตอง

1) 1 ลกบาศกฟต เทากบ ........................ ลกบาศกนว

3 ลกบาศกฟต เทากบ ........................ ลกบาศกนว

2) 27,000 ลกบาศกเซนตเมตร เทากบ ........... ลกบาศกฟต

......................ลกบาศกเซนตเมตร เทากบ 11 ลกบาศกฟต

3) ...................ลกบาศกเซนตเมตร เทากบ 1 ลกบาศกเมตร

20,000,000 ลกบาศกเซนตเมตร เทากบ .........ลกบาศกเมตร

4) ...................ลตร เทากบ 1,000 มลลลตร

7 ลตร เทากบ .........................มลลลตร

5) 1 ถง เทากบ ........................ ลตร

..................... ถง เทากบ 100 ลตร

2. จงน าตวอกษรทางขวามอเตมลงหนาขอความทางซายมอ

ใหถกตอง

................. 1) 1,000,000 cm3 a. 5 เกวยน

................. 2) 60 ลตร b. 10,000,000 cm3

................. 3) 2 กโลลตร c. 270 ลกบาศกฟต

................. 4) 10 ลกบาศกเมตร d. 3 ถง

................. 5) 500 ถง e. 40 ลกบาศกเมตร

................. 6) 2 เกวยน f. 2 ลกบาศกฟต

................. 7) 54,000 มลลลตร g. 2,000 ลตร

................. 8) 1 ลกบาศกฟต h. 1 ลกบาศกเมตร

................. 9) 5 ลกบาศกวา i. 4 ลกบาศกเมตร

................. 10) 10 ลกบาศกหลา j. 1,728 ลกบาศกนว

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ปรซมเปนรปทรงสามมตทมฐานทงสองเปนรปเหลยมทเทากน

ทกประการ และฐานทงสองอยในระนาบทขนานกน และมดานขาง

แตละดานเปนรปสเหลยมมมฉาก การเรยกชอปรซมจะเรยกตาม

ลกษณะของฐานปรซมนน

พจารณาปรซมสเหลยมมมฉาก ดงรปตอไปน

การหาพนทผวของรปเรขาคณตสามมตใด ๆ เปนการหาพนท

ของพนทผวทงหมดของรปเรขาคณตสามมตนน ดงเชน

พนทผวของปรซม คอ พนทรปคลทงหมดของรปคลปรซม

นนคอ พนทผวของปรซม = พนทฐานทงสอง + พนทผวขาง

หรอ

จงหาปรมาตรและพนทผวของปรซมสเหลยมจตรสทม

ฐานยาวดานละ 4 นว และสง 10 นว ดงรป

วธท า ปรมาตร = พนทฐาน สวนสง

= (4 นว 4 นว) 10 นว

= 16 นว210 นว

= 160 นว3

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ถาน าปรซมสเหลยมจตรสทมฐานยาวดานละ 4 นว และสง

10 นว มาคลออกจะไดดงน

= (4 นว 4 นว) +(4 นว 4 นว) + (16 นว 10 นว)

= 16 นว2 + 16 นว2 + 160 นว2

= (16 + 16 + 160) นว2

= 132 นว2

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พนทฐาน = ความกวาง ความยาว

และ h แทน สวนสงของปรซม

แบงปรซมออกเปนแผนบาง ๆ ใหม

ความหนาแผนละ 1 หนวย จะได

1 แผนมปรมาตร

= (ความกวาง ความยาว) 1

h แผน มปรมาตร

= (ความกวาง ความยาว) h

ปรซมสเหลยมจตรส

ปรซมสง 10 นว

ฐานเปนรป สเหลยมจตรส

พนทสเหลยมจตรส=ดานดาน

ฐานยาวดานละ 4 นว

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จากรปทก าหนดใหจงหาปรมาตรและพนทผว

(ก าหนดหนวยความยาวเปน เมตร (m))


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ถาน าปรซมทก าหนดใหมาคลออกจะไดรปดงน

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(ก าหนดหนวยความยาวเปน เซนตเมตร (cm))


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ปรซม................................................

ปรซมสง ..........................................

ฐานเปนรป........................................

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กวาง ........................................

ยาว ..........................................

ปรซม................................................

ปรซมสง ..........................................

ฐานเปนรป........................................

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กวาง ........................................

ยาว ..........................................

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ปรซม................................................

ปรซมสง ..........................................

ฐานเปนรป........................................

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ฐานยาว ..................................

ฐานสง ......................................


(ก าหนดหนวยความยาวเปน เซนตเมตร (cm))


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ปรซม.........................................

ปรซมสง ...................................

ฐานเปนรป.................................

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ฐานยาว ..................................

ฐานสง ......................................

8





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(ก าหนดหนวยความยาวเปน ฟต (f))


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ปรซม................................................

ปรซมสง ..........................................

ฐานเปนรป........................................

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ฐานสง ......................................

ผลบวกดานคขนาน ...................

ปรซม................................................

ปรซมสง ..........................................

ฐานเปนรป........................................

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ฐานยาว ..................................

ฐานสง ......................................

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ขนตอนสการแกปญหาโจทยปญหาอยางมประสทธภาพ

นกเรยนตองมความรดงตอไปน

1. มความสามารถในการหาพนทผวและปรมาตรของปรซม

2. นกเรยนตองเขาใจในขนตอนการแกโจทยปญหา

1) โจทยถามเกยวกบอะไร

2) โจทยตองการใหหาอะไร

3) โจทยใหขอมลอะไรมาบาง

1) เปลยนขอความใหเปนสญลกษณ หรออน

2) ระบขนตอนการแกปญหา

ด าเนนการตามขนตอนทระบไวใน ขนท 2 ขอ 2)

1) ตรวจสอบตามขนตอนการด าเนนงานทกขนตอน

2) ตรวจสอบความถกตองหรอความสมเหตสมผลของ

ค าตอบทได

ปรมาตรของปรซม = พนทฐาน สวนสง

พนทผวของปรซม

= พนทฐานทงสอง + (ความยาวรอบฐาน สง)

ถงน าทรงสเหลยมมมฉากฐานเปนสเหลยมผนผา

ขนาดกวาง 2 เมตร ยาว 4 เมตร สง 5เมตร อยากทราบวา พนทผว

ทงหมดของถงใบนวาเปนเทาใด


ตอบ พนทผวของถงน าทรงสเหลยมมมฉากฐานเปน

สเหลยมผนผา


ตอบ พนทผวของถง

3) โจทยใหขอมลอะไรมาใหบาง

ตอบ ถงน าทรงสเหลยมมมฉากฐานเปนสเหลยมผนผา

มความกวาง 2 เมตร, ยาว 4 เมตร และสง 5 เมตร ดงรป

แทนคา ตามสตรพนทผวของปรซม

= พนทฐานทงสอง + (ความยาวรอบฐาน สง)

วธท า พนทผวของปรซม

= (2 m 4 m) +(2 m 4 m) + (12 m 5 m)

= 8 m2 + 8 m2 + 60 m2

= (8 + 8 + 60) m2

= 76 m2

พจารณาความสมเหตสมผล พนทผวของปรซม

76 = 60+16

76 = 76 จรง

พนทผวทงหมดของถงใบนคอ 76 ตารางเมตร

10


อางเกบน าทรงสเหลยมมมฉากยาว 20 เมตร

กวาง 12 เมตร ถาตองการเกบน าไวในอาง 1,920 ลกบาศกเมตร

ระดบน าจะตองสงจากกนอางเทาไร


ตอบ ปรมาตรของอางเกบน าทรงสเหลยมมมฉาก


ตอบ ระดบน าจะตองสงจากกนอางหรอความสง(h)


ตอบ อางเกบน าทรงสเหลยมมมฉาก มความยาว 20 เมตร,

ความกวาง 12 เมตร ,ปรมาตรของน าในอางเกบน าเปน

1,920 ลกบาศกเมตร

แทนคาตามสตรปรมาตรของปรซม = พนทฐาน สวนสง

ปรมาตรของปรซม = พนทฐาน สวนสง

1,920 m3 = (20m 12m) h

(20m 12m) h = 1,920 m3

h = 8 m

จะได ระดบน าสงจากกนอาง 8 เมตร

พจารณาความสมเหตสมผล

ปรมาตรของปรซม สงพนทฐาน

1,920 82012?

1,920 920,1?

1,920 920,1 จรง

ตอบ ระดบน าสงจากกนอาง 8 เมตร

ถงน าสแตนเลสฐานสเหลยมผนผา กวาง 2.3 เมตร

ยาว 4.8 เมตร สง 5.2 เมตร อยากทราบวาพนทผวทงหมดของ

ถงใบนวาเปนเทาใด


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กลองนมทรงสเหลยมมมฉาก กนกลองเปนรปสเหลยม

จตรสยาวดานละ 6 เซนตเมตร กลองนมความสง 15 เซนตเมตร

จะตองใชกระดาษท ากลองคดเปนพนททงหมดอยางนอยเทาไร โดย

ไมคดสวนทเปนรอยพบส าหรบตดกาว


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หองเรยนกวาง 5.5 เมตร ยาว 6 เมตร สง 3.5 เมตร

ตองการทาสหองเรยนใหม จะตองจายเงนคาทาสเทาไร เมอคาทาส

ตารางเมตรละ 50 บาท


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12


เสาตนหนงสง 10 ฟต เปนเสาหนาตดรปสเหลยม

จตรส ซงยาวดานละ 8 ฟต คดเปนปรมาตรของเนอไมของเสาตนน

คดเปนเทาไร


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ขนมปงแถวหนงมหนาตดเปนรปสเหลยมผนผา

มขนาดของดานกวางและดานยาวเปน 9 และ 10 เซนตเมตร

ตามล าดบ ถาขนมปงมปรมาตร 450 ลกบาศกเซนตเมตร

จงหาความสงของขนมปงแถวน


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13


บอเกบน าทรงสเหลยมมมฉาก ส าหรบน าน าเขานากง

ทมความยาว 15 เมตร และกวาง 12 เมตร ถาตองการเกบน าไวใน

อาง 1,080 ลกบาศกเมตร ระดบน าจะสงกวากนบอเทาไร


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อสระตองการซอดนมาถมทส าหรบปลกบาน ซงม

พนทหนาตดเปนรปสเหลยมคางหม มดานคขนานยาว 5 เมตร และ

4 เมตร ตามล าดบ หางกนอย 4 เมตร ตองการถมทใหสงจากเดม 3

เมตร จะตองใชดนทงหมดกลกบาศกเมตร


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14


สวนสาธารณะแหงหนง เปนรปสเหลยมผนผายาว

229.5 เมตร กวาง 27.5 เมตร ตองการท าทางเดนส าหรบให

ประชาชนใชวงออกก าลงกาย กวาง 2.5 เมตร รอบนอก

สวนสาธารณะและใหสงจากพนดน 10 เซนตเมตร อยากทราบวา

จะตองใชซเมนตคอนกรตทางเดนเปนปรมาตรเทาใด


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มงกรตองการซอทรายมาถมสนามหนาบาน ซงเปน

รปสเหลยมมมฉาก ยาว 7.5 เมตร กวาง 5.5 เมตร โดยยกระดบสง

กวาเดม 15 เซนตเมตร จะตองซอทรายมาถมอยางนอยกคน

ถารถบรรทกทรายคนหนงมกระบะบรรทกยาว 3 เมตร

กวาง 2 เมตร สง 0.75 เมตร


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15


รปเรขาคณตสามมตทมฐานสองฐานเปนรปวงกลมทเทากนทก

ประการและอยบนระนาบทขนานกน เมอตดรปเรขาคณตสามมตนน

ดวยระนาบทขนานกบฐานแลวจะไดหนาตดเปนวงกลมทเทากนทก

ประการกบฐานเสมอเรยกรปเรขาคณตสามมตนนวา

“ทรงกระบอก”

พจารณาทรงกระบอก ดงรปตอไปน

ทรงกระบอกมพนทฐาน = r2

และ h แทน ความสงของทรงกระบอก

เมอแบงทรงกระบอกออกเปนแผนบาง ๆ ใหมความหนาแผนละ

1 หนวย

จะได 1 แผน มปรมาตร = r21

ดงนน h แผน มปรมาตร = r2h

ปรมาตรของทรงกระบอกน = r2h

ดงนน ปรมาตรของทรงกระบอก = r2h

เมอ r แทน ความยาวของรศมของฐาน

h แทน ความสงของทรงกระบอก

การหาพนทผวของรปเรขาคณตใด ๆ เปนการหาพนทของ

พนทผวทงหมดของรปเรขาคณตสามมตนน พนทผวของ

ทรงกระบอก กคอ หาพนทดานขางทงหมด รวมกบพนทฐานทงสอง

ตวอยางดงรปคลน

รปคลของทรงกระบอกทมความสง h หนวย

และมรศมของฐานยาว r หนวย

พนทผวของทรงกระบอก คอ พนทรปคลทงหมดของรปคลของ

ทรงกระบอก

นนคอ พนทผวของทรงกระบอก= พนทฐานทงสอง + พนทผวขาง

= (พนทวงกลม+ พนทวงกลม)+ (พนทสเหลยมผนผา)

= (พนทวงกลม+ พนทวงกลม)+ (กวางยาว)

= r2 + r2 + (2rh)

= 2r2 + 2rh

= 2r (r+h)

ดงนน พนทผวของทรงกระบอก = 2r2 + 2rh

หรอ = 2r (r+h)

เมอ r แทน ความยาวของรศมของฐาน

h แทน ความสงของทรงกระบอก

16


จงหาปรมาตรและพนทผวของทรงกระบอก ทมรศม

ของฐาน 14 เซนตเมตร และความสง 100 เซนตเมตร (22/7)

วธท า

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จงหาปรมาตรและพนทผวของทรงกระบอกทมรศม


วธท า

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ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

17


จงหาปรมาตรและพนทผวของทรงกระบอกทมเสน

ผานศนยกลางของฐาน 35 เซนตเมตร และความสง 28 เซนตเมตร

(22/7)

วธท า

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ผานศนยกลางของฐาน 14 เซนตเมตร และความสง 18 เซนตเมตร

(22/7)

วธท า

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ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

18



ผานศนยกลางของฐาน 7 เซนตเมตร และมความสง 6 เซนตเมตร

(22/7)

วธท า

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จงหาปรมาตรและพนทผวของทรงกระบอก ทมรศม


วธท า

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ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

ทรงกระบอกสง (h) ................

ฐานเปนรป............................

r = ................................

= ..............................

19



นกเรยนตองมความร ดงน ขนท 1 เขาใจปญหา ขนท 2 วางแผน

การแกปญหา ขนท 3 ด าเนนการแกปญหา และขนท 4 ตรวจสอบ

ยอนกลบ

โลหะทรงกระบอกมรศมทฐานยาว 14 เซนตเมตร

สง 10 เซนตเมตร จะมพนทผวทงหมดเทาไร


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โลหะทรงกระบอกมรศมทฐานยาว 3 เซนตเมตร

สง 14 เซนตเมตร จะมพนทผวทงหมดเทาไร


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20


เสาคอนกรตทรงกระบอกมรศมยาว 28 เซนตเมตร

สง 9 เมตร จงหาพนทผวขางของเสาคอนกรตน


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ไมพลองของลกเสออนหนงยาว 1.5 เมตร วด

เสนผาศนยกลางหนาตดหวทายไดยาว 3.5 เซนตเมตร เทากน

ถาจะทาสผวของไมพลองน จะตองทาสคดเปนพนทเทาไร


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21


สมชายตองการหลอเสาคอนกรตทรงกระบอกมเสน

ผานศนยกลางยาว 56 เซนตเมตร สง 7 เมตร สมชายจะตองเตรยม

คอนกรตไวประมาณกลกบาศกเมตร เมอตองการหลอเสาขนาด

เดยวกนน จ านวน 4 ตน


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บอเกบน ารปทรงกระบอกวดเสนผานศนยกลาง

ภายในได 2.8 เมตร สง 5.5 เมตร จะสามารถจน าไดเทาไร


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22


ทอระบายน ารปทรงกระบอกวดเสนผานศนยกลาง

ภายนอกได 35 นว หลอปนหนา 3 นว ยาว 5 ฟต จงหาปรมาตรของ

ปนทใชท าทอระบายน าน


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ถงเกบปนซเมนตทรงกระบอกขนาดใหญ มรศมยาว

7 เมตร เกบปนซเมนตได 1,540 ลกบาศกเมตร ถงเกบปนซเมนตน

สงเทาไร


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23


รปเรขาคณตสามมตทมผวโคงเรยบและจดทกจดบนผวโคงอย

หางจากจดคงทจดหนงเปนระยะเทากนเรยกวา “ทรงกลม”

การหาปรมาตรของทรงกลมสามารถทดลองไดดงน

1. เตรยมถงทรงกระบอกทมรศมทฐานยาว r หนวย และมความ

สง 2r หนวย โดยใสน าไวเตมถง และเตรยมทรงกลมทมเสนผาน

ศนยกลางเทากบ 2r หนวย

2. น าทรงกลมใสในถงทรงกระบอก จะไดน าทลนออกมาคอ

ปรมาตรของทรงกลม

3. น าน าทลนออกมาใสในถงทรงกระบอกทมรศมทฐานยาว

r หนวย และมความสง 2r หนวย วดระดบน าในถงจะได

2 ใน 3ของถง

จากปรมาตรทรงกระบอก = r2h

= r2(2r) r2h;

= 2r3

เนองจากปรมาตรทรงกลมเปน 32ของปรมาตรทรงกระบอก

3r23

2

3r3

4

ดงนน ปรมาตรของทรงกลม 3r3

4

เมอ r แทน ความยาวของรศมวงกลม

จงหาปรมาตรของทรงกลมรศมยาว 7 เซนตเมตร

(22/7)

วธท า ปรมาตรของทรงกลม 3r3

4

=.......................................................................

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จงหาปรมาตรของทรงกลมรศมยาว 14 เซนตเมตร

(22/7)


4

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24


จงหาปรมาตรของทรงกลม เมอรศมของทรงกลม

21 เซนตเมตร (22/7)


4

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3 ฟต (22/7)


4

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6 นว (22/7)


4

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10.5 เซนตเมตร (22/7)


4

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25





ยอนกลบ

ทรงกลมมรศม 6 เซนตเมตร จะมปรมาตรประมาณ

เทาใด


ตอบ ............................................................................... ......

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ทรงกลมมปรมาตรประมาณ 38,808 ลกบาศกนว

จะมรศมประมาณเทาใด

1) โจทยถามเกยวกบอะไร ตอบ ...............................

2) โจทยตองการใหหาอะไร ตอบ ...............................


ตอบ .....................................................................................

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ลกบอลพลาสตกทรงกลมมรศม 14 นว จะมปรมาตร

ประมาณเทาใด




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26


หลอลกทมน าหนกดวยเหลก ใหมเสนผานศนยกลาง

ยาว 14 เซนตเมตร ลกทมน าหนกนจะหนกเทาใด เมอ 1 ลกบาศก

เซนตเมตร หนก 7.8 กรม




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ทรงกลมมปรมาตรประมาณ 310,464 ลกบาศก

เซนตเมตร จะมรศมประมาณเทาใด




ตอบ .....................................................................................

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ถงเกบน ามนแหงหนงเปนรปทรงกลมมรศม 1.5 เมตร

จะจน ามนไดประมาณเทาใด




ตอบ .....................................................................................

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ซอน าผลไมซงบรรจกลองทรงกลมจ านวน 10 อน

น ามาวดเสนผานศนยกลางได 4 เซนตเมตร อยากทราบวาน าผลไมม

ปรมาตรทงหมดประมาณเทาใด




ตอบ .....................................................................................

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27


รปเรขาคณตสามมตทมฐานเปนรปวงกลมมยอดแหลมทไมอย

บนระนาบเดยวกนกบฐานและเสนทตอระหวางจดยอดและจดใด ๆ

บนขอบของฐานเปนสวนของเสนตรง เรยกรปเรขาคณตสามมตนน

วา “กรวย”

สวนตาง ๆ ของกรวย ประกอบดวย

ใหนกเรยนพจารณาภาพตอไปน

จะสงเกตเหนวารปท 1 เปนกรวย และรปท 2 เปนทรงกระบอก

ทมรศมและความสงเทากน ทดลองน าทรายใสกรวยในรปท 1 จน

เตม จากนนน าทรายในรปท 1 เทใสทรงกระบอกในรปท 2 จนหมด

วดความสงของทรายในรปท 2 ได ของความสงรปท 2 นน

หมายความวา ปรมาตรของกรวยเทากบ 31

ของปรมาตรทรงกระบอกทมรศมและความสงเทากน

ดงนน ปรมาตรของกรวย = 31ปรมาตรทรงกระบอก

= 31r2h

เมอ r แทน ความยาวของรศมของฐานกรวย

h แทน ความยาวของสงตรง

จงหาปรมาตรของกรวยทมรศมของฐานยาว 3 ฟต

สงตรงยาว 7 ฟต (22/7)

วธท า ปรมาตรของทรงกรวย = 31r2h

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จงหาปรมาตรของกรวยทมเสนผานศนยกลางของ

ฐานยาว 24 นว สงเอยง 20 นว (22/7)


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28


กรวยอนหนงมความสง 12 นว มเสนผานศนยกลาง

ภายในยาว 7 นว มปรมาตรเทาใด (22/7)


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กรวยอนหนงมความสง 21 นว มเสนผานศนยกลาง

ภายในยาว 10 นว มปรมาตรเทาใด (22/7)


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กรวยอนหนงมสงเอยงยาว 5 นว มรศมยาว 4 นว จง

หาปรมาตรเทาใด (22/7)


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กรวยอนหนงมสงเอยงยาว 15 นว มเสนผาน

ศนยกลางยาว 24 นว จงหาปรมาตรของกรวย (22/7)


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29





ยอนกลบ

ตองการท าททบกระดาษรปทรงกรวยตนไวแจก

งานปใหมจ านวน 200 อน โดยททบกระดาษมเสนผานศนยกลาง

5 เซนตเมตร สง 7 เซนตเมตร จะตองเตรยมวสดในการท าททบ

กระดาษเทาใด

1) โจทยถามเกยวกบอะไร ตอบ .....................................


ตอบ .....................................................................................

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แกวน าส าหรบใสน าผลไมเปนรปกรวยกลม มเสน

ผานศนยกลางทปากแกว 7 เซนตเมตร ถาแกวลก 14 เซนตเมตร

จะจน าผลไมไดเทาใด




ตอบ .....................................................................................

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กระโจมอนเดยแดงเผาหนงมลกษณะเปนรปกรวยสง

2 เมตร ฐานของกระโจมมรศมยาว 1.5 เมตรผาท ากระโจมมพนท

เทาใด (ไมคดสวนทเกยกน)




ตอบ .....................................................................................

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30


ถงเกบน ารปกรวยสง 3 เมตร และมเสนรอบวงของ

ปากกรวยยาว 14 เมตร ถาตองการใสน าไวในถงนเพยงของความจ

ของถง จะตองใสน าลงถงนกลกบาศกเมตร




ตอบ .....................................................................................

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กรวยอนหนงมสงเอยงยาว 10 นว มรศมยาว 8 นว

จงหาปรมาตรเทาใด




ตอบ .....................................................................................

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ถวยกรวยกระดาษส าหรบใสน าดมขนาดเสนผาน

ศนยกลาง 6 เซนตเมตร สง 8 เซนตเมตร เตรยมไวส าหรบนกเรยน

จ านวน 200 คน โดยประมาณวาดมคนละถวยจะตองเตรยมน าไว

ประมาณเทาใด




ตอบ .....................................................................................

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กรวยกลมซงเปนอนสรณสถาน 14 ตลา ทถนนราช

ด าเนน กรงเทพมหานคร มเสนผานศนยกลางของฐาน ยาว

2 เมตร มปรมาตร 9.42 ลกบาศกเมตรเมตร มความสงกเมตร




ตอบ .....................................................................................

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31


รปเรขาคณตสามมตทมฐานเปนรปเหลยมใด ๆ มยอดแหลมท

ไมอยบนระนาบเดยวกนกบฐาน และหนาทกหนาเปนรปสามเหลยม

ทมจดยอดรวมกนทยอดแหลมนนเรยกวา “พระมด”

สวนตาง ๆ ของพระมด ประกอบดวย

ใหนกเรยนพจารณาภาพตอไปน

จะสงเกตเหนวารปท 1 เปนพระมดสเหลยมผนผา และรปท 2

เปนปรซมสเหลยมผนผาทงสองรปมพนทฐานและสวนสงเทากน

ทดลองน าทรายใสรปท 1 จนเตม จากนนน าทรายในรปท 1 เทใสรป

ท 2 จนหมด วดสวนสงของทรายในรปท 2 ได 31ของสวนสงรปท 2

นนหมายความวา ปรมาตรของพระมดสเหลยมผนผา เทากบ

31ของปรมาตรปรซมสเหลยมผนผา และมสวนสงเทากน

ในท านองเดยวกนเมอเปลยนการทดลองโดยใชฐานเปนรป

เหลยมตางๆ ผลการทดลองยงเชนเดม

ดงนน ปรมาตรพระมด = 31 ปรมาตรของพระมด

= 31 พนทฐาน สงตรง

จงหาปรมาตรของพระมดฐานสเหลยมจตรสทมฐาน

ยาวดานละ 4 นว และสงตรง 9 นว

วธท า ปรมาตรพระมด = 31 พนทฐาน สงตรง

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จงหาปรมาตรของพระมดสเหลยมจตรส ทมฐานยาว

ดานละ 8 เซนตเมตร และสงเอยง 5 เซนตเมตร


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พระมดมสงตรง .....................

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พระมดมสงตรง .....................

32


พระมดฐานเปนรปสามเหลยมดานเทา ยาวดานละ

6 ฟต สง 10 ฟต มปรมาตรเทาใด


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พระมดฐานเปนรปหกเหลยมดานเทามมเทา ยาวดาน

ละ 2 เมตร และสง 10 เมตร พระมดนมปรมาตรเทาใด


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พระมดฐานเปนรปสเหลยมจตรส ยาวดานละ 12 นว

และสงเอยง 10 นว พระมดนมปรมาตรเทาใด


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พระมดฐานเปนรปหกเหลยมดานเทามมเทา

ยาวดานละ 6 ฟต และมสนยาว 10 ฟต มปรมาตรเทาใด


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ฐานเปนรป...............................

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ฐานเปนรป...............................

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พระมดมสงตรง .....................

ฐานเปนรป...............................

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พระมดมสงตรง .....................

33





ยอนกลบ

ตองการหลอพลาสตกเหลวเปนพระมดตนฐานเปน

รปสามเหลยมดานเทา ยาวดานละ 12 เซนตเมตร สง 20 เซนตเมตร

จะตองใชพลาสตกเหลวกลกบาศกเซนตเมตร

1) โจทยถามเกยวกบอะไร ตอบ .....................................


ตอบ .....................................................................................

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พระมดฐานเปนรปสเหลยมจตรสมดานยาวดานละ

40 นว และสง 21 นว ตองใชปนพลาสเตอรหลอกลกบาศกนว




ตอบ .....................................................................................

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จงหาปรมาตรของเหลกทใชหลอพระมด

เปนรปหกเหลยมดานเทามมเทา ซงมความยาว ดานละ 4 นว

และสง 10 นว




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34


จงหาปรมาตรของทรายทจะเทลงไปบรรจในพระมด

ฐานเปนรปสเหลยมจตรสยาวดานละ 30 เซนตเมตร ทสงเอยงยาว

25 เซนตเมตร




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พระมดท าดวยไมฐานเปนรปสามเหลยมดานเทา ม

ดานยาวดานละ 20 เซนตเมตร สวนสงยาว 30 เซนตเมตร จงหา

ปรมาตรของเนอไมทใชท าพระมดอนน




ตอบ .....................................................................................

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ตองการใชปนหลอพระมดฐานเปนรปสเหลยมจตรส

ยาวดานละ 18 นว สง 17 นว จงหาปรมาตรของปนทใชหลอพระมด




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ตองการท าปายจารกเปนรปพระมดฐานสเหลยม

จตรส ซงดานยาวดานละ 40 เซนตเมตร และสง 21 เซนตเมตร โดย

ใชวสดเหลวใสเลซนสหลอ จงหาวาตองใชเปนปรมาณเทาใด




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35





ยอนกลบ

ดวงจนทรมเสนผานศนยกลางยาวประมาณ 1/4 ของ

ความยาวเสนผานศนยกลางของโลก จงหาวาปรมาตรของโลกเปนก

เทาของดวงจนทร


ตอบ ....................................................................................


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ทองแดงรปทรงกลมตน 3 ลก รศม 6, 8 และ 10 นว

ตามล าดบ เมอหลอมเปนลกเดยวกนจะไดทรงกลมใหมมรศมยาว

กนว


ตอบ ....................................................................................

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36


ถงน าทรงกระบอกและกรวยมความสง และเสนผาน

ศนยกลางเทากน โดยมความสง 15 เซนตเมตร ถาใชกรวยตกน าให

เตมพอดแลวเทใสลงในถงทรงกระบอก จงหาวาระดบน าในถง

ทรงกระบอกจะสงกเซนตเมตร


ตอบ ....................................................................................

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ใสน าลงในอางน าทรงสเหลยมมมฉากกวาง

25 เซนตเมตร ยาว 50 เซนตเมตร และมความสง 38 เซนตเมตร

ถาระดบน าต ากวาขอบบนของอางอย 8 เซนตเมตร จงหาวามน าอย

ในอางดงกลาวกลกบาศกเซนตเมตร


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กรวยกลมและทรงกระบอกมฐานเทากนและมสง

เทากน ถากรวยมปรมาตร 9 ลกบาศกเซนตเมตร แลวทรงกระบอก

มปรมาตรเทาไร


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กรวยกลมและทรงกระบอกมฐานเทากนและมสง

เทากน ถากรวยมปรมาตร 7 ลกบาศกเซนตเมตร แลวทรงกระบอก

มปรมาตรเทาไร


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พระมดฐานสเหลยมจตรสตงอยบนปรซมซงฐาน

ของพระมดทบกบฐานของปรซมไดสนทพอด ซงปรซมมพนทฐาน

9 ตารางเมตร และสง 2 เมตร ถาปรมาตรของพระมดและปรซม

เทากน แลวพระมดจะมความสงตรงกบขอใด


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ขนมเคกกอนหนงท าเปน 2 ชน หนาเปนรปวงกลม

วางซอนกนแตละชนหนา 5 เซนตเมตร เสนผานศนยกลางของ

วงกลมชนบน และชนลางยาว 14 เซนตเมตร และ 28 เซนตเมตร

ตามล าดบ เคกกอนนมปรมาตรเทาใด


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โครงงาน เรอง ……………………………………………………….

1. ความเปนมาและความส าคญ

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2. จดประสงค

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3. ระยะเวลาด าเนนงาน ......... ชวโมง

4. สาระคณตศาสตรทเกยวของ

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5. วธการด าเนนงาน

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6. วสดอปกรณ

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7. วธการศกษา

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8. สรปผล

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9. ขอเสนอแนะ

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