Triangle-ception

Geometry Level 2

How many triangles are there in the image above?

20 26 27 30

10 solutions

Calvin Lin Staff
May 1, 2015

There are 10 length 1 upright equilateral triangles (the blue ones),
6 length 2 upright equilateral triangles,
3 length 3 upright equilateral triangles,
1 length 4 upright equilateral triangle.

There are 6 length 1 inverted equilateral triangles (the red ones),
1 length 2 inverted equilateral triangle (green below).

Hence, there are at total of $10 + 6 + 3 + 1 + 6 + 1 = 27$ triangles in the image.

Did you miss the 7 inverted triangles, or the 1 big inverted triangle?

Moderator note:

How would you generalize this to a triangle of side length n?

Can it be generalised for eq. triangle with side "n"?

Archit Boobna - 6 years, 1 month ago

There is an actual formula for this, namely $\lfloor \dfrac{n(n + 2)(2n + 1)}{8} \rfloor$ , as found here .

A discussion on this problem, with a variety of approaches, can be found here .

Brian Charlesworth - 6 years, 1 month ago

Yes, this seems to work. I was trying n^2+((n-1)^2-(n-2))+((n-2)^2-(n-3))....+(1^2-0). This fitted everything till n=4, but for n=5, the answer should be 48 while my formula gives 49.

Ajan Sen Roy - 6 years, 1 month ago

i agree.. n=4

Peggy Pang - 6 years ago

what is n actually???

Kush Joshi - 5 years, 11 months ago

I missed the big inverted one. Rushing the answer, I forgot about Morley triangles. It is the same configuration of three sets of four parallel lines forming 27 triangles.

Maria Kozlowska - 6 years, 1 month ago

Illuminati confirmed! Calvin Lin!

Aryan Gaikwad - 6 years, 1 month ago

I almost missed the length 2 inverted triangle. To my eye having the triangles colored made this triangle less discernible when compared to a more 'standard' uncolored line drawing.

Brian Charlesworth - 6 years, 1 month ago

There are 1 + 3 + ... + (2n-1) = n^2 equilateral triangles of unit length of each side. (in the spec. case shown, n=4 but this formula holds in general.)

In the special case n=4: - there are 7 triangles with each side of length 2; - there are 3 triangles with each side of length 3; - and of course, 1 triangle w/ sides of length 4.

It would be nice to derive a general formula for the side lengths of 2, ..., up to n (where obviously there's exactly one triangle with each side of length n).

Predrag Tosic - 6 years, 1 month ago

One big inverted.

Hiew Soon - 6 years, 1 month ago

the big inverted one...

Sai Krishna Venky - 6 years, 1 month ago

The inverted one

Michael Long - 5 years, 8 months ago

I missed the one inverted big triangle made up of 4 smaller ones -_-

Sabrina Q - 6 years, 1 month ago

big inverted

Mahfuz AL Hasan Sakin - 5 years, 8 months ago

but i found 31 triangles in that triangle

Milan Varghese - 6 years, 1 month ago

maybe you counted some triangles twice!

Asha Gupta - 6 years, 1 month ago

Can you explain how you counted 31 triangles?

Calvin Lin Staff - 6 years, 1 month ago

I dont know how i can explain it to you. we can find may triangular disigns in it.

Milan Varghese - 6 years, 1 month ago

@Milan Varghese – Can you add a diagram? Which triangle of yours am I not counting?

Calvin Lin Staff - 6 years, 1 month ago

@Calvin Lin – how can i do that? sorry for the late replay. i don't visit the site very frequesntly

Milan Varghese - 6 years ago

Enrico Basuki
May 9, 2015

Nice picture. Thanks for including it!

Calvin Lin Staff - 6 years, 1 month ago

Since you're from the staff, I thought it would be a good idea to tell you this, even if the problem is bit old. I did choose option 27, and yet it shows that the correct answer is 30.

Thinking Sherlock - 5 years, 8 months ago

Daniel Schnoll
May 10, 2015

Count all individual triangles for a total of 16. Now group 4 triangles together to make one larger triangle. Count them up and you get 6 that are right side up and one in the center which is inverted, leading to a total of 7. 16+7 is 23. Now group 9 triangles together and count those triangles. You get 3, so now you're up to 26. Count the whole triangle as one more and you get 27

Good job!

Calvin Lin Staff - 6 years, 1 month ago

Mark White
May 9, 2015

I just counted them... is that not right or do I have to do all this crazy equation stuff? :/

I somehow multiplied them together in my head.... I have no idea how I do the math haha. Must have been luck.

Martin Manweiler - 6 years, 1 month ago

Simply counting is great!

Calvin Lin Staff - 6 years, 1 month ago

Brian Lee
May 9, 2015

1+3+5+7 = 16 single triangles 1+2+4 = 7 triangles from 4 single triangles 1+2 = 3 triangles from 10 single triangles And the big triangle as a whole

Therefore, 16+7+3+1 = 27 triangles.

That's right!

Calvin Lin Staff - 6 years, 1 month ago

Sandeep Dara
May 9, 2015

16 small triangles blue and red ones. 7 medium triangles(each one with4 small triangles). 3 big triangles(each one with 9 small ttriangles). 1 huge one (with all the small triangles). 16+7+3+1=27

Jun Villa
Jun 7, 2015

Illuminati confirmed! Epic maymays!

Catherine John
Jun 4, 2015

i counted it

How can we be certain that we counted all of them correctly?

Calvin Lin Staff - 6 years ago

Daniel Chris Lago
May 10, 2015

There are basically 27 triangles. Make bigger triangles from smaller triangles. Even the triangle that holds all the triangles. And look for some very cheeky triangles that are hiding. That's basically the answer. No Algebra Stuff and things. Just believe in your 2-D Shapes.

Miguel Rodriguez
May 9, 2015

I just kinda counted them and squinted my eyes so i could see past the repetitions that confuse me regularly.

Triangle-ception

10 solutions

Moderator note:

0 pending reports