Selina Solutions Concise Mathematics Class 6 Chapter 28 Polygons Exercise 28(B) provides students with a clear idea of concepts which are important from the latest syllabus pattern. The solutions created by the experts are comprehensive, according to the current ICSE exam. Finding the number of sides in a regular polygon is the main concept discussed under this exercise. Frequent practice of solutions helps students to solve difficult problems within a short span of time. To ace the annual exam, students can access Selina Solutions Concise Mathematics Class 6 Chapter 28 Polygons Exercise 28(B) PDF, from the links provided here with a free download option.

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**Exercise 28(B)**

**1. Fill in the blanks:**

**In case of regular polygon, with**

Number of sides |
Each exterior angle |
Each interior angle |

(i) 6 |
â€¦â€¦â€¦â€¦. |
â€¦â€¦â€¦â€¦.. |

(ii) 8 |
â€¦â€¦â€¦â€¦. |
â€¦â€¦â€¦â€¦.. |

(iii) â€¦â€¦â€¦â€¦ |
36^{0} |
â€¦â€¦â€¦â€¦.. |

(iv) â€¦â€¦â€¦â€¦ |
20^{0} |
â€¦â€¦â€¦â€¦.. |

(v) â€¦â€¦â€¦â€¦ |
â€¦â€¦â€¦â€¦â€¦ |
135^{0} |

(vi) â€¦â€¦â€¦.. |
â€¦â€¦â€¦â€¦… |
165^{0} |

**Solution:**

Number of sides | Each exterior angle | Each interior angle |

(i) 6 | 60^{0} |
120^{0} |

(ii) 8 | 45^{0} |
135^{0} |

(iii) 10 | 36^{0} |
144^{0} |

(iv) 18 | 20^{0} |
160^{0} |

(v) 8 | 45^{0} |
135^{0} |

(vi) 24 | 15^{0} |
165^{0} |

(i) Each exterior angle = 360^{0} / 6

= 60^{0}

Each interior angle = 180^{0} – 60^{0}

= 120^{0}

(ii) Each exterior angle = 360^{0} / 8

= 45^{0}

Each interior angle = 180^{0} – 45^{0}

= 135^{0}

(iii) Given that, each exterior angle = 36^{0}

So, number of sides = 360^{0} / 36^{0}

= 10 sides

Each interior angle = 180^{0} â€“ 36^{0}

= 144^{0}

(iv) Given that, each exterior angle = 20^{0}

Hence, number of sides = 360^{0} / 20^{0}

= 18 sides

Each interior angle = 180^{0} – 20^{0}

= 160^{0}

(v) Given that, each interior angle = 135^{0}

Hence, exterior angle = 180^{0} – 135^{0}

= 45^{0}

Therefore, number of sides = 360^{0} / 45^{0}

= 8 sides

(vi) Given that, each interior angle = 165^{0}

Hence, exterior angle = 180^{0} – 165^{0}

= 15^{0}

Therefore, the number of sides = 360^{0} / 15^{0}

= 24 sides

**2. Find the number of sides in a regular polygon, if its each interior angle is:**

**(i) 160 ^{0}**

**(ii) 150 ^{0}**

**Solution:**

(i) 160^{0}

Let the number of sides of a regular polygon = n

Each interior angle = 60^{0}

The sum of interior angle of polygon can be calculated as,

(2n â€“ 4) Ã— 90^{0} = 160^{0} Ã— n

180^{0}n â€“ 360^{0} = 160^{0}n

180^{0}n â€“ 160^{0}n = 360^{0}

20^{0}n = 360^{0}

n = 360^{0} / 20^{0}

We get,

n = 18

Hence, the number of sides = 18

(ii) 150^{0}

Let us consider the number of sides of regular polygon be n

The sum of the interior angle of polygon = (2n â€“ 4) **Ã— **90^{0}

Each interior angle = 150^{0}

The sum of the interior angle of polygon can be calculated as,

(2n â€“ 4) Ã— 90^{0} = 150^{0} Ã— n

180^{0}n â€“ 360^{0} = 150^{0}n

180^{0}n â€“ 150^{0}n = 360^{0}

30^{0}n = 360^{0}

n = 360^{0} / 30^{0}

We get,

n = 12

Hence, the number of sides = 12

**3. Find number of sides in a regular polygon, if its each exterior angle is:**

**(i) 30 ^{0}**

**(ii) 36 ^{0}**

**Solution:**

(i) 30^{0}

Let us assume the number of sides be n

Each exterior angle = 30^{0}

Each exterior angle of polygon = 360^{0} / n

Now, we have

360^{0} / n = 30^{0}

n = 360^{0} / 30^{0}

We get,

n = 12

Hence, the number of sides = 12

(ii) 36^{0}

Let us assume the number of sides be n

Each exterior angle = 36^{0}

Each exterior angle of polygon = 360^{0} / n

Now, we have

360^{0} / n = 36^{0}

n = 360^{0} / 36^{0}

We get,

n = 10

Hence, the number of sides = 10

**4. Is it possible to have a regular polygon whose each interior angle is:**

**(i) 135 ^{0}**

**(ii) 155 ^{0}**

**Solution:**

(i) 135^{0}

Let the number of sides of regular polygon be n

The sum of the interior angle of polygon = (2n â€“ 4) Ã— 90^{0}

Each interior angle = 135^{0}

The sum of interior angle of polygon can be calculated as,

(2n â€“ 4) Ã— 90^{0} = 135^{0} Ã— n

180^{0}n â€“ 360^{0} = 135^{0}n

180^{0}n â€“ 135^{0}n = 360^{0}

45^{0}n = 360^{0}

n = 360^{0} / 45^{0}

We get,

n = 8

Since, it is a whole number

Therefore, it is possible to have a regular polygon whose interior angle is 135^{0}

(ii) 155^{0}

Let the number of sides of a regular polygon is n

The sum of the interior angle of polygon is (2n â€“ 4) **Ã— **90^{0}

Each interior angle = 155^{0}

The sum of the interior angle of polygon can be calculated as,

(2n â€“ 4) Ã— 90^{0} = 155^{0} Ã— n

180^{0}n â€“ 360^{0} = 155^{0}n

180^{0}n â€“ 155^{0}n = 360^{0}

25^{0}n = 360^{0}

n = 360^{0} / 25^{0}

We get,

n = 72 / 5

Since, it is not a whole number

Therefore, it is not possible to form a regular polygon whose interior angle is 155^{0}

**5. Is it possible to have a regular polygon whose each exterior angle is:**

**(i) 100 ^{0}**

**(ii) 36 ^{0}**

**Solution:**

(i) 100^{0}

Let the number of sides be n

Each exterior angle = 100^{0}

Each exterior angle of a polygon is calculated as,

360^{0} / n

So,

360^{0} / n = 100^{0}

n = 360^{0} / 100^{0}

We get,

n = 18 / 5

Since, it is not a whole number

Therefore, it is not possible to form a regular polygon

(ii) 36^{0}

Let us consider the number of sides be n

Each exterior angle = 36^{0}

Each exterior angle of polygon = 360^{0} / n

So,

360^{0} / n = 36^{0}

n = 360^{0} / 36^{0}

We get,

n = 10

Since, it is a whole number

Therefore, it is possible to form a regular polygon

**6. The ratio between the interior angle and the exterior angle of a regular polygon is 2: 1. Find:**

**(i) each exterior angle of this polygon.**

**(ii) number of sides in the polygon.**

**Solution:**

(i) Given

Interior angle: exterior angle = 2: 1

Let us assume the interior angle = 2x^{0} and the exterior angle = x^{0}

The sum of the interior angle and exterior angle is 180^{0}

Hence,

2x^{0} + x^{0} = 180^{0}

3x = 180^{0}

x = 180^{0} / 3

We get,

x = 60^{0}

Therefore, each exterior angle = 60^{0}

(ii) Let us assume the number of sides be n

Each exterior angle = 60^{0}

Each exterior angle of polygon = 360^{0} / n

So,

360 / n = 60^{0}

n = 360^{0} / 60^{0}

We get,

n = 6

Hence, the number of sides = 6

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