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Polygons

Key Concepts

Solved Examples: 35

Practice Questions: 5

Solved Examples

Question 1: The sum of the interior angles of a polygon is 11 times the sum of the interior angles of a triangle. Find the number of sides of the polygon.

Sum of the interior angles of a triangle = 180°
Therefore, the sum of the interior angles of the polygon = 11 × 180° = 1980°
Using the formula,
(n − 2) × 180° = 1980°
or, n − 2 = 11
or, n = 13

Question 2: The sum of the interior angles of one polygon is 540° more than that of another polygon. If the smaller polygon has 6 sides, find the number of sides of the larger polygon.

Sum of the interior angles of the smaller polygon
= (6 − 2) × 180°
= 720°
Therefore, the sum of the interior angles of the larger polygon
= 720° + 540°
= 1260°
Now,
(n − 2) × 180° = 1260°
or, n − 2 = 7
or, n = 9

Question 3: The ratio of the sums of the interior angles of two polygons is 7 : 4. If the smaller polygon has 6 sides, find the number of sides of the larger polygon.

Sum of the interior angles of the smaller polygon
= (6 − 2) × 180°
= 720°
Since the ratio is 7 : 4,
Sum of the interior angles of the larger polygon
= 720° × 7/4
= 1260°
Now,
(n − 2) × 180° = 1260°
or, n − 2 = 7
or, n = 9

Question 4: The sum of the interior angles of a polygon exceeds the sum of the interior angles of a decagon by 900°. Find the number of sides of the polygon.

Sum of the interior angles of a decagon
= (10 − 2) × 180°
= 1440°
Therefore, the sum of the interior angles of the required polygon
= 1440° + 900°
= 2340°
Now,
(n − 2) × 180° = 2340°
or, n − 2 = 13
or, n = 15

Question 5: A polygon has four more sides than another polygon. The sum of the interior angles of the smaller polygon is 1080°. Find the number of sides of each polygon.

Let the smaller polygon have n sides.
Sum of interior angles of a polygon:
(n − 2) × 180° = 1080°
or, n − 2 = 6
or, n = 8
Therefore, the smaller polygon has 8 sides.
The larger polygon has four more sides:
= 8 + 4
= 12 sides
Answer: The smaller polygon has 8 sides and the larger polygon has 12 sides.

Question 6: The sum of the interior angles of a polygon is 1440°. If all the interior angles are equal, find the measure of each interior angle.

Sum of interior angles:
(n − 2) × 180° = 1440°
or, n − 2 = 8
or, n = 10
The polygon has 10 sides.
Each interior angle:
= 1440° ÷ 10
= 144°

Question 7: The sum of the interior angles of a polygon is twice the sum of its exterior angles. Find the number of sides of the polygon.

Sum of exterior angles of any polygon = 360°
Therefore,
Sum of interior angles = 2 × 360°
= 720°
Using the formula:
(n − 2) × 180° = 720°
or, n − 2 = 4
or, n = 6

Question 8: The interior angles of a polygon are in the ratio 2 : 3 : 4 : 5 : 6 : 7. Find the sum of the interior angles of the polygon.

The polygon has 6 interior angles.
Therefore,
Number of sides (n) = 6
Sum of interior angles:
(n − 2) × 180°
= (6 − 2) × 180°
= 4 × 180°
= 720°

Question 9: Find the measure of each interior angle of a regular hexagon.

A hexagon has 6 sides.
Each interior angle of a regular polygon:
= [(n − 2) × 180°] ÷ n
= [(6 − 2) × 180°] ÷ 6
= (4 × 180°) ÷ 6
= 720° ÷ 6
= 120°

Question 10: Each interior angle of a regular polygon is 135°. Find the number of sides of the polygon.

Each exterior angle:
= 180° − 135°
= 45°
For a regular polygon,
Each exterior angle = 360° ÷ n
Therefore,
45° = 360° ÷ n
or, n = 360° ÷ 45°
= 8

Question 11: The measure of each interior angle of a regular polygon is 30° more than that of a regular hexagon. Find the number of sides of the polygon.

Each interior angle of a regular hexagon:
= 120°
Interior angle of required polygon:
= 120° + 30°
= 150°
Exterior angle:
= 180° − 150°
= 30°
Number of sides:
n = 360° ÷ 30°
= 12

Question 12: Each interior angle of a regular polygon is equal to 7/8 of a straight angle. Find the number of sides of the polygon.

A straight angle = 180°
Each interior angle:
= 7/8 × 180°
= 157.5°
Exterior angle:
= 180° − 157.5°
= 22.5°
Number of sides:
n = 360° ÷ 22.5°
= 16

Question 13: The ratio of the interior angle to the exterior angle of a regular polygon is 5 : 1. Find the number of sides of the polygon.

Let the interior angle be 5x and the exterior angle be x.
Since, Interior angle + Exterior angle = 180°
Therefore,
5x + x = 180°
or, 6x = 180°
or, x = 30°
Exterior angle = 30°
For a regular polygon,
Number of sides:
n = 360° ÷ Exterior angle
= 360° ÷ 30°
= 12

Question 14: The interior angle of a regular polygon exceeds its exterior angle by 120°. Find the number of sides of the polygon.

Let the exterior angle be x°.
Interior angle = (x + 120)°
Since, Interior angle + Exterior angle = 180°
(x + 120) + x = 180°
2x + 120 = 180°
2x = 60°
x = 30°
Exterior angle = 30°
Number of sides:
n = 360° ÷ 30°
= 12

Question 15: Find the measure of each exterior angle of a regular decagon.

A decagon has 10 sides.
Each exterior angle:
= 360° ÷ n
= 360° ÷ 10
= 36°

Question 16: Each exterior angle of a regular polygon is 24°. Find the number of sides of the polygon.

For a regular polygon:
Each exterior angle = 360° ÷ n
Therefore,
24° = 360° ÷ n
n = 360° ÷ 24°
= 15

Question 17: The number of sides of two regular polygons are in the ratio 3 : 5. Find the ratio of their exterior angles.

For a regular polygon,
Exterior angle = 360° ÷ Number of sides
Let the number of sides be 3x and 5x.
Exterior angles:
First polygon:
= 360° ÷ 3x
Second polygon:
= 360° ÷ 5x
Ratio:
= (360° ÷ 3x) : (360° ÷ 5x)
= 5 : 3

Question 18: The exterior angle of one regular polygon is 12° more than the exterior angle of another regular polygon. If the first polygon has 20 sides, find the number of sides of the second polygon.

Exterior angle of the first polygon:
= 360° ÷ 20
= 18°
Exterior angle of the second polygon:
= 18° − 12°
= 6°
Number of sides of the second polygon:
= 360° ÷ 6°
= 60

Question 19: The sum of the interior angles of a polygon is 720° more than that of a pentagon. Find the number of sides of the polygon.

Sum of interior angles of a pentagon:
= (5 − 2) × 180°
= 540°
Sum of interior angles of the required polygon:
= 540° + 720°
= 1260°
Using the formula:
(n − 2) × 180° = 1260°
n − 2 = 7
n = 9

Question 20: The number of sides of a regular polygon is twice the number of sides of another regular polygon. The exterior angle of the first polygon is 12° less than the exterior angle of the second polygon. Find the number of sides of both polygons.

Let the smaller polygon have n sides.
The larger polygon has 2n sides.
Exterior angle of smaller polygon:
= 360° ÷ n
Exterior angle of larger polygon:
= 360° ÷ 2n
According to the question:
360° ÷ n − 360° ÷ 2n = 12°
Taking LCM:
(720° − 360°) ÷ 2n = 12°
360° ÷ 2n = 12°
180° ÷ n = 12°
n = 15
Therefore,
Smaller polygon = 15 sides
Larger polygon = 30 sides

Question 21: The number of diagonals of a polygon is 20. Find the number of sides of the polygon.

Number of diagonals:
= n(n − 3) ÷ 2
Therefore,
n(n − 3) ÷ 2 = 20
n(n − 3) = 40
n² − 3n − 40 = 0
(n − 8)(n + 5) = 0
n = 8

Question 22: A polygon has 5 more sides than a hexagon. Find the sum of its interior angles.

A hexagon has 6 sides.
Number of sides of the polygon:
= 6 + 5
= 11
Sum of interior angles:
= (n − 2) × 180°
= (11 − 2) × 180°
= 9 × 180°
= 1620°

Question 23: The sum of the interior angles of a regular polygon is 12 times the measure of each exterior angle. Find the number of sides of the polygon.

Let the number of sides be n.
Sum of interior angles:
= (n − 2) × 180°
Each exterior angle:
= 360° ÷ n
According to the question:
(n − 2) × 180° = 12 × (360° ÷ n)
or, 180n(n − 2) = 4320
Dividing by 180:
n(n − 2) = 24
n² − 2n − 24 = 0
(n − 6)(n + 4) = 0
n = 6

Question 24: Find the number of diagonals in a decagon.

A decagon has 10 sides.
Number of diagonals:
= n(n − 3) ÷ 2
= 10(10 − 3) ÷ 2
= 10 × 7 ÷ 2
= 35

Question 25: A polygon has 44 diagonals. Find the number of sides of the polygon.

Number of diagonals:
n(n − 3) ÷ 2 = 44
or, n(n − 3) = 88
or, n² − 3n − 88 = 0
or, (n − 11)(n + 8) = 0
n = 11

Question 26: How many diagonals can be drawn from one vertex of a dodecagon?

A dodecagon has 12 sides.
Diagonals from one vertex:
= n − 3
= 12 − 3
= 9

Question 27: A hexagon and an octagon are given. How many more diagonals does the octagon have compared to the hexagon?

For hexagon: n = 6
Number of diagonals:
= 6(6 − 3) ÷ 2
= 6 × 3 ÷ 2
= 9
For octagon: n = 8
Number of diagonals:
= 8(8 − 3) ÷ 2
= 8 × 5 ÷ 2
= 20
Difference: = 20 − 9
= 11

Question 28: A polygon has 27 diagonals. Find the sum of its interior angles.

Number of diagonals:
n(n − 3) ÷ 2 = 27
n(n − 3) = 54
n² − 3n − 54 = 0
(n − 9)(n + 6) = 0
n = 9
Sum of interior angles:
= (n − 2) × 180°
= (9 − 2) × 180°
= 7 × 180°
= 1260°

Question 29: The number of diagonals of a polygon is twice the number of sides. Find the number of sides of the polygon.

Number of diagonals:
= n(n − 3) ÷ 2
According to the question:
n(n − 3) ÷ 2 = 2n
n(n − 3) = 4n
n² − 3n = 4n
n² − 7n = 0
n(n − 7) = 0
n = 7

Question 30: A polygon has 8 sides. If two more sides are added to it, how many additional diagonals will be formed?

Original polygon: n = 8
Diagonals:
= 8(8 − 3) ÷ 2
= 20
New polygon: n = 10
Diagonals:
= 10(10 − 3) ÷ 2
= 35
Additional diagonals:
= 35 − 20
= 15

Question 31: The number of diagonals of a polygon is 8 times the number of sides minus 24. Find the number of sides of the polygon.

Number of diagonals:
= n(n − 3) ÷ 2
According to the question:
n(n − 3) ÷ 2 = 8n − 24
n(n − 3) = 16n − 48
n² − 3n − 16n + 48 = 0
n² − 19n + 48 = 0
(n − 16)(n − 3) = 0
n = 16

Question 32: The sum of the interior angles of a regular polygon is 5 times the sum of its exterior angles. Find the number of diagonals of the polygon.

Sum of exterior angles: = 360°
Sum of interior angles:
= 5 × 360°
= 1800°
Using:
(n − 2) × 180° = 1800°
n − 2 = 10
n = 12
Number of diagonals:
= n(n − 3) ÷ 2
= 12 × 9 ÷ 2
= 54

Question 33: The ratio of the number of sides of two regular polygons is 3 : 5. The difference between their exterior angles is 48°. Find the number of sides of both polygons.

Let the sides of the polygons be:
3x and 5x
Exterior angle of first polygon:
= 360° ÷ 3x
Exterior angle of second polygon:
= 360° ÷ 5x
Difference:
360°/3x − 360°/5x = 48°
Taking LCM:
(600 − 360) ÷ x = 48°
240/x = 48°
x = 5
Therefore,
Number of sides:
First polygon = 3 × 5 = 15
Second polygon = 5 × 5 = 25

Question 34: The number of diagonals of a polygon is equal to three times the number of sides. Find the number of sides and the sum of its interior angles.

Number of diagonals:
= n(n − 3) ÷ 2
According to the question:
n(n − 3) ÷ 2 = 3n
n(n − 3) = 6n
n² − 3n = 6n
n² − 9n = 0
n(n − 9) = 0
n = 9
Sum of interior angles:
= (9 − 2) × 180°
= 1260°

Question 35: The number of diagonals of a polygon is 5 times the number of sides. Find the number of sides and the sum of interior angles of the polygon.

Number of diagonals:
= n(n − 3) ÷ 2
According to the question:
n(n − 3) ÷ 2 = 5n
n(n − 3) = 10n
n² − 3n − 10n = 0
n² − 13n = 0
n(n − 13) = 0
n = 13
Sum of interior angles:
= (13 − 2) × 180°
= 1980°
Answer: The polygon has 13 sides and the sum of interior angles is 1980°.

What are Polygons?

A polygon is a closed two-dimensional (2D) figure formed by joining three or more straight line segments end to end. Each line segment is called a side, and the point where two sides meet is called a vertex. Polygons are named according to the number of sides they have, such as a triangle (3 sides), quadrilateral (4 sides), pentagon (5 sides), and hexagon (6 sides). Polygons are classified based on the number of sides they have and whether all their sides and angles are equal or not.

Parts of a Polygon

A polygon consists of the following parts:
Side: A straight line segment that forms the boundary of the polygon.
Vertex (Plural: Vertices): The point where two sides meet.
Interior Angle: The angle formed inside the polygon by two adjacent sides.
Diagonal: A line segment joining two non-adjacent vertices of a polygon.

Classification of Polygons by Number of Sides

Polygons are named according to the number of sides they have.

Number of Sides	3	4	5	6	7	8	9	10
Name	Triangle	Quadrilateral	Pentagon	Hexagon	Heptagon	Octagon	Nonagon	Decagon

Regular and Irregular Polygons

A regular polygon has all sides equal and all interior angles equal.
An irregular polygon has sides or angles that are not all equal.
Examples:
A square is a regular polygon.
A rectangle is an irregular polygon because its sides are not all equal.

Convex and Concave Polygons

A convex polygon has all interior angles less than 180°. Every diagonal lies completely inside the polygon.
A concave polygon has at least one interior angle greater than 180°. At least one diagonal lies outside the polygon.

Sum of Interior Angles of a Polygon

The sum of the interior angles of an n-sided polygon is:
Sum of Interior Angles = (n − 2) × 180°
where n is the number of sides.

Each Interior Angle of a Regular Polygon

Since all interior angles of a regular polygon are equal,
Each Interior Angle = [(n − 2) × 180°] ÷ n
where n is the number of sides.

Sum of Exterior Angles of a Polygon

The sum of one exterior angle at each vertex of any polygon, taken in the same direction, is always:
Sum of Exterior Angles = 360°
This is true for both regular and irregular polygons.

Each Exterior Angle of a Regular Polygon

Since all exterior angles of a regular polygon are equal,
Each Exterior Angle = 360° ÷ n
where n is the number of sides.

Number of Diagonals in a Polygon

The number of diagonals in an n-sided polygon is:
Number of Diagonals = n(n − 3) ÷ 2
This formula counts all line segments joining non-adjacent vertices.

Number of triangles from one Vertex

If all possible diagonals are drawn from one vertex of an n-sided polygon, the polygon is divided into:
Number of Triangles = n − 2
This property is used to derive the formula for the sum of interior angles of a polygon.

Summary of Polygons

Concept	Summary
Polygon	A closed two-dimensional figure formed by three or more straight line segments.
Parts	Every polygon has sides, vertices, interior angles and diagonals.
Regular Polygon	All sides and all interior angles are equal.
Irregular Polygon	Sides or angles are not all equal.
Convex Polygon	All interior angles are less than 180°.
Concave Polygon	At least one interior angle is greater than 180°.
Sum of Interior Angles	(n − 2) × 180°
Each Interior Angle (Regular Polygon)	[(n − 2) × 180°] ÷ n
Sum of Exterior Angles	360°
Each Exterior Angle (Regular Polygon)	360° ÷ n
Number of Diagonals	n(n − 3) ÷ 2
Triangles from One Vertex	n − 2

Common Mistakes

Confusing the number of sides with the number of angles. A polygon has the same number of sides, vertices and interior angles.
Using the wrong formula for the sum of interior angles. The correct formula is:(n − 2) × 180°where n is the number of sides of the polygon.
Forgetting that the sum of exterior angles of any polygon is always 360°. This rule applies to both regular and irregular polygons.
Using 360° ÷ n for interior angles. This formula gives the measure of each exterior angle of a regular polygon, not the interior angle.
Confusing regular and irregular polygons. A regular polygon must have both equal sides and equal angles. Having only equal sides or only equal angles is not sufficient.
Applying the exterior angle formula to irregular polygons. The formula: Each Exterior Angle = 360° ÷ n is applicable only when all exterior angles are equal, as in a regular polygon.
Forgetting the relationship between interior and exterior angles. The interior angle and exterior angle at the same vertex always add up to:180°
Using the wrong formula for diagonals. The total number of diagonals of an n-sided polygon is:n(n − 3) ÷ 2. Do not use n(n − 1) ÷ 2, as this counts all possible line segments between vertices.
Including sides connected to the same vertex while counting diagonals from one vertex. A vertex can form diagonals only with non-adjacent vertices.Diagonals from one vertex = n − 3
Assuming that every polygon with equal sides is regular. A polygon is regular only when all sides and all angles are equal.
Forgetting to verify whether the calculated number of sides is possible. The value of n must always be a whole number greater than or equal to 3.

Practice Questions

Question 1: A regular polygon has the same number of diagonals as the sum of its sides and vertices multiplied by 5. Find the number of sides of the polygon.

Question 2: The measure of each interior angle of a regular polygon is 7 times the measure of its exterior angle. Find the number of sides of the polygon.

Question 3: The number of diagonals of a polygon is 42 more than the number of its sides. Find the number of sides and the sum of its interior angles.

Question 4: The exterior angle of a regular polygon is 15° less than the exterior angle of another regular polygon. The first polygon has 24 more sides than the second polygon. Find the number of sides of both polygons.

Question 5: The number of sides of a regular polygon is increased by 5. The difference between the number of diagonals of the new polygon and the original polygon is 75. Find the number of sides of the original polygon.