Step by step
The question
Question: a square pyramid has base 12 and height 5: find the angle between a slant edge and the base.
1 Find the right triangle
The line is the slant edge A V, from a corner of the base up to the top, V.
Its shadow on the base goes from A to M, the middle of the base, right under the top.
M V is the height, straight up.
So A M V is a right triangle, right angle at M.
2 Half the diagonal
First the whole diagonal of the square base, A C: 12 squared plus 12 squared is 288.
A M is half of A C.
Half a length gives a quarter of its square: 288 divided by 4 is 72.
So A M is the square root of 72.
That is about 8.5.
Careful!
M is the middle of the base.
From a corner to M is half of the diagonal, not half of a side.
3 The angle
θ is at A.
The side across from it goes up: 5.
The side next to it is the shadow, AM.
Tan is opposite over adjacent.
To get the angle back, we use tan to the minus 1 on a calculator: about 30.5°, to the nearest tenth.
4 Check
Let's check another way.
The whole line AV is the square root of 72 plus 25.
Sin θ is 5 over that.
Sin to the minus 1 gives 30.5° again.
So it is right!
Let's look back.
First: find the right triangle.
Then: half the diagonal.
Then: the angle.
Last: check.