How many triangles in this diagram?

Can you count them?

triangle diagram

Yeah, me neither.

The funny thing is, this was Grant's homework. Yes, that would be KINDERGARTEN homework.

How is a 5-year-old supposed to figure out how many triangles are in that figure?


If i was a kid, i would use my imagination and i probabily would say that there are so mutch triangles as stars. In the sky.

If the teacher get nervous, i would say: keep going and count yourself to teach me.

Posted by Giselle on November 11, 2008 at 11:19 AM CST #

Ha! I love it, Giselle. :)

It's funny, Grant and I tried to count the triangles together and I eventually said "Just write down 20." My husband said we had to put the right answer, otherwise the teacher would mark it wrong on his homework. I said, "Do you seriously think the teacher even knows how many triangles are on this thing?"

Posted by Melanie Gao on November 11, 2008 at 12:07 PM CST #

30 is the answer. Not 20.

Posted by SwitchBL8 on November 11, 2008 at 02:00 PM CST #

The answer is: a lot.

Posted by qyjohn on November 11, 2008 at 02:04 PM CST #

Yeah, yeah, SwitchBL8 where were you Sunday night when I could have used you? ;)

John, you are exactly right. You're welcome in my house any time! :)

Posted by Melanie Gao on November 11, 2008 at 02:11 PM CST #

I came up with 18. I think I just failed Kindergarten.

Posted by christin on November 11, 2008 at 02:57 PM CST #

Wow, in kindergarten really? At least the answer (10) is small so you can draw them all in different colors, but I can't really see the point of asking this to such young kids... It seems much better suited to university students to teach them about combinatorics (5 chose 3 equals 10).

Posted by Marc on November 12, 2008 at 12:28 PM CST #

Welcome to Asian schools! :D

Seriously, though i studied here, i can hardly comprehend the logic behind school & college syllabus.

Posted by Maddy on November 13, 2008 at 07:33 AM CST #

30 is not the answer either - i count 45 - not sure if I got them all

Posted by guest on November 13, 2008 at 06:45 PM CST #

It's so difficult to children like Grant.

A great deal of stresses on students in Chinese, especially during elementary school and middle school - everyone knows this. Now, the pressure spreads to kindergarten ...

Posted by Kelly on November 14, 2008 at 03:24 AM CST #

Marc the answer's gotta be higher than 10, I can count 15 and know there are more that I'm not counting. And I'm not counting them because I run out of patience. :), I wish you'd left your email address so I could follow up on how you got 45. I don't see that many.

Maddy, he's just in Kindergarten. How will I survive 12 more years of this? And I presume things will get harder though that's hard to imagine!

Christin, isn't it a good thing these skills aren't required to succeed in every day life?

Posted by Melanie Gao on November 14, 2008 at 03:25 AM CST #

Kelly, fortunately Grant is immune to the stress. He couldn't care less how many triangles are in that diagram. And he doesn't care what the letters of the alphabet look like or how to add and subtract. Maybe if he cared then he would do better at school!

Posted by Melanie Gao on November 14, 2008 at 03:30 AM CST #

Oups, you are right, I misinterpreted the question. I now count 35. There are 10 that have all their vertices on the outside, 15 that have 2 consecutive vertices outside and one inside, 5 that have 2 non-consecutive vertices on the outside and one inside (you can group those with the previous ones by saying that each of the 5 4-tuples of external vertices generates 4 triangles around one internal vertex), 5 that have only one vertex outside (and thus need all five external vertices for their construction), and none with only inside vertices. An other way to get this number is to count only the triangles that contain one given external vertex, giving each of them a weight of 1/(number of external vertices of this triangle). This gives 7, and 7\*5=35.

But I am not happy with not finding an elegant mathematical way to solve this, something that would generalize to larger polygons for instance.

Posted by Marc on November 14, 2008 at 12:48 PM CST #

Well I think the answer is 40. Here's how - there are 5 vertices each of which have 4 lines emanating from them. How many ways can you choose 2 lines from 4 = (4 choose 2) = 6 (the way the diagram is constructed you just need 2 - the third will always be there). Thus 5 \* 6 = 30. These account for big triangles.

The inscribed star has 5 small triangles and there are 5 more small ones "between" the star and the pentagon.

Moral of story: not a problem for Grant.

Posted by Shubho on November 15, 2008 at 05:07 AM CST #

Grant this is too tough for me dear :-). Let me write an algorithm and get the answer for you:-)

Posted by Deepak K Goyal on November 17, 2008 at 03:18 PM CST #

What I love about this comments section is that so far we have 15 comments from 10 different adults and we still don't know how many triangles are in that diagram. Gang, we are not off to a good start here. We've somehow got to sustain Grant through high school and college so please get out those old textbooks and start studying up. ;)

Posted by melanie gao on December 17, 2008 at 08:09 AM CST #

The answer is 35...

Posted by anonymous on January 04, 2009 at 07:21 PM CST #

i think it is 35 we r only in 6th grade so i would not trust us.

Posted by bob notrom on January 07, 2009 at 07:21 PM CST #

There are definitely 35 triangles, and I don't think there's a simple mathematical algorithm to shorten this.

Subho, I'm not sure how you can do this with the "4 choose 2" logic because when you multiply it by 5 vertices, there will be overlap (for example, a triangle that can be made with lines 3 and 4 on one vertice can also be made by lines 1 and 2 on the next). Also, more than one triangle can be made by two lines eminating from one vertice.

In any case, I'm glad I'm not in kindergarten! I'm also glad that the "simple math question" before you can post on this site doesn't require any advanced counting!!

Posted by Charlie on January 26, 2009 at 09:54 PM CST #

I'm pretty sure that there is 36 triangles in this drawing! maybe the home work is not to figure out the correct answer but just to encourage the kids minds to develop an idea of challenges! nothing wrong with a challenge is there? and i don't mean competition!

Posted by tivon creager on February 18, 2009 at 12:38 AM CST #

I like that idea very much. And yes, I think kids should learn to take on a challenge from a young age.

Plus the homework provided hours of entertainment for us adults here on my blog. :)

Posted by Melanie Gao on February 18, 2009 at 12:56 PM CST #

i counted 31. but im not really sure. im kinda blitzed. hahaha

Posted by Blazed on March 06, 2009 at 08:47 AM CST #

take that back. haha i counted 34.

Posted by Blazed on March 06, 2009 at 08:53 AM CST #


Posted by Poddle on March 08, 2009 at 12:23 AM CST #


Posted by Poddle on March 08, 2009 at 12:23 AM CST #

I think the easiest way to figure this is to draw it, and count as you draw. This simplifies and keeps you from counting triangles multiple times.

First draw a star, how many triangles in a star? 10, 5 little triangle (obvious) then 5 more from each of the inward dip points.(sorry, trying to make it easy for others to understand)

Now draw your first line from point to point. How many new triangles does this make? 4, 1 big(new line is bottom+star), 1 litle (drawn line, and 2 points), and 2 mediums (Point to inward dip off backside)

So, we have 10+4 so far, now every line you draw from point to point will add 5 triangle, the same 4, plus one goes back into the last area we looked.

Now, the last line you draw creates a 6th triangle going into the area of the first line we had drawn. So....

10+4+5+5+5+6 = 35

This is your answer, I realize that it may be quite late. (by a few months), but at least now you can sleep in peace. :)

Posted by Zhang Liao on March 10, 2009 at 05:09 PM CST #

the answer is 46 i think... im not sure... i gave up after 46, this is one of my problems for summer hw, im going into the 10th grade

Posted by 10th grader on August 06, 2009 at 10:37 PM CST #

lol never mind its 35 fer sure... sorry

Posted by 10th grader on August 06, 2009 at 10:41 PM CST #

Lol, I got 35, I had this question for advanced math and i came here first, after i figured it out i came back and awnted to tell you that the answer IS IN FACT... 35. I checked my answers so many times you can kill yourself with laughter..... kind of.

Posted by Atrain1020 on August 31, 2009 at 01:59 AM CST #

If we were a democracy then I think the 35s would win here.

I'm going to leave the discussion open though. I'm amazed at how many comments this entry gets even though it's been almost a year since the original post. Thanks to all of you for your participation in this! :)

Posted by Melanie Gao on August 31, 2009 at 03:35 AM CST #

Dear Melanie, Right now this seems the easiest way to write to you.
I loved the triangles for a 5 year old. Wow! I guess I was WAY low at about 15. I thought it would be a lot, but didn't stay with it long enough to be sure, or to color code it as one commented.

Will you be in Beijing between Dec. 11-13? I will be with Alissa from about Dec. 4 to Jan 7. We are planning for a 12 day stay in an
Ashram in Southern India. then time other places. I know you go to Alabama for Christmas, so i wonder about your total plans. I would love to see you, Xhin, Audrey and Grant. Plans are somewhat fexible, but I am excited to make my reservations in a day or two. Love-------Nelda

Posted by nelda Kilguss on September 24, 2009 at 09:15 PM CST #

i would not trust us so i think it is 35 we r only in 6th grade .

Posted by chan sll on September 01, 2010 at 07:36 AM CST #

The answer is in fact, 35. Someone did explain how they got that number, but I am still confused. FWIW, it's a question in my COLLEGE math book.

Posted by meghan on September 23, 2010 at 12:37 AM CST #

I love this post. The commentary is great. :-)

I stumbled across your blog because I'm looking at a few career opportunities in China. When I found this post, I couldn't resist responding.

The answer is indeed 35. There are 7 types of triangles and 5 different ways to place them in the diagram, so 7 \* 5 = 35.


Posted by Mack on March 06, 2011 at 10:32 PM CST #

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