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You could present the concept that, ignoring scale, a right triangle can be defined entirely by a single number: its smallest interior angle. Call this the triangle's "describing number." If you want to dramatize this, you could specify some angle like 30 degrees and ask students to cut triangles out of paper that have this describing number. Comparing all ...

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I've always seen it referred to as the "Right Triangle Altitude Theorem". It may take a bit of practice, but I start by declaring $\angle M$ and $\angle K$ complementary. Then I draw a mark to indicate $\angle NLM$ and show that it is also complementary to $\angle M$. Thus $\angle NLM$ is congruent to $\angle K$ and so on. I do this visually by getting 2 ...

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Another mnemonic attempt (despite not having much sense): The cones in my pie are squared, because their age is over three. Getting together, we have that the volume of a cone is $\pi r^2\cdot \dfrac h3$.

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I'm not sure if I would class this as a mnemonic, except in the most basic definition of "a way to remember": A cone is basically a three-dimensional triangle. Triangles have an area of $\frac12bh$; but triangles are two-dimensional, so it makes sense that a three-dimensional version would use $3$ instead of $2$, yielding $\frac13bh$. The "base" of a cone ...

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The comments give great suggestions for how to understand and remember the formula, but since you asked for a mnemonic specifically, how's this? Henry won third in the creative desserts contest because his pie are squared. $h\cdot (1/3) \pi r^2$ Edit: someone "fixed" this by changing "are" to "is." The mnemonic falls apart if you do that, so please leave ...

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Let the two original lines be $AB$, $AC$, with the angle to recreate as $\angle BAC$. Draw a line parallel to $AC$ that passes through $AB$ and is not $AC$. Call this line $DE$. Where $DE$ and $AB$ intersect ($F$), drop the perpendicular of $DE$ to $AC$, and call this point $G$. $AG$ and $GF$ can be used to recreate the angle.

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I have found this site to be an invaluable online resource for commentary. Long story short, nobody in history ever wrote an authoritative second edition of Elements, so we're still "stuck" with Euclid's original thoughts. Undeniably a work of genius that deserves its place in history, but it does have a few rough patches. Here are David Joyce's thoughts ...

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