Perimeter of Polygon	Formula
Perimeter of Triangle	Sum of Three Sides
Perimeter of Parallelogram	2(Sum of Adjacent Sides)
Perimeter of Rectangle	2(length + breadth)
Perimeter of Square	4 × Side
Perimeter of Rhombus	4 × Side
Perimeter of Trapezium	Sum of Parallel Sides + Sum of Non-Parallel Sides
Perimeter of Pentagon	5 × Side or (S1+S2+S3+S4+S5)

Diagonals of Polygon

Formula for number of diagonal of polygon is given as:

Number of Diagonals in a Polygon = n(n – 3)/2

Solved Practice Problems on Polygons

Problem 1: A convex polygon has 20 diagonals. Find the number of sides of that polygon.

Solution:

No. Of diagonals of n sides = n(n – 3)/2

Now,

20 = n(n – 3)/2

40 = n(n – 3)

40 = 8(8 – 3)

Then, n = 8.

Therefore, the number of sides of that polygon = 8.

Problem 2: Find the number of sides of a polygon whose sum of all interior angle is 2160°.

Solution:

Given, sum of all interior angle = 2160°

We know, sum of all interior angles of polygon = (n – 2)×180°.

(Where n = number of sides of the polygon)

Now,

(n – 2) × 180° = 2160°

(n – 2) = 12

n = 14

Therefore, the number of sides are =14

Problem 3: If the external angle of a polygon is 45°, then find the number of diagonal in this polygon.

Solution:

Given angle = 45°

The formula for external angle = (360°)/n

Number of diagonal = (n2 – 3n)/2

(Where n = no. of sides)

Now,

45° = 360°/n

n = 8

Again, putting the value of n

(n2 – 3n)/2

=(64 – 24)/2

=40/2

=20

Therefore, the number of diagonals of the polygon is 20.

Problem 4: Find the measure of each angle of a regular nonagon.

Solution:

Number of sides of a nonagon = 9

Formula for each interior angle

{ (2n – 4) × 90}/n

={(2 × 9 – 4) × 90}/9

={14 × 90}/9

=1260/9

=140°

Therefore, the measure of each angle of a nonagon = 140°

Problem 5: If the area of a trapezium is 200 cm² and the measures of parallel sides are 6 cm and 4 cm, respectively, then find the height of the trapezium.

Solution:

Let h is the height or distance between the parallel sides of a trapezium.

Given, the parallel sides: 6 cm and 4 cm, sum = 10cm

We know, the area of a trapezium = (½) × Sum of parallel sides × Distance between parallel sides or height.

(½) × (6 + 4) × h = 200 cm² (given)

(½) × 10 × h = 200

h = (200 × 2)/10

h = 40

Therefore, the height of the trapezium is 40 cm.

Problem 6: Find each interior angle of a regular polygon of 15 sides.

Solution:

Given sides = 15

Formula for the sum of interior angles = (n – 2)×180°

Now,

(n – 2)×180° = 13 × 180° = 2340°

Now, each interior angle = 2340° ÷ 15 = 156°.

Problem 7: A parallelogram has 25 cm of length and width is 30 cm. Find out the perimeter of the given parallelogram.

Solution:

Given, length = 25cm and width = 30cm.

Perimeter = 2×(length + width)

= 2 × (25 + 30)

= 2×55 = 110cm

Therefore, the perimeter of the parallelogram = 110cm.

Problem 8: Find the area of a right angle triangle with base = 9 cm and hypotenuse = 15 cm.

Solution:

Given base = 6cm and hypotenuse = 15cm.

The triangle is a right angle triangle, so we have to use Pythagoras Theorem to find its height.

∴ height = √(15² – 9² ) = 12cm.

Now, the area of the triangle = 1/2 × b × h = 1/2 × 9 × 12 = 54 cm² .

Problem 9: Find out the area of a rhombus having diagonals equal to 5 cm and 6 cm.

Solution:

Given, diagonal1 = 5cm and diagonal2 = 6cm

We know that the area of rhombus = 1/2 × diagonal1 × diagonal2

= 1/2 × 5 × 6

= 15cm²

∴ The area of the rhombus = 15cm² .

Problem 10: If the sides of a pentagon are 3cm, 3.5cm, 4cm, 5cm and 6cm , then find out the perimeter of the pentagon.

Solution:

We know that the perimeter of a pentagon = (S₁ + S₂ + S₃ + S₄ + S₅)

Now, putting the values of sides in the formula, we get

Perimeter of Pentagon = (3 + 3.5 + 4 + 5+ 6) = 21.5cm

Therefore, the perimeter of the pentagon = 21.5 cm

Practice Questions on Polygons

Q1: One angle of a hexagon is 90° and all the remaining five angles are equal. Then find out the measure of the other angle.

Q2: Find the sum of all the interior angles having 16 sides .

Q3: The perimeter of a rectangle is 120 cm, and the width is 10 cm. What is the length of this rectangle? .

Q4: The sum of all the interior angles of a polygon is 1440°

Q5: The length and width of a rectangle are 15 cm and 25 cm, respectively. What is the perimeter of this rectangle?.

Q6: The number of sides of a regular polygon is 24, what is the interior angle of the polygon ?

Q7: Find the area and perimeter of a regular pentagon whose side is 5 cm and apothem length is 6 cm. ??

Q8: One of the internal angle of a regular polygon is 135°. Find the number of sides in the polygon.

Q9: Find the area of a rhombus in which the diagonal lengths are 10 cm and 8 cm respectively.

Also Check,

• Types of Polygons
• Sum of interior angles of a polygon
• Area of 2-D shapes
• Diagonal of a Polygon Formula

FAQs on Polygons

What is a 7 sided polygon called?

A heptagon is a seven-sided polygon. It is also known as a septagon.

What is called smallest polygon?

The smallest polygon Triangle. It is formed using three lines. It has three sides and three interior angles.

What is a 100-sided polygon called?

In geometry, a hectogon or hectonta-gon or 100-gon is a hundred-sided polygon. The sum of all hectogon’s interior angles are 17640 degrees.

What are the first 7 polygons?

• Triangle. (3 sides).
• Quadrilateral. (4 sides).
• Pentagon. (5 sides).
• Hexagon. (6 sides).
• Heptagon. (7 sides).
• Octagon. (8 sides).
• Nonagon. (9 sides).

What are Properties of a Polygon?

Some important properties of the polygon are:

• Sum of all the exterior angles of a polygon is always equal to 360°
• If at least one of the interior angles of a polygon is greater than180°, it is known as a concave polygon.
• If all the angles of a polygon are less than 180° then it is a convex polygon.

Can a Circle be Classified as a Polygon?

Polygon is a closed shape made up of straight-line segments. The circle is a closed figure, but it is made of a curve. So, a circle is not a polygon.