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What is the slant height x of the square pyramid?

 Express your answer in radical form.

 May 14, 2018

In a 30 - 60 - 90, triangle, the side opposite of the 30 degree angle is half of the hypotenuse.


We don’t have to treat this problem as 3D geometry, we can just pay attention to the side facing us. Now it is just a 2D geometry problem. 


Using the rule I stated at the top of this post, the hypotenuse is twice the side opposite of the 30 degree angle, which is 3 meters long. 


Therefore, the hypotenuse is 3*2 = 6 


Since we know 2 sides of a right triangle, we can find the third. 


using the Pythagorean theorem, x^2 = 6^2 - 3^2 = 30 


x = √30


I hope this helped,



 May 14, 2018

Gavin, it appears as if you made a slight error with some basic arithmetic. It happens to all of us.


You correctly identified a special feature of the special 30-60-90 triangle and applied that knowledge correctly: The side length across from the 30° angle is half the length of the hypotenuse. 


You, then, decided to apply your knowledge of the famous Pythagorean Theorem, and this step is where the error is.


\(x^2=\textcolor{red}{27}\) Do you see the discrepancy? Of course, the rest of the algebra is fairly simple. 
\(x=\sqrt{27}\text{ or }-\sqrt{27}\) In the context of geometry, a negative side length is nonsensical, so we will reject this answer. The only thing left to do is rationalize. 
\(x=\sqrt{27}=3\sqrt{3}\text{ m}\) Do not forget the unit of measure!



I was able to identify this error quickly because of another feature of 30-60-90 triangles. The side length across from the 30° angle multiplied by \(\sqrt{3}\) equals the length of the angle across from the 60º angle. 

TheXSquaredFactor  May 14, 2018

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