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[잡담] 각기둥 부피 공식 일반화
별명은최대열자리이다 2020.03.22 00:38 조회 551
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

일반적으로 다각기둥의 부피는

V=S imes h \cdots (1) 로 구한다. (S는 밑넓이, h는 높이)

 

하지만 위의 공식만으로

모든 다각기둥의 부피를 구하는 것에는 좀 무리가 있다.

 

예를 들어보자.

 

위의 그림에서 (1)공식으로 부피를 구할 수 있을까?

구할 수는 있겠지만 쪼개고, 붙이고, 빼는 등 상당히 번거로울 것이다.

실제로 이렇게 구해보면

V=\frac{1}{3}(a+b+c)S

라는 식이 나올 것이다. 

여기서 눈여겨 볼 것은

이 삼각기둥의 부피가 윗면과 아랫면의 넓이와 관계가 없다는 점이다.

 

그러면 이렇게 추론할 수 있다.

"임의의 n각형을 두 밑면으로 하는 다각기둥을 생각하자.

이 다각기둥의 옆면의 길이가  x_{1}, x_{2}, \cdots , x_{n}이고,

다각기둥의 옆면에 모두 수직하며 각 변들이 모두 다각기둥의 옆면 위에 있는 n각형의 넓이를 S라 했을 때,

이 다각기둥의 부피는 

V=\frac{1}{n}\sum_{k=1}^{n}x_{k}S 로 표현된다."

--------------------------------------------------------------------------------------------------------------------------------------

위에서 제가 쓴 것이 맞나요?

삼각기둥 공식 구하다가 갑자기 생각난 거라서 충분한 고민은 하지 않은 채로 올렸네요;;

상당히 직관적인 거라 왠지 틀렸을 것 같네요.

 

부탁1. 마지막에 쓴 가설이 틀렸다면 간단한 반례를 보여주세요.

부탁2. 마지막에 쓴 가설이 맞았다면 증명도 해주세요.(너무 궁금합니다!)

부탁3. 이런 것에 대한 선행자료가 있으면 잊으시지 말고 꼭! 댓글로 남겨주세요. (url 등등)

 

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  •  
    giplee Lv.8 2020.03.22 00:51

    흠;; 

    상당히 배우신분 같네요. 

    이런문제는 함풀문에 올리시는게 어떨까요??

    댓글 작성하기 좋아요0 댓글수2
    •  
      giplee Lv.8 2020.03.22 00:53

      아... 비꼬는건 아니고 잡담보단 함풀문에 있으면 사람들이 더 많은 관심을 가질 것같아서....

      좋아요0
    •  
      별명은최대열자리이다 Lv.3 2020.03.22 00:53

      네!

      좋아요0
  •  
    리프 Lv.7 2020.03.22 01:18

    근데 이 방식으로 삼각기둥의 부피를 구하는 것을 어디서 알게 되셨나요? 

    댓글 작성하기 좋아요0 댓글수0
  •  
    GUN.007 Lv.11 2020.03.22 04:31

    잘 정리해서 결론나시면 매스포터 올리셔도 될 듯

    댓글 작성하기 좋아요0 댓글수0
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