Step by step
The question
Question: a cuboid is 5 by 8 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.
Shine a light straight down from above.
The shadow of A G on the base is A C, right under it.
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 at A, between A G and its shadow A C.
We call it θ, a letter for an angle.
2 Base diagonal
A C is the diagonal of the bottom rectangle.
Pythagoras: 5 squared plus 8 squared is 89.
So A C is the square root of 89, about 9.4.
We keep √89 to stay exact.
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 straight up: C G, the height of the box, 6.
The side next to it is the shadow, AC.
We know both.
Tan is opposite over adjacent, so we use tan.
The tan⁻¹ button on the calculator turns the ratio back into the angle: about 32.5°, to the nearest tenth.
We put the exact √89 in, not 9.4, so the answer is not off.
4 Check
Let's check another way.
The whole line AG is the longest side: Pythagoras again.
AC squared is 89, 6 squared is 36.
So AG is the square root of 89 plus 36: √125.
Sin θ is 6 over √125.
Sin to the minus 1 gives 32.5° again.
So it is right!
Let's look back.
First: find the right triangle.
Then: base diagonal.
Then: the angle.
Last: check.