Posers and Puzzles

Posers and Puzzles

  1. Standard memberHurricaneConway125
    SUPREMO OF SOMERSET
    SOMERSET
    Joined
    26 Oct '07
    Moves
    47416
    17 Mar '08 20:12
    I was set a question in my maths GCSE class about finding the volume of an octagonal prism, i know whow i would do this; area of the end face times the length but my problem is how on earth do you find the area of that end face. All i know is that its a regular octagon with all sides being 0.8 metres long.
    Any prods in the right direction would be very helpful thanks
  2. Joined
    20 Dec '05
    Moves
    37635
    17 Mar '08 20:23
    well octogon is made up of 8 isos triangles. inter angles of oct add to 1080 ((sides - 2)x180). therefor each int angle is 144. so base angles of each isos tringle are 72. using trig you can find ht of isos triangle (0.4tan72). Area of triangle half base x ht. Multiplied by 8 gives area of octogon. hope helped.
  3. Montgomery
    Joined
    17 Mar '06
    Moves
    7336
    17 Mar '08 20:26
    Originally posted by HurricaneConway125
    I was set a question in my maths GCSE class about finding the volume of an octagonal prism, i know whow i would do this; area of the end face times the length but my problem is how on earth do you find the area of that end face. All i know is that its a regular octagon with all sides being 0.8 metres long.
    Any prods in the right direction would be very helpful thanks
    the area of any regular polygon can be calculated by the formula

    1/2 * (Apothem) * (Perimeter of the polygon)

    The apothem is calculated by dropping a perpendicular bisector to one of the sides from the center of the polygon then you can complete the triangle by drawing another line from the center to a vertex of the same side that was bisected. Then you can use the sin laws to find your apothem and you are home free...
  4. Standard memberHurricaneConway125
    SUPREMO OF SOMERSET
    SOMERSET
    Joined
    26 Oct '07
    Moves
    47416
    17 Mar '08 20:29
    Originally posted by jim9
    well octogon is made up of 8 isos triangles. inter angles of oct add to 1080 ((sides - 2)x180). therefor each int angle is 144. so base angles of each isos tringle are 72. using trig you can find ht of isos triangle (0.4tan72). Area of triangle half base x ht. Multiplied by 8 gives area of octogon. hope helped.
    That looks interesting thankyou very much i will just have to try it for myself
  5. Joined
    20 Dec '05
    Moves
    37635
    17 Mar '08 20:30
    very nice but not sure if too advanced for GCSE course. dont know though.
  6. Standard memberHurricaneConway125
    SUPREMO OF SOMERSET
    SOMERSET
    Joined
    26 Oct '07
    Moves
    47416
    17 Mar '08 20:39
    Originally posted by jim9
    very nice but not sure if too advanced for GCSE course. dont know though.
    No its fine im predicted an A* at GCSE so my maths teacher was giving me something outside of the syllabus to practice for A-level, i can do trig and interior angles etc. just wasnt sure of the way to divide the thing up!
  7. Joined
    11 Nov '05
    Moves
    43938
    17 Mar '08 20:47
    Draw it on a paper and you can see that ia regular octagon can have a grid of 3 by 3. Five of them is quadratic, four of them is triangles. If you pair the triangles together you get two quadrats of a different size. Then add the areas toghether.
  8. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
    Moves
    52874
    17 Mar '08 22:45
    Originally posted by FabianFnas
    Draw it on a paper and you can see that ia regular octagon can have a grid of 3 by 3. Five of them is quadratic, four of them is triangles. If you pair the triangles together you get two quadrats of a different size. Then add the areas toghether.
    But he said the VOLUME not the area. So its a 3d thing, not a 2d.
  9. Standard memberAgerg
    The 'edit'or
    converging to it
    Joined
    21 Aug '06
    Moves
    11458
    17 Mar '08 23:083 edits
    Take a square of side length 1.6m
    from each corner mark a point .4m both sides of it, join these by a line and then lop em off your square and you have yourself an octagon with sides .8m each

    So essentially you have subtracted 2 squares of side length .4m from a square with 1.6m length, that will be the area of your face which you then multiply by perpendicular length.
  10. Joined
    28 Aug '07
    Moves
    3178
    18 Mar '08 02:23
    Originally posted by HurricaneConway125
    I was set a question in my maths GCSE class about finding the volume of an octagonal prism, i know whow i would do this; area of the end face times the length but my problem is how on earth do you find the area of that end face. All i know is that its a regular octagon with all sides being 0.8 metres long.
    Any prods in the right direction would be very helpful thanks
    When you get it, try to do with 12 sides, 30 sides, and "x" sides.
    Then try to find the area of the polygon with infinite sides (or a large enough number).
    What do you get?
  11. Joined
    26 Nov '07
    Moves
    1085
    18 Mar '08 10:20
    Originally posted by sonhouse
    But he said the VOLUME not the area. So its a 3d thing, not a 2d.
    It's an octagonal prism - find the area of the octagon then multiply by the length of the prism-bit. He was wondering how you found the area of the octagon.
  12. Standard memberwolfgang59
    Mr. Wolf
    at home
    Joined
    09 Jun '07
    Moves
    46027
    18 Mar '08 11:00
    Originally posted by Agerg
    Take a square of side length 1.6m
    from each corner mark a point .4m both sides of it, join these by a line and then lop em off your square and you have yourself an octagon with sides .8m each

    So essentially you have subtracted 2 squares of side length .4m from a square with 1.6m length, that will be the area of your face which you then multiply by perpendicular length.
    This is manifestly rubbish.

    Your octagon will not be regular.

    Four sides will have length 0.8

    Four sides will have length sqroot(0.32) (by Pythagoras)
  13. Standard memberAgerg
    The 'edit'or
    converging to it
    Joined
    21 Aug '06
    Moves
    11458
    18 Mar '08 12:58
    good point...I made a mistake
  14. Joined
    07 Sep '05
    Moves
    35068
    18 Mar '08 13:09
    Originally posted by serigado
    When you get it, try to do with 12 sides, 30 sides, and "x" sides.
    Then try to find the area of the polygon with infinite sides (or a large enough number).
    What do you get?
    n sides, distance from centre to a corner is r.

    Internal angle = 180 - 360/n
    Dividing up into triangles OAB (where O is the centre, A and B are vertices), you have an isosceles triangle with angles x = 90 - 180/n.

    The area of the triangle is r^2 sin x cos x, or (r^2 sin 2x)/2,
    = 0.5 r^2 sin (180 - 360/n)
    = 0.5 r^2 sin (360/n)

    Total area of the polygon is 0.5 n r^2 sin (360/n)

    Now take the limit as n -> infinity.
    sin (360/n) ~ 2pi/n, because sin x ~ x if x is measured in radians.

    Total area ~ pi r^2

    I.e. the area of a circle with this radius, which is what you'd expect.
  15. SubscriberAThousandYoung
    iViva la Hispanidad!
    tinyurl.com/y2c6j2t3
    Joined
    23 Aug '04
    Moves
    24791
    18 Mar '08 14:25
    Originally posted by sonhouse
    But he said the VOLUME not the area. So its a 3d thing, not a 2d.
    That's why you multiply that area by the height...
Back to Top