## How many triangles #8

## 1.) How many triangles do you see this picture ?

### Answer with solution

**How?**

Let’s say the smallest triangle side is 1 cms, then

**Number of triangles with side 1cms**: 12(3+4+3+2 upward facing) + 12(downward facing)

**Number of triangles with side 2cms**: 6(3+2+1 upward facing) + 6(downward facing)

**Number of triangles with side 3cms**: 1(upward facing) + 1(downward facing)

### Thus, the total number of triangles in above puzzle are 38.

## How many triangles #9

## 1.) How many triangles do you see this picture ?

### Answer with solution

**How?**

Let’s say the smallest triangle side is 1 cms, then

**Number of triangles with side 1cms**: 10(1+2+3+4 upward facing) + 6(0+1+2+3 downward facing)

**Number of triangles with side 2cms**: 6(1+2+3 upward facing) + 1(downward facing)

**Number of triangles with side 3cms**: 3(upward facing)

**Number of triangles with side 4cms**: 1(upward facing)

### Thus, the total number of triangles in above puzzle are 27.

## How many triangles #7

## 1.) How many triangles are there in this hexagon?

### Answer with solution

Again it is good to use symmetry here, we can brake this image into six small triangles each formed by one of the side of the hexagon and each of the triangle is divided in half by a line. we will count the number of triangles formed by each part and by taking two or more such parts together.

**Number of triangles formed by one part:** 3 so total 3*6 = **18**.

**Number of triangles formed by taking two adjacent parts together:** 2*3(three such cases as for 3 cases it is zero) so total** 6**.

**Number of triangles formed by taking three adjacent triangles:** 2*6 ( for each case 2 triangles are possible) so total **12**

**Number of triangles formed by taking 4 or 5 parts together:** 0

**Number of triangles formed by taking all 6 parts together:** **1**

### Thus, total number of triangles possible in above puzzle are: 18+6+12+1= 37

## How many triangles #6

## 1.) How many triangles are there in this pentagon?

See Answer### There are a total of 35 triangles

**How?**

Lets count the number of triangles where side of the pentagon is one of the side of the triangle, for each one such side there are 6 triangles. but two of triangle has other side of the pentagon, those will be counted twice so we will count only one triangle for each side, so for five sides, total 5*5 = 25 such triangles.

Now, lets count the triangles where side of the pentagon is not the side of the triangle: there are 5+5 = 10 such triangles, this will be clear by the below diagram(5 green and 5 red triangles).

### Thus, there are a total of 25+ 10= 35 triangles

## How many triangles #5

## 1.) How many triangles are there in this triangle ?

See Answer### Answer: There are a total of 47 triangles

**How?**

Lets count…

By looking at it, first impression is … its very simple, But mind you! its not that easy !!!

So lets stick to the divide and conquer approach, we will divide this triangle into 3 equal triangles(triangle formed by connecting outer side to the center of the circle) and will count the number of triangles in each part and then by taking two or more parts together.

**Number of triangles in one part**: 4(non overlapping) + 3(overlapping) = **7 * 3 = 21**

**Number of triangles by taking two parts together**: 8 = **8 * 3 = 24**

**Number of triangles by taking all three parts together**: **2**

### Thus, total number of triangles in this puzzle are : 21+24+2= 47

## How many squares #4

## 1.) How many squares do you see in this picture ?

See Answer### Answer with solution

As all the sides are either horizontal or vertical we can say squares diagonals should be parallel. Now to see if the diagonals are parallel or not, i have drawn the diagonals where its not clear if it is a square or rectangle in the image below.

You can see i have labelled few sides as x and y to help you.

So,

Number of squares with side x/2 = 4

Number of squares with side x = 4

Number of squares with side y = 3

Number of squares with side x+y = 6

Number of squares with side x+y = 0

Number of squares with side x+2y = 1

Number of squares with side 2x+2y = 1

#### Thus, Total number of squares in above puzzle are 19.

## How many triangles #4

## 1.) How many triangles do you see in this image ?

See Answer### Answer with Solution

Looking at the symmetry, we can divide this triangle in 5 equal parts(square) and we will first count the number of triangles in each part and then number of triangles formed by combining 2 or more parts together.

**Number of triangles in one square**: 8(non overlapping triangles) + 8(overlapping triangles with middle lines a part of the triangle) =**16*5 = 80**

**Number of triangles by taking two squares(one middle and one other) at a time:** 4*4(there are four such combinations) = **16**

**Number of triangles by taking (only) three squares(one middle and two others) at a time:** 4(all squares should be a part of the triangle) *4(four such combinations) = **16**

**Number of triangles by taking 4 squares together: ** 2 *4(four such combinations) = **8**

### Thus total number of triangles = 80 + 16 + 16 + 8 = 120.

## How many triangles #3

## 1.) How many triangles do you see in this picture ?

See Answer### There are a total of 84 triangles

**How?**

Lets count…

Looking at the symmetry, we can divide this triangle in 4 equal parts(square) and we will first count the number of triangles in each part and then number of triangles formed by combining 2 or more parts together.

**Number of triangles in one square**: 8(non overlapping triangles) + 4(overlapping triangles with middle lines a part of the triangle) =**12**

**Number of triangles by taking two squares at a time:** 6(try a little harder, you will fidn these) * 4(there are four such combinations) = **24**

**Number of triangles by taking (only) three squares at a time:** 1(all squares should be a part of the triangle) *4(four such combinations) = **4**

**Number of triangles by taking all squares together = 8**

### Thus total number of triangles = 12*4 + 24 + 4 + 8 = 84.

## How many triangles #2

## 1.) How many triangles are there in this picture ?

See Answer### There are a total of 92 triangles

**How?**

Lets count…

Looking at the symmetry, we can divide this triangle in 4 equal parts(square) and we will first count the number of triangles in each part and then number of triangles formed by combining 2 or more parts together.

**Number of triangles in one square**: 8(non overlapping triangles) + 8(overlapping triangles with diagonal a part of one of the side) + 3(diagonal is perpendicular to one of the side of the triangle) =**19(You can get it correct after a little hard try)**

**Number of triangles by taking two squares at a time: **3(common side of square is perpendicular) * 4(there are four such combinations) = **12**

**Number of triangles by taking three squares at a time:** 1(all squares should be a part of the triangle) *4(four such combinations) = **4**

**Number of triangles by taking all squares together = 0**