# How Many Triangles 1

How many triangles can you count in this diagram?

Click the yellow button to find out.

## A Mathematics Lesson Starter Of The Day

Topics: Starter | Combinations | Problem Solving | Shape

• Judy, Chatsmore CHS
•
• This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles.
• Jo Melville, Aberdeen
•
• All my S1 - S4 classes enjoyed this activity at different levels. S3 have managed to write a formula for the number of triangles in an n-row triangle. Excellent!
• Guy Broster, Cumbria, UK
•
• A great all-purpose starter for any lesson. Got my little ones on task and ready for their lesson.
• Mr R. Ditta, Stokesley School, North Yorkshire
•
• It is a great starter that could be improved if the demonstration counted along with the shading of the different triangles.
• Year 8 and Miss D, Wade Deacon High school, Widnes
•
• Good starter, gets the mind working. Challenging for a class of year 8's, but fun.
• Mrs Myers' Year 8 gang!, Woldgate 'the best' College
•
• My Year 8 class got 27!!! It did take us a while though!
• Mr Sambhi, Solihull
•
• Excellent starter. I have used with every class I teach.
• Mr Cansdale, 7H3
•
• We really enjoyed this activity, we want more of these please!
• Miss Pankhurst, Norfolk
•
• Excellent starter! I have used it with nearly all of my groups and they all got focussed on the task in hand really quickly. I also used a similar one with a 5x5 square grid and asked "how many squares?". Highly recommended!
• Steven, University of South Carolina
•
• This web page is wrong there are actually 2 more than listed.
• Mrs. Happily, Norfold State Primary
•
• This was a very good activity. The students enjoyed it.
• Bill Loyd, Marysville, CA
•
• I agree with Steve, there are 29! There are 2 coming from the 2 vertical sides that were not counted.
• Matthew, Anchorage, AK
•
• 2 more than listed?
A great mental exercise.
• Enrique Martin, Tyler Texas
•
• Are you sure there are two more triangles? It might be a same counted triangle looked from another perspective.
• Mike C, Grand Forks, ND
•
• I still only count 27. Where exactly are the two missing triangles?
• Matt, Jacksonville, Florida
•
• Actually...there are in fact only 27. The two "vertical triangles" that you were counting...they were actually counted, but in a different perspective than it appeared.
• Rich, Florida
•
• Steve and Bill: How do you get 29?
• Julian, Brevard University
•
• There are 27. You've made the mistake of counting a "rotated" segment that when counted was viewed top to bottom. The math is true.
• Zach, Redding, Cali
•
• Multiply the number of lines by the number of external sides.
• Dylan, Shasta High, Redding, Ca
•
• I happen to have written a formula for this as well. I'd post it, but I'm afraid some of the work would get quite confusing if put into text. If anyone is interested or has one to compare, Email me at YuletidePirate@hotmail.com and I'll scan mine and send it to you.
P.S. Zach, I'm also in Redding and that's really weird that I should stumble across you. You go to Shasta?
• Danny, Freeport, NY
•
• It's the number of internal segments or lines (do not include the sides of the whole) cubed.
• Godwin Amarikwa-Obi, St. Mary's CofE School, Hendon, London UK
•
• I used this starter as part of group qiz and my students did enjoy it. They were really thrilled when the computer counted the triangles. It is even more fun when you tell them to count 1,2,3,... at the same time as the computer. Transum.org thank you again for having wonderful 'starter of the day'.
• Tom Morley, NC
•
• There are in fact 29. They most certainly did miss 2.
• Jared Huseman, Amarillo, TX
•
• There are only 27 triangles. Not only do I just know this from looking at it, but come on guys. We live in the 21 century. Search Google and you'll find nothing to support this having 29 triangles.
•
• There are 29. Math is not spelled with an "s".
•

• Robert And Emily Bernal, California
•
• An advertisement ask the very same question (for the triangle) but did not tell us the answer. Get this, it wanted our phone numbers. Instead we closed that page and typed in "how many triangles" into Google and found THIS page complete with the great visual answer which allowed us to count the very first time. Awesome work!
• Hazel, London UK
•
• Great puzzle. Maths is always spelt with an s in England - it is short for mathematics - Math is an americanism.
• Ron M, Cincinnati, Ohio
•
• Steve and Bill came up with 29, but if you used their same theory to count each base of a triangle, they should come up with 81 triangle bases. Which is 27 times the 3 sides for each triangle. I believe the "colour" "maths" diagam is correct with 27 triangles.
• Wil Mashburn, Oregon
•
• I thought there were 29 also... but then I looked again and the center "3x3" triangles were already counted just rotated once.
• Megan Gibson Year 5, Marthan Primary School
•
• My brain wasn't switched on because I only found 20.
• Miss Jabeen, Rochdale
•
• My kids enjoyed this activity very much it was a great start and quite so very challenging!
• Robbert Van Der Kruk, The Netherlands
•
• n=4 --> 27
n=5 --> 48
n=6 --> 78
In general: (n(n+1(2n+1) -n.mod(2))/8
Proef: email robbertvdkruk@live.nl.
• S1 Lockerbie Academy, Lockerbie Scotland
•
• We think there are 27.
• Lionel Morbelli, My Home
•
• 2n--->n(n+1)(4n+1)/2
2n+1->(n+1)(4n2+7n+2)/2
which gives:
1---->1
2---->5
3---->13
4---->27
5---->48
6---->78
7---->118
8---->170
Maybe less elegant that Robbert Van Der Kruk
email: takelion@yahoo.com.sg
I looked by recurrence at how many triangles are added when moving from k to k+1 "floors", where the "floors" are added below. Let S(n) be the number of triangle for n "floors":
S(n)=S(n-1)
+1+2+...+n <--- these are the triangles pointing up
+ (if n even) 1+2+...+n/2 +1+2+...+(n/2-1) <--- pointing down
+ (if n odd) 2(1+...+(n-1)/2) <--- pointing down
S(n)=S(n-1) + (n even) n(3n+2)/4 + (n odd) (n-1)(n+1)/4
where S(0)=0 and S(1)=1
By recurrence, knowing that 1+2+...+k=k(k+1)/2 and 1+4+9+...+k2=k(k+1)(2k+1)/6, one may find the formulas of the top of this post:
S(2n)=n(n+1)(4n+1)/2
S(2n+1)=(n+1)(4n2+7n+2)/2.
• Sanya, Laos
•
• Sn=Si+Sj, Si=1+3+...[n^(2)-n+1] ,i=1,2,3,.(n-1)
Sj=(n+1)^2 ,j=1,2,3,..(n-1)
form 4 line , then n=3
Sn=S3=Si(3)+Sj(3)
Sn=1+3+...+[n^(2)-n+1]+(n+1)^2 ,n=1,2,3,4..(n-1)
Sn=1+3+7+16 =27.
• Snowy White, UK
•
• There are in fact 28 triangles - everybody seems to have missed out the triangle in the word "triangles"! This adds a nice twist if you give a hint half way through that there are 28, since the quickest pupils will be racking their brains looking for the 28th while the others catch up and reach 27...
• Jeffrey, London
•
• There are two more triangles if the triangle is drawn in a square
Secondly, the method is not 81 but rather the number of lines touching the largest triangle multiplied by sides which in this case is 3x9 for three horizontal and 6 diagonal lines.
...and maths is correct, an abbreviation for mathematics as a plural noun.
• Rambabu Joshi, Nepal
•
• If n is the number of dots in the base then....
for even, number of triangles are:
n(2n^2-n-2)/8
if n=4 then
4(2*4^2-4-2)/8 = 13
for odd numbers of dot, number of triangles are:
(n-1)(n+1)((2n-1)/8)
if n=5 then
(5-1)(5+1)((2*5-1)/8) = 27
rambabu.joshi@yahoo.com.
• 10R4, Devon
•
• George in my class 10R4 thinks there are 36, we aren't sure but he has quite a convincing arguement.
• 7Y3, Wintringham Grimsby
•
• 7Y3 the majority think 25.
• Bahraoui, Morocco
•
• In 1979 I found this problem in a red french pocket book named JEUX "GAMES" with many others enigmas, the solution was in this form ( I can't remember exactly what it was):
Ckn = n!/k!.(n−k)! or something like this.
I was looking for this book on the market and in the net but I can not find it.
• Niamh, UK
•
• I really enjoyed this one. I think there are 27 triangles!
• 8a/Ma2, Little Ilford School
•
• We found 27 triangles, it was really easy!
• Ms Simpson, Priory School UK
•
• Fantastic - loved that you showed how to do this problem for the answer !
• Quikquik, Iowa
•
• It's an optical illusion: there are really just 7 triangles seen here. I figured that out by counting all of the angles (borders) and dividing by 3. there are 21 borders and, since a triangle is a grouping of three angles hence its name, when we divide 21 by 3 we get 7 so therefore there are 7 triangles.
• Gurpreet_IQ148, India
•
• There are in total 27 triangles.
Total no. Of Little triangles = 16
Total no. Of triangles with 4 smaller triangles inside them = 7
Total no. Of triangles with 9 smaller triangles inside them = 3
The largest outer triangle = 1
Therefore, total number of triangles = 16+7+3+1
= 27.
• Lionel Morbelli, My Home
•
• My apologies, there was a typo: for n odd, a 3 was missing that I have corrected below. I have also added a few more explanations.
2n--->n(n+1)(4n+1)/2
2n+1->(n+1)(4n2+7n+2)/2
which gives:
1---->1
2---->5
3---->13
4---->27
5---->48
6---->78
7---->118
8---->170
Maybe less elegant that Robbert Van Der Kruk
email: takelion@yahoo.com.sg
I looked by recurrence at how many triangles are added when moving from k to k+1 "floors", where the "floors" are added below. Let S(n) be the number of triangle for n "floors":
S(n)=S(n-1)
+1+2+...+n <--- these are the triangles pointing up
+ (if n even) 1+2+...+n/2 +1+2+...+(n/2-1) <--- pointing down
+ (if n odd) 2(1+...+(n-1)/2) <--- pointing down
S(n)=S(n-1) + (n even) n(3n+2)/4 + (n odd) (3n-1)(n+1)/4
where S(0)=0 and S(1)=1
From there:
S(2n)=S(2n-2) + 6n2 - n
S(2n+1)=S(2n-1) + 6n2 + 5n +1
Then, by recurrence or by summing up knowing that 1+2+...+k=k(k+1)/2 and 1+4+9+...+k2=k(k+1)(2k+1)/6, one may find the formulas of the top of this post:
S(2n)=n(n+1)(4n+1)/2
S(2n+1)=(n+1)(4n2+7n+2)/2.
• Edmond, Lake Worth, TX
•
• Be sure to accommodate those students who are smart enough to see past the foolishness of this little puzzle game and the lines provided, when they give you the correct answer which is actually this: "There are an infinite amount of triangles here."
Keep in mind that any three points in space form a triangle, and that points have no height, width or depth. Therefore, on any given plane or plane segment, such as this, there are infinite triangles.
• Malcolm, London
•
• Just a quick point, maths IS spelt with a 'S' because it is a shortening of mathematics....which ends with a S.
• Raman Unni, India
•
• There are 27 isosceles triangles. In fact I have devised a formula for any number for the side, like 5 here.
•

• Gene Ryan, Orlando, FL
•
• Snowy White has is right! There's no reason at all not to include the "triangle" in the problem's question from the information given. So, the correct answer is (Lionel Morbelli + 1) or 28.
•
• The answer is not 27 and it will never be 27. The correct answer is 42. I can prove it.
• Laura Pascoe, Macaulay Primary
•
• Wow my class found it really hard so did I really.
•
• Pretty good we liked learning and problem solving.
• Pimbilibom, U Tube
•
• Given the length of the side of﻿ the triangle n,
which in this case n = 4 and the formula equals 27.
• Mr Naylor, Springwell Community College, Chesterfield
•
• Most students initially said 24, 25 or 26. After doing it together we sorted it. Jake got 27 straight away.
• Mr Naylor, Springwell Community College, Chesterfield
•
• Most students put 24 but 1 student said 27. we all got it when we broke it down into smaller triangles.
• Petromir Panaiotov, Bulgaria
•
• Here is a general formula for every natural n (even and odd):
T(n)=(n(n+2)(2n+1)-1+(-1)^n)/16
The proof is very elegant, but is too long to post it here. Very nice problem for solving by students, by the way - it really gives you idea about the way of their mathematical thinking.
• Petromir Panayotov, Bulgaria
•
• Correction: Messed up in a hurry: the right formula is: T(n)= (2n(n+2)(2n+1)-1+(-1)^n)/16.
• Jackie Ford, Somerset
•
• This looks great - any chance of something similar for KS2?
• Mrs Wise Old Eagle Owl, Eagle Town
•
• Our class thoroughly enjoyed this challenging morning wake up.... they are bright eyed and bushy tailked ready for the week now!!
•
• You can easily find by visiting www.dillber.com/findtriangles.
• Ray Alfie Banta, Philippines
•
• ,.actually,there are a total of three formulas for that problem,the first formula is for the sum of normal triangles,the next two is for the sum of inverted triangles, depending on the number of rows if its an even number or an odd.,i formulated this formula about a week ago from this date. for the sum of normal triangles, the formula is the summation of quantity X over 2 multiplied by X plus 1 ((x/2)(X+1)) starting from limit of 1 to n number of layers. for the sum of inverted triangles for even number of layer, the formula is the summation of quantity 2X squared minus X (2X exp.2, - X) and the limit is from 1 to n/2. for odd number of layers, the formula is the summation of quantity 2X squared minus X (2X exp.2 - X) and the limit is from 1 to (n-1)/2 and added with the square of n minus one all over 4 ((n exp. 2, minus 1)/4). for the total triangles,just add the number of normal and inverted triangles:)i applied the concept of arithmetic progression in deciphering this formula.
• Review, Kuwait
•
• I see 27 but maybe I'm wrong... I've been looking at it for days and still see 27. In the middle there is a 3X3 shape that could be considered a triangle but if it is it does not comply to the general patterns.
• Kellett School Year 6 - MH, Hong Kong
•
• We found 27 in about 5 minutes!
Good fun, keep these openers coming!
• Nigel Fern,
•
• Hi, good resource. I use it a lot. Also, in the same diagram how many trapeziums are there?
• Year 9, WKGS
•
• We got 27. Some of us got it in one minute!
• Miss Hallsby, Uk
•
• Our class got 32!

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

Previous Day | This starter is for 23 September | Next Day

The important thing is not the answer but the method you used to arrive at the answer. Did you have a structured approach? Were you sure that you had the right answer?

There is a printable worksheet to go with this activity for Transum subscribers.

Can you make up a puzzle like this for others to solve?

## Extension Challenge

How many triangles are on the front of this hotel reception desk?

Christmas Present Ideas

It is often very difficult choosing Christmas presents for family and friends but so here are some seasonal, mathematics-related gifts chosen and recommended by Transum Mathematics.

## Equate board game

Here's a great board game that will give any family with school-aged kids hours of worthwhile fun. Christmas is a time for board games but this one will still be useful at any time of year. Games can be adapted to suit many levels of Mathematical ability.

For Maths tutors working with just one or small groups of pupils this game has proved to be an excellent activity for a tutorial. Deciding on the best moves can spark pertinent discussions about mathematical concepts.

Equate looks a bit like Scrabble--for aspiring mathematicians, that is. Designed by a real mathematician, it works like this: You put down tiles on a board and make points by correctly completing simple equations. Your nine tiles include both numbers and mathematical symbols; you can add on to previous plays both vertically and horizontally. more...

## How Not To Be Wrong

The maths we learn in school can seem like an abstract set of rules, laid down by the ancients and not to be questioned. In fact, Jordan Ellenberg shows us, maths touches on everything we do, and a little mathematical knowledge reveals the hidden structures that lie beneath the world's messy and chaotic surface. In How Not to be Wrong, Ellenberg explores the mathematician's method of analyzing life, from the everyday to the cosmic, showing us which numbers to defend, which ones to ignore, and when to change the equation entirely. Along the way, he explains calculus in a single page, describes Gödel's theorem using only one-syllable words, and reveals how early you actually need to get to the airport.

What more could the inquisitive adult want for Christmas? This book makes a cosy, interesting read in front of the fire on those cold winter evenings. more...

## Graphic Display Calculator

This handheld device and companion software are designed to generate opportunities for classroom exploration and to promote greater understanding of core concepts in the mathematics and science classroom. TI-Nspire technology has been developed through sound classroom research which shows that "linked multiple representation are crucial in development of conceptual understanding and it is feasible only through use of a technology such as TI-Nspire, which provides simultaneous, dynamically linked representations of graphs, equations, data, and verbal explanations, such that a change in one representation is immediately reflected in the others.

For the young people in your life it is a great investment. Bought as a Christmas present but useful for many years to come as the young person turns into an A-level candidate then works their way through university. more...

The analytics show that more and more people are accessing Transum Mathematics via an iPad as it is so portable and responsive. The iPad has so many other uses in addition to solving Transum's puzzles and challenges and it would make an excellent gift for anyone.

The redesigned Retina display is as stunning to look at as it is to touch. It all comes with iOS, the world's most advanced mobile operating system. iPad Pro. Everything you want modern computing to be. more...

## Aristotle's Number Puzzle

It’s a bit of a tradition to give puzzles as Christmas Gifts to nieces and nephews. This puzzle is ideal for the keen puzzle solver who would like a challenge that will continue over the festive period (at least!).

This number puzzle involves nineteen numbers arranged into a hexagon. The goal of the puzzle is to rearrange the numbers so each of the fifteen rows add up to 38. It comes in a wooden style with an antique, aged look.

Keep the Maths in Christmaths with this reasonably priced stocking filler. more...

## The Story Of Maths [DVD]

The films in this ambitious series offer clear, accessible explanations of important mathematical ideas but are also packed with engaging anecdotes, fascinating biographical details, and pivotal episodes in the lives of the great mathematicians. Engaging, enlightening and entertaining, the series gives viewers new and often surprising insights into the central importance of mathematics, establishing this discipline to be one of humanity s greatest cultural achievements. This DVD contains all four programmes from the BBC series.

Marcus du Sautoy's wonderful programmes make a perfect Christmas gift more...

## Christmas Maths

This book provides a wealth of fun activities with a Christmas theme. Each photocopiable worksheet is matched to the Numeracy Strategy and compatible with the Scottish 5-14 Guidelines. This series is designed for busy teachers in the late Autumn term who are desperate for materials that are relevant and interesting and that can be completed with minimun supervision.

All the activities are suitable for use by class teachers, supply teachers, SEN teachers and classroom assistants and cover topics such as 'How many partridges did the true love give all together?' and 'Filling a sleigh with presents by rolling a dice!'. Children will have lots of fun working through the Christmas Maths themes but also gain valuable skills along the way.

A great source of ideas and another reasonably priced stocking filler. more...

## A Compendium Of Mathematical Methods

How many different methods do you know to solve simultaneous equations? To multiply decimals? To find the nth term of a sequence?

A Compendium of Mathematical Methods brings together over one hundred different approaches from classrooms all over the world, giving curious mathematicians the opportunity to explore fascinating methods that they've never before encountered.

If you teach mathematics to any age group in any country, you are guaranteed to learn lots of new things from this delightful book. It will deepen your subject knowledge and enhance your teaching, whatever your existing level of expertise. It will inspire you to explore new approaches with your pupils and provide valuable guidance on explanations and misconceptions. more...

I had been tutoring the wonderful Betsy for five years. When the day came for our last ever session together before the end of her Year 13, I received this beautiful book as a gift of appreciation.

This a very readable book by Ben Orlin. I'm really enjoying the humour in the writing and the drawings are great.

Ben Orlin answers maths' three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that mathematics should belong to everyone. more...

Click the images above to see all the details of these gift ideas and to buy them online.

## Maths T-Shirts

Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items I have chosen and recommend to get you started. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.

Other shape counting starters:

 Teacher, do your students have access to computers?Do they have iPads or Laptops in Lessons? Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Transum.org/go/?Start=September23

Here is the URL which will take them to a related student activity.

Transum.org/go/?to=howmany

For Students:

For All: