[Math] Calculating angle between the face of a pyramid and the base

trigonometry

I have the question "Calculate the angle between the face BCE and the base ABCD in the pyramid pictured below, giving your answer to 1 decimal place."

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I assume the angle we are trying to find is B and so I draw a triangle DBE and use the cosine rule to get the answer of 74.6 degrees. However the answer should be 79 degrees.

Here is my working:

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Is the angle B the correct angle which I need to find or is it a different angle ? Because I know the method to use but the angle B is giving an incorrect value so it can't be this.

Best Answer

Let the midpoint of $AC$ be $M$ and the midpoint of $AD$ be $N$. Then $AC$ is perpendicular to $EM$. We can find the length of $NE$ using Pythagoras:

$NE = \sqrt{8^{2} - 1.5^{2}} = \frac{\sqrt{247}}{2}$

Now we need to find $\angle ENM$

Again, since $ENM$ forms a right angled triangle, $\angle ENM = arccos(\frac{1.5}{\frac{\sqrt{247}}{2}}) = 78.9955... \approx 79$degrees

I can include a diagram if this is hard to follow.

Edit: diagram included:

enter image description here

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