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Math Help - Volume question

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    Volume question

    I have to find the volume of a pyramid with height h and base an equilateral triangle with side a (a tetrahedron). I know that volume = integral of area, but I'm having trouble coming up with a formula for the area in terms of one variable. Help please!
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    Quote Originally Posted by experiment00005 View Post
    I have to find the volume of a pyramid with height h and base an equilateral triangle with side a (a tetrahedron). I know that volume = integral of area, but I'm having trouble coming up with a formula for the area in terms of one variable. Help please!
    Draw a horizontal line on each face of the pyramid at height y. Since the sides are lines, the length of those lines is a linear function of y: L= my+ b. When y= 0, that length is a so L= m(0)+ b= a or b= a. When y= h, that length is 0 so L= m(h)+ a= 0 and m= -\frac{a}{h}.

    That is, slicing the pyramind horizontally at height y gives a cross section of an equilateral triangle with side length b- \frac{ay}{h}.

    What is the area of such a triangle, A(y)? Imagining a thin layer, of height dy, the volume of such a layer is A(y)dy. Integrate that from y= 0 to y= h to find the volume of the pyramid.
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