Sum of the Measures of Angles of a Polygon: Definitions, Types, and Examples

Sum of the Measures of Angles of a Polygon: A closed curve or figure set up by the line segments such that no two line segments intersect excluding at their end-points, and no two line segments with a common end-point are coinciding is called a polygon. In simple words, a polygon is a simple closed curve made up of only line segments.

A polygon whose all sides and each angle are equal is known as a regular polygon. Therefore, a regular polygon is equiangular and equilateral.

This article will study some of the polygons’ interior and exterior angles and solve some example problems.

Learn About Convex Polygon Here

Polygon

A polygon is a simple closed curve made up of only line segments. Each side is a straight line in a polygon. A triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon are known as a polygon as it comprises \(3,\,4,\,5,\,6,\,7,\,8,\,9,\,10\) sides, respectively.

If two sides have a common end-point (vertex), it is called the adjacent side of a polygon.

The end-points of the same side of a polygon are known as the adjacent vertices of a polygon.

The line segment obtained by joining vertices that are not adjacent is called the diagonals of a polygon.

Different Types of Polygons

There are four types of polygons, namely:

1. Convex Polygon
2. Concave Polygon
3. Regular Polygon
4. Irregular Polygon

1. Convex Polygon

A convex polygon is one in which each angle is less than \({180^{\rm{o}}}\).

In the above figure, \(PQRS\) is a convex polygon.

2. Concave Polygon

A concave polygon has at least one angle more than \({180^{\rm{o}}}\).

In the above figure, \(ABCD\) is a concave polygon. \(\angle BCD\) is more than \({180^{\rm{o}}}\), as shown.

3. Regular Polygon

A polygon with all sides equal and each one of the angles equal is called a regular polygon. An equilateral triangle and a square are better examples of a regular polygon.

Since both have the same length and measure an equal angle, they are called regular polygons.

4. Irregular Polygons

Polygons which is not regular are called irregular polygons. In simple words, a polygon in which the sides and angles differ is known as an irregular polygon.

Some of the common examples of irregular polygon are a rectangle and a rhombus.

Regular Polygon Formula

For a regular polygon of \(n\) sides, we have:

(i) The sum of the measure of exterior angles \( = {360^{\rm{o}}}\)
(ii) So, the measure of each exterior angle of a regular polygon \( = \frac{{{{360}^{\rm{o}}}}}{n}\)
(iii) The sum of the measure of exterior angles \( = (n – 2) \times {180^{\rm{o}}}\)
(iv) So, the measure of each interior angle \( = \frac{{(n – 2) \times {{180}^{\rm{o}}}}}{n}\)

Sum of the Measures of Interior Angles of some Regular Polygon

Equilateral triangle: An equilateral triangle is one with all sides and all angles equal.

Interior angle of an equilateral triangle \( = \frac{{(3 – 2) \times {{180}^{\rm{o}}}}}{3} = {60^{\rm{o}}}\)

So, the sum of interior angles of an equilateral triangle \( = {60^{\rm{o}}} + {60^{\rm{o}}} + {60^{\rm{o}}} = {180^{\rm{o}}}\)

Square: A square is a plane figure with four equal straight sides and four equal angles.

Interior angle of a square \( = \frac{{(4 – 2) \times {{180}^{\rm{o}}}}}{4} = {90^{\rm{o}}}\)

So, the sum of interior angles of a square \( = 4 \times {90^{\rm{o}}} = {360^{\rm{o}}}\)

Regular Pentagon: A pentagon is a polygon that has five sides. A regular pentagon is one in which all of the sides and angles are the same.

Interior angle of a regular pentagon \( = \frac{{(5 – 2) \times {{180}^{\rm{o}}}}}{5} = {108^{\rm{o}}}\)

So, the sum of interior angles of a regular pentagon \( = 5 \times {108^{\rm{o}}} = {540^{\rm{o}}}\)

Regular Hexagon: A hexagon is a polygon that has six sides. A regular Hexagon is one in which all of the sides and angles are the same.

Interior angle of a regular hexagon \( = \frac{{(6 – 2) \times {{180}^{\rm{o}}}}}{6} = {120^{\rm{o}}}\)

Therefore, the sum of interior angles of a hexagon \( = 6 \times {120^{\rm{o}}} = {720^{\rm{o}}}\)

Regular Heptagon: A heptagon is a polygon that has seven sides. A regular heptagon is one in which all of the sides and angles of a heptagon are the same.

Interior angle of a regular heptagon \( = \frac{{(7 – 2) \times {{180}^{\rm{o}}}}}{7} = \left( {\frac{{900}}{7}} \right)^\circ \cong {128.57^{\rm{o}}}\)

Therefore, the sum of the interior angles of a regular heptagon \( = 7 \times {128.57^{\rm{o}}} = {899.99^{\rm{o}}} \cong {900^{\rm{o}}}\)

Regular Octagon: An octagon is a  polygon that has eight sides. A regular Octagon is one in which all of the sides and angles of an octagon are the same.

Interior angle of a regular octagon \(n = \frac{{(8 – 2) \times {{180}^{\rm{o}}}}}{8} = {135^{\rm{o}}}\)

So, the sum of interior angles of a regular octagon \( = 8 \times {135^{\rm{o}}} = {1080^{\rm{o}}}\)

Regular Nonagon: A nonagon is a polygon that has nine sides. A regular nonagon is one in which all of the sides and angles of a nonagon are the same.

Interior angle of a nonagon \( = \frac{{(9 – 2) \times {{180}^{\rm{o}}}}}{9} = {140^{\rm{o}}}\)

Therefore, the sum of the interior angle of a nonagon \( = 9 \times {140^{\rm{o}}} = {1260^{\rm{o}}}\)

Regular Decagon: A decagon is a polygon that has ten sides. A regular decagon is one in which all of the sides and angles of a decagon are the same.

Interior angle of a decagon \( = \frac{{(10 – 2) \times {{180}^{\rm{o}}}}}{{10}} = {144^{\rm{o}}}\)

Therefore, the sum of the interior angle of a decagon \( = 10 \times {144^{\rm{o}}} = {1440^{\rm{o}}}\)

In other words, we can find the sum of an interior angle of a \(n\) sided polygon \( = (n – 2) \times {180^{\rm{o}}}\)

Sum of the Measures of Exterior Angles of some Regular Polygon

Square: The exterior angle of a square \( = \frac{{{{360}^{\rm{o}}}}}{4} = 90\)

So, the sum of exterior angles of a square \( = 4 \times {90^{\rm{o}}} = {360^{\rm{o}}}\)

Regular Pentagon: The exterior angle of a regular pentagon \( = \frac{{{{360}^{\rm{o}}}}}{5} = {72^{\rm{o}}}\)

Therefore, the sum of exterior angles of a regular pentagon \( = 5 \times 72^\circ  = 360^\circ \)

Regular Hexagon: The exterior angle of a regular hexagon \( = \frac{{{{360}^{\rm{o}}}}}{6} = {60^{\rm{o}}}\)

Therefore, the sum of exterior angles of a regular hexagon \( = {60^{\rm{o}}} \times 6 = {360^{\rm{o}}}\)

Regular Heptagon: The exterior angle of a regular heptagon \( = \left( {\frac{{{{360}^{\rm{o}}}}}{7}} \right) \cong {51.43^{\rm{o}}}\)

Therefore, the sum of exterior angles of a regular heptagon \( = 7 \times {51.43^{\rm{o}}} \cong {360^{\rm{o}}}\)

Regular Octagon: The exterior angle of a regular octagon \( = \frac{{{{360}^{\rm{o}}}}}{8} = {45^{\rm{o}}}\)

Therefore, the sum of exterior angles of a regular octagon \( = 8 \times {45^{\rm{o}}} = {360^{\rm{o}}}\)

Regular Nonagon: The exterior angle of a nonagon \( = \frac{{360^\circ }}{9} = 40^\circ \)

Therefore, the sum of exterior angles of a regular nonagon \( = 9 \times {40^{\rm{o}}} = {360^{\rm{o}}}\)

Regular Decagon: The exterior angle of a decagon \( = \frac{{360^\circ }}{{10}} = 36^\circ \)

Therefore, the sum of exterior angles of a regular decagon \(= 10 \times 36^\circ  = 360^\circ \)

So, from all the above regular polygons, we can see that the sum of exterior angles of a regular polygon will always be \({360^{\rm{o}}}\).

Solved Examples – Sum of the Measures of Angles of a Polygon

Q.1. Find the sum of measures of all interior angles of a polygon with the number of sides \(12\).
Ans: We know that the sum of an interior angle of a \(n\) sided polygon \( = (n – 2) \times {180^{\rm{o}}}\)
Here, \(n = 12\)
So, interior angle of \(12\) sided polygon \( = (12 – 2) \times {180^{\rm{o}}}\)
\( = 10 \times {180^{\rm{o}}}\)
\( = {1800^{\rm{o}}}\)
Therefore, the sum of an interior angle of a \(12\) sided polygon is \({1800^{\rm{o}}}\)

Q.2. The four angles of a regular pentagon are \({30^{\rm{o}}},\,{65^{\rm{o}}},\,{115^{\rm{o}}}\) and \({125^{\rm{o}}}\). Find the fifth angle.
Ans: Given: The four angles of a pentagon are \({30^{\rm{o}}},\,{65^{\rm{o}}},\,{115^{\rm{o}}}\) and \({125^{\rm{o}}}\)
We know that the sum of the interior angles of a regular pentagon is \({540^{\rm{o}}}\).
Let the fifth angle be \(x\).
Therefore, \({30^{\rm{o}}} + {65^{\rm{o}}} + {115^{\rm{o}}} + {125^{\rm{o}}} + x = {540^{\rm{o}}}\)
\( \Rightarrow {335^{\rm{o}}} + x = {540^{\rm{o}}}\)
\( \Rightarrow x = {540^{\rm{o}}} – {335^{\rm{o}}}\)
\( \Rightarrow x = {205^{\rm{o}}}\)
Therefore, the fifth angle of a given regular polygon is \({205^{\rm{o}}}\).

Q.3. What is the sum of measures of an interior angle in a regular polygon with \(13\) sides?
Ans: We know that the sum of an interior angle of a \(n\) sided polygon \( = (n – 2) \times {180^{\rm{o}}}\)
Here, \(n = 13\)
So, interior angle of \(13\) sided polygon \( = (13 – 2) \times {180^{\rm{o}}}\)
\( = {1980^{\rm{o}}}\)
Therefore, the sum of an interior angle of a \(13\) sided polygon is \({1980^{\rm{o}}}\)

Q.4. What is the sum of the interior angles of a regular polygon where a single exterior angle measures \({72^{\rm{o}}}\)?
Ans: If the regular polygon has an exterior angle of \({72^{\rm{o}}}\) there would be \(5\) sides to the polygon and a pentagon.
The exterior measures of a polygon must add up to \({360^{\rm{o}}}\), therefore, \(\frac{{{{360}^{\rm{o}}}}}{{{{72}^{\rm{o}}}}}\) means \(5\) exterior angles, five interior angles, so \(5\) sides to the polygon.
The exterior angle and the interior angle of any polygon are supplementary.
\( \Rightarrow {180^{\rm{o}}} – {72^{\rm{o}}} = {108^{\rm{o}}}\)
So, each interior angle of a pentagon \( = {108^{\rm{o}}}\)
Therefore, the sum of an interior angle of a pentagon \( = 5 \times {108^{\rm{o}}} = {540^{\rm{o}}}\)

Q.5. The sum of a polygon’s interior angle measures is \({2340^{\rm{o}}}\). What kind of polygon is it?
Ans: We know that the sum of an interior angle of a \(n\) sided polygon \( = (n – 2) \times {180^{\rm{o}}}\)
Here the sum of interior angels is \({2340^{\rm{o}}}\).
\( \Rightarrow (n – 2) \times {180^{\rm{o}}} = {2340^{\rm{o}}}\)
\( \Rightarrow n – 2 = 13\)
\( \Rightarrow n = 15\)
Therefore, the polygon is \(15\) sided.

Summary

In this article, we learned the definition of polygons, different types of polygons, and some regular polygons having different numbers of sides. Also, we have learned the sum of interior angles of a polygon and the formula to find the sum of interior angles of a polygon and the sum of exterior angles of a polygon and solved some example problems.

Frequently Asked Questions (FAQs) – Sum of the Measures of Angles of a Polygon

Q.1. What is the sum of the interior angle measures of a polygon with n sides?
Ans: The sum of the interior angle measures of a polygon with \(n\) sides is \((n – 2) \times {180^{\rm{o}}}\).

Q.2. What will be the sum of all the interior angles of a polygon having 14 sides?
Ans: We know that the sum of an interior angle of a \(n\) sided polygon \( = (n – 2) \times {180^{\rm{o}}}\)
Here, \(n = 14\)
So, interior angle of \(12\) sided polygon \( = (14 – 2) \times {180^{\rm{o}}}\)
\( = {2160^{\rm{o}}}\)
Therefore, the sum of an interior angle of a \(14\) sided polygon is \({2160^{\rm{o}}}\).

Q.3. How do I find the sum of the measures of the interior angles of a polygon?
Ans: We can find the sum of the measures of the interior angles of a polygon using the formula \((n – 2) \times {180^{\rm{o}}}\). Where \(n\) is the number of sides.

Q.4. What is the sum of angles of a polygon with 6 sides?
Ans: The polygon with six sides is a hexagon. The sum of interior angles of a hexagon \( = (6 – 1) \times {180^{\rm{o}}} = {720^{\rm{o}}}\)
The sum of exterior angles of a hexagon \( = 360^\circ \).

Q.5. Do all polygon’s angles add up to 360 degrees?
Ans: The sum of exterior angles of all polygons will add up to \({360^{\rm{o}}}\).

Learn About Types Of Angles Here

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