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Math Help - Surface area of a cone

  1. #1
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    Surface area of a cone

    Find the lateral area of a cone. Why is it

    Pi * r * l

    The book I am reading says it is 1/2 the circumference of the base times the slant height. I cannot picture how or why this would get you the lateral area. Can someone help me visualize and understand this?

    Looking at the net of the lateral area it looks like 1/3 of a whole circle where the slant height represents that circles radius. So (1/3)(pi * l ^2) would make sense to me.
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  2. #2
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    Re: Surface area of a cone

    Quote Originally Posted by Jman115 View Post
    Find the lateral area of a cone. Why is it

    Pi * r * l

    The book I am reading says it is 1/2 the circumference of the base times the slant height. I cannot picture how or why this would get you the lateral area. Can someone help me visualize and understand this?

    Looking at the net of the lateral area it looks like 1/3 of a whole circle where the slant height represents that circles radius. So (1/3)(pi * l ^2) would make sense to me.
    note the diagram of a cone sliced along its slant height, L , and laid flat.

    the area of a circle sector is A = \frac{\theta}{2\pi} \cdot \pi R^2 = \frac{\theta}{2} \cdot R^2

    for the major sector shown in the diagram, \theta = \frac{\2\pi r}{L}

    substitute for \theta in the sector area formula ...

    A = \frac{2 \pi r}{2L} \cdot L^2 = \pi r L
    Attached Thumbnails Attached Thumbnails Surface area of a cone-cone.jpg  
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    Re: Surface area of a cone

    I appreciate the response but I am not fully understanding it.

    I recall an explanation a long time ago that made sense to me that used ratios. I believe it had something to do with the ratio of r:l I am not quite sure.

    I think part of my issue is I am looking to understand why pi r l works for area of the lateral surface and not having any experience with A of a circle sector, now my mind is wondering where that formula came from if you follow me.
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    Re: Surface area of a cone

    I understand that the area of a circle sector A = \frac{\theta}{2\pi} \cdot \pi R^2

    is taking the area of the whole circle and by multiplying it by \frac{\theta}{2\pi} and gets you the portion of the circle you have. But I am not sure where  \frac{\theta}{2\pi} comes from now.

    I assume that  \frac{\theta}{2\pi} represents the fraction portion of the circle. How do we know that  \frac{\theta} represents 2pi(r) over L aka the circumference of the circle sector divided by it's radius.




    EDIT:

    Just remembered a piece of the ratio thing I was talking about earlier. I was told something about the proportion of the missing area is equal to the proportion of the missing circumference. Ill see if i can remember in full what I was supposed to do.
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  5. #5
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    Re: Surface area of a cone

    \theta is an angle in radians

    there are 2\pi radians in circle

    I recommend you research the topic further.
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  6. #6
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    Re: Surface area of a cone

    Just remembered what I heard before, tell me if this makes sense. Seems to make sense to me.

    Since the proportion of the original circle's area that is left in the circle sector equal to the proportion of the circle sector's circumference (for example, if i have half the area, i have half the circumference as well)

    I can do the following:

    A = \frac{2\pi r}{2\pi L} \cdot  \frac{\pi L^2}{1} =  \pi r L


    Sorry if you viewed that mid edit, i was playing with what I assume is the LATEX stuff i heard about. When I quoted your post it gave me the code you used. Much more convenient
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