Square based pyramid. Need help.

[dummy123]

Here is a problem I'm facing:

“A symmetrical pyramid stands on a square base of side 8cm. The slant height of the pyramid is 20 cm. Find the angle between the slant edge and the base, and the angle between a slant face and the base.”

So, I use pythag. 8²+8² = 128. Then, I find the square root of 128 and cut it in half. I get 5.657.
Then, I use inverse tan (20/5.657) and get 74.2

That's where I stopped because it's wrong and I don't understand why.
Please help!!

 

[Dr.Peterson]

How are you defining "slant height" and "slant edge"?

It sounds like you are thinking the former means the height of the pyramid itself. It does not.

If the latter means a slanted edge of the pyramid, then your work will be correct once you correct it by finding the height of the pyramid.

 

[Deleted member 4993]

Here is a problem I'm facing:

“A symmetrical pyramid stands on a square base of side 8cm. The slant height of the pyramid is 20 cm. Find the angle between the slant edge and the base, and the angle between a slant face and the base.”

So, I use pythag. 8²+8² = 128. Then, I find the square root of 128 and cut it in half. I get 5.657.
Then, I use inverse tan (20/5.657) and get 74.2

That's where I stopped because it's wrong and I don't understand why.
Please help!!

You are supposed to calculate two angles.

Which one is 74.2?

 

[pka]

“A symmetrical pyramid stands on a square base of side 8cm. The slant height of the pyramid is 20 cm. Find the angle between the slant edge and the base, and the angle between a slant face and the base.”

Here is what is standard for a Square based pyramid.
\(\displaystyle \ell\) is the length of a side of the square base and \(\displaystyle h\) is the perpendicular height .
Then the angle between the base and a side is \(\displaystyle \arctan\left(\dfrac{2h}{\ell}\right)\)

 

[dummy123]

How are you defining "slant height" and "slant edge"?

It sounds like you are thinking the former means the height of the pyramid itself. It does not.

If the latter means a slanted edge of the pyramid, then your work will be correct once you correct it by finding the height of the pyramid.

Honestly, I'm not 100% sure what the question is asking me. I've just been calculating angles to see if I can understand and I can't get it.

and you're right. I have been reading the problem wrong. I thought 20 was the height, and it's not.

You are supposed to calculate two angles.

Which one is 74.2?

The answer at the back of the book says, 73.57, 78.22
I don't know which is which. I'm just looking for angles and am trying to figure it out, but I can't get either of those two numbers in any of my calculations.

Here is what is standard for a Square based pyramid.
\(\displaystyle \ell\) is the length of a side of the square base and \(\displaystyle h\) is the perpendicular height .
Then the angle between the base and a side is \(\displaystyle \arctan\left(\dfrac{2h}{\ell}\right)\)

The book I'm reading did not give that equation. Let me work with it right now and see if I can get it.

 

[Dr.Peterson]

“A symmetrical pyramid stands on a square base of side 8cm. The slant height of the pyramid is 20 cm. Find the angle between the slant edge and the base, and the angle between a slant face and the base.”

So, I use pythag. 8²+8² = 128. Then, I find the square root of 128 and cut it in half. I get 5.657.
Then, I use inverse tan (20/5.657) and get 74.2

First, I hope you know the terms. The slant height is the altitude of the triangle forming a side of the pyramid; from that you can use the Pythagorean theorem to find the height of the pyramid.

Then, for the first question (angle between edge and base), do what you did, but using the correct height.

Finally, for the second question (angle between slanted face and base, that will be the angle between the altitude of the face and the base; for that you will be using the same triangle you will have used to find the height of the pyramid. You will likely be doing the same calculation pka gave you, but you shouldn't bother looking up that formula -- it's far better to see why you do that. (And I would actually use an arcsin, myself.)

Now, I don't get the exact angles you say the book gives, but close. I don't know what might cause the difference.

 

[pka]

Here is what is standard for a Square based pyramid.
\(\displaystyle \ell\) is the length of a side of the square base and \(\displaystyle h\) is the perpendicular height .
Then the angle between the base and a side is \(\displaystyle \arctan\left(\dfrac{2h}{\ell}\right)\)

The answer at the back of the book says, 73.57, 78.22
I don't know which is which. I'm just looking for angles and am trying to figure it out, but I can't get either of those two numbers in any of my calculations. The book I'm reading did not give that equation. Let me work with it right now and see if I can get it.

The square has diagonals of length \(\displaystyle 8\sqrt 2\). The diagonals bisect each other.
The perpendicular height of the pyramid is \(\displaystyle \sqrt{(20)^2-(4\sqrt{2})^2}=4\sqrt{23}\) SEE HERE
Thus applying the formula \(\displaystyle \theta=\arctan\left(\frac{2(4\sqrt{23}}{8}\right)=78.22^o\) SEE HERE

 

[Dr.Peterson]

The square has diagonals of length \(\displaystyle 8\sqrt 2\). The diagonals bisect each other.
The perpendicular height of the pyramid is \(\displaystyle \sqrt{(20)^2-(4\sqrt{2})^2}=4\sqrt{23}\) SEE HERE
Thus applying the formula \(\displaystyle \theta=\arctan\left(\frac{2(4\sqrt{23}}{8}\right)=78.22^o\) SEE HERE

That's true if the edge is 20 cm. Taking it as the slant height, as stated, I get a different value for the height, and therefore for the angle.

This is, in fact, why I initially asked about the OP's definition of terms. I'm using this.

Here's a page that defines both "slant height" and "slant edge", and agrees with my answers:

Pyramid Calculator

 

[dummy123]

Thank you! That helped me. I understand what I did wrong now.