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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$\begingroup$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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