Step by step
The question
Question: a cuboid is 9 by 12 by 6: find the angle between the diagonal A G and the base.
1 Find the right triangle
The line is the diagonal A G, from a bottom corner to the opposite top corner.
Its shadow on the base is A C: right under it, along the floor.
C G goes straight up from the base.
So A C G is a right triangle, with the right angle at C.
The angle between the line and the base is θ, at A, between A G and its shadow A C.
2 Base diagonal
A C is the diagonal of the bottom rectangle.
Pythagoras: 9 squared plus 12 squared is 225.
So A C is 15.
Careful!
Not the angle with the bottom edge.
The angle with the base is the angle with the shadow on the base.
3 The angle
θ is at A.
The side across from it goes up: 6.
The side next to it is the shadow, AC.
Tan is opposite over adjacent.
To get the angle back, we use tan to the minus 1 on a calculator: about 21.8°, to the nearest tenth.
4 Check
Let's check another way.
The whole line AG is the square root of 225 plus 36.
Sin θ is 6 over that.
Sin to the minus 1 gives 21.8° again.
So it is right!
Let's look back.
First: find the right triangle.
Then: base diagonal.
Then: the angle.
Last: check.