Cuboid 3 by 4 by 12, angle between the diagonal and the base Easy

The question

Question: a cuboid is 3 by 4 by 12: 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: 3 squared plus 4 squared is 25.

So A C is 5.

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: 12.

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 67.4°, to the nearest tenth.

4 Check

Let's check another way.

The whole line AG is the square root of 25 plus 144.

Sin θ is 12 over that.

Sin to the minus 1 gives 67.4° again.

So it is right!

Let's look back.

First: find the right triangle.

Then: base diagonal.

Then: the angle.

Last: check.

▶ Watch it solved

Cuboid 2 by 3 by 6, angle between the diagonal and the base Medium

The question

Question: a cuboid is 2 by 3 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: 2 squared plus 3 squared is 13.

So A C is the square root of 13, about 3.6.

We keep √13 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 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 59.0°, to the nearest tenth.

4 Check

Let's check another way.

The whole line AG is the square root of 13 plus 36.

Sin θ is 6 over that.

Sin to the minus 1 gives 59.0° again.

So it is right!

Let's look back.

First: find the right triangle.

Then: base diagonal.

Then: the angle.

Last: check.

▶ Watch it solved

Square pyramid base 6 height 4, angle between a slant edge and the… Harder

The question

Question: a square pyramid has base 6 and height 4: 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: 6 squared plus 6 squared is 72.

A M is half of A C.

Half a length gives a quarter of its square: 72 divided by 4 is 18.

So A M is the square root of 18.

That is about 4.2.

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: 4.

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 43.3°, to the nearest tenth.

4 Check

Let's check another way.

The whole line AV is the square root of 18 plus 16.

Sin θ is 4 over that.

Sin to the minus 1 gives 43.3° again.

So it is right!

Let's look back.

First: find the right triangle.

Then: half the diagonal.

Then: the angle.

Last: check.

▶ Watch it solved