# Diagonals on the Face of a Cube. What’s the Angle?

It might be that if you are in a particularly mathematical frame of mind you are trying to use a three dimensional co-ordinate system and a matrix transform to project the 45° angles in the corners in to a different plane. By all means. But it’s not necessary, essentially this is a 2D problem, it’s just about finding the right two dimensions to work in. Let’s add another line to the diagram and see how we go.

Any clearer? The dotted line completes a triangle. What’s more since all it’s edges are made up of diagonals from the squares, they are all the same length, it is an equilateral triangle. Like all equilateral triangles it’s internal angles are each 60°

A problem I have found with these 3D nets is not everyone sees them the same, it’s like an optical illusion. So at some point over the next few images it should become clear what I’m driving at…

And finally, just because we can.

## Vector Maths

As I’ve said before it’s my intention that anyone using this site does not need formal maths training. But where there is an obvious maths solution I will try to include it.

The above diagram shows a standard right handed co-ordinate system with the label on the z axis hidden. We will calculate the angle of the two red lines that meet at the origin. The first being defined as (1,1,0) and the second (1,0,1)

The angle between to vectors is given by:

*cos θ = (u•v) / (|u||v|)
cos θ = (1•1 + 1•0 + 0•1) / (√2√2)
cos θ = 1/2
⇒ θ = π/3 or 60°*

### Your Answer

## More puzzles to try-

### What is the logic behind these ?

3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »### Defective stack of coins puzzle

There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »### Which clock works best?

Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »### (Advanced) Cheryl’s Birthday Puzzle

Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers ...Read More »### Five greedy pirates and gold coin distribution Puzzle

Five puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »### Magical flowers!!

A devotee goes to three temples, temple1, temple2 and temple3 one after the other. In front of each temple, there ...Read More »### Tuesday, Thursday what are other two days staring with T?

Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »### How could only 3 apples left

Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »### How Many Eggs ?

A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over ...Read More »### HARD MATHS – How much faster is one train from other Puzzle

Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination ...Read More »### Most Analytical GOOGLE INTERVIEW Question Revealed

Let it be simple and as direct as possible. Interviewer : Tell me how much time (in days) and money would ...Read More »### Lateral thinking sequence Puzzle

Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »### How did he know?

A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »### Pizza Cost Math Brain Teaser

Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for $10. It turned out ...Read More »### Which letter replaces the question mark

Which letter replaces the question markRead More »### Which room is safest puzzle

A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »### Richie’s Number System

Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »### Srabon wanted to pass

The result of math class test came out. Fariha’s mark was an even number. Srabon got a prime!! Nabila got ...Read More »### Become Normal!!

Robi is a very serious student. On the first day of this year his seriousness for study was 1 hour. ...Read More »### Sakib Knows The Number!

Ragib: I got digits of a 2 digit number Sakib: Is it an odd? Ragib: Yes. Moreover, the sum of ...Read More »