In this chapter we will learn about interior angles of polygon along with some important formulas related to the topic.
What are interior angles of polygons ?
The angles which lie inside the polygon are called interior angles.
The interior angles are formed at the intersection of two sides of polygon.
Polygon classification and interior angles
Polygons are broadly classified into two categories;
(a) Regular Polygons
These polygons have equal sides and equal interior angles.
(b) Irregular Polygons
These polygons have different sides and interior angle measurement.
Interior angle formula for regular polygons
The measure of interior angle for regular polygons is given by following formula;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}
Where;
n = number of sides of polygon
Sum of all interior angle formula for polygon
The measure of sum of all interior angle is given by following formula;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}
Where;
n = number of sides of polygon
Calculating interior angles of regular polygons
I hope you understood the above mentioned concept. Let us now calculate the interior angle values of different polygons.
Interior Angles of Triangle
It’s a polygon with 3 sides.
Given below is triangle ABC with interior angles ∠A, ∠B & ∠C
Calculating interior angle of Triangle
The measure of interior angle is given by following calculation;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 3-2) \ \times 180}{3}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{1\ \times 180}{3}}\\\ \\ \mathtt{Interior\ angle\ =\ 60\ degree}
Hence, each angle of regular triangle measures 60 degree.
Sum of all angle of Triangle
The sum of all interior angle of triangle is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 3\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 180\ }
Interior angle of Quadrilateral
It is a polygon with 4 sides.
Given below is the regular quadrilateral with interior angles ∠A, ∠B, ∠C and ∠D.
Calculating Interior angles of Quadrilateral
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 4-2) \ \times 180}{4}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{2\ \times 180}{4}}\\\ \\ \mathtt{Interior\ angle\ =\ 90\ degree}
Hence, each angle of regular quadrilateral measures 90 degree.
Sum of all Interior angle of Quadrilateral
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 4\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 360\ degree}
The sum of all interior angle of quadrilateral measures 360 degree.
Interior angles of Pentagon
It’s a polygon with 5 sides.
Given below is the regular polygon with interior angles marked in red color.
Calculating Interior angle of Pentagon
The interior angles of regular pentagon is calculated as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 5-2) \ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{3\ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ 108\ degree}
Hence, each angle of pentagon measures 108 degree.
Sum of all interior angle of Pentagon
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 5\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 540\ degree}
Hence, sum of all interior angles of pentagon measures 540 degree.
Interior angle of Hexagon
It is a polygon with six sides.
Given below is the image of regular hexagon.
Calculating Interior angle of Hexagon
The interior angle of hexagon is calculated as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 6-2) \ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{4\ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ 120\ degree}
Hence, each interior angle of regular polygon measure 120 degree.
Sum of all interior angle of Hexagon
The sum of all interior angles of hexagon is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 6\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 720\ degree}
Interior angles of Heptagon
It’s a polygon with 7 sides.
Given below is the image of regular heptagon.
Calculating Interior angle of Heptagon
The interior angles of heptagon is measured as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 7-2) \ \times 180}{7}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{5\ \times 180}{7}}\\\ \\ \mathtt{Interior\ angle\ =\ 128.5\ degree}
Hence, each angle of regular heptagon measures 128.5 degree.
Sum of all interior angles of Heptagon
The sum of all interior angles of heptagon is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 7\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 600\ degree}
Interior angles of Octagon
It’s a polygon with 8 sides.
Given below is the image of regular octagon.
Calculating Interior angle of Heptagon
The interior angle of octagon is calculated as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 8-2) \ \times 180}{8}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{6\ \times 180}{8}}\\\ \\ \mathtt{Interior\ angle\ =\ 135\ degree}
Hence, each angle of regular octagon measures 135 degree.
Sum of all interior angles of Heptagon
The sum of interior angles of octagon is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 8\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1080\ degree}
Interior angles of Nonagon
It’s a polygon with 9 sides.
Given below is the image of regular nonagon.
Calculating Interior angle of Nonagon
The interior angle of Nonagon is calculated as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 9-2) \ \times 180}{9}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{7\ \times 180}{9}}\\\ \\ \mathtt{Interior\ angle\ =\ 140\ degree}
Hence, each interior angle measures 140 degree.
Sum of all interior angles of Nonagon
Sum of all interior angles of Nonagon is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 9\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1260\ degree}
Interior angles of Decagon
It’s a polygon with 10 sides.
Given below is the image of regular decagon with angled marked in red color.
Calculating interior angles of Decagon
The interior angle of regular decagon is calculated as;
\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 10-2) \ \times 180}{10}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{8\ \times 180}{10}}\\\ \\ \mathtt{Interior\ angle\ =\ 144\ degree}
Hence. each angle measures 144 degree.
Sum of angles of Decagon
Sum of all interior angles is calculated as;
\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 10\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1440\ degree}