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I am working on volumes of prisms, using $\pi$, diameter, and radius; however, I haven't done any problems with cross-sectional areas before... Also, I'm only 14 and am finding this really hard!

It is my first piece of math homework on this subject as I have been in the hospital and missed lessons> How do I do it?

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If $S$ is a solid whose cross-sectional area is a constant $A$ and whose length perpendicular to these cross-sections is $\ell$, the volume of $S$ is simply $A\ell$.

Added: This is a generalization of a couple of formulas that you already know. Suppose that you have a right circular cylinder of radius $r$ and length $\ell$; then its volume is $2\pi r\ell$. But each cross-section perpendicular to the axis of the cylinder is a disk of radius $r$, so each cross-section has area $A=2\pi r$, and the formula $V=2\pi r\ell$ can be rewritten $V=A\ell$, where $A$ is the area of each cross-section.

Similarly, the volume of a rectangular solid of height $h$, width $w$, and length $\ell$ is $hw\ell$. If you take cross-sections perpendicular to the length, each is a rectangle of height $h$ and width $w$, so each has area $A=hw$. Again, the volume formula $V=hw\ell$ can be rewritten as $V=A\ell$, where $A=hw$ is the cross-sectional area perpendicular to the length of the block.

It turns out that these two familiar volume formulas generalize: as long as all of the cross-sections have the same area $A$, the volume is given by $V=A\ell$, where $\ell$ is the length of the object in the direction perpendicular to the cross-sections.

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Hi, thank you for answering but would you mind explaining a bit more simply? Thank you – user58405 Jan 15 '13 at 21:22
@user58405: You have a prism $250$ cm long. No matter where you slice it perpendicular to its long axis, the cross-section has area $40\text{ cm}^2$. Thus, in your problem $A=40$ and $\ell=250$. Can you go from there? – Brian M. Scott Jan 15 '13 at 21:24

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