Understanding Interior Angles in Polygons
Explore the concept of interior angles in polygons, including definitions of polygons, convex and concave polygons, regular and irregular polygons, as well as the sum of interior angles in triangles and quadrilaterals. Discover the naming convention for polygons based on their number of sides and learn how to calculate the sum of interior angles in different polygons.
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21 September 2024 The interior angles of a polygon. LO: To calculate the interior angles in a polygon. www.mathssupport.org
Polygons A polygon is a 2-D shape made when line segments enclose a region. The line segments are called sides. D The points where the sides meet are called vertices. Vertices are labelled with capital letters. C E One of these is called a vertex. B A These two dimensions 2-D stands for two-dimensional. are length and width. A polygon has no thickness. www.mathssupport.org
Polygons A polygon could be regular. A regular polygon has equal sides and equal angles. www.mathssupport.org
Polygons A polygon could be irregular. An irregular polygon has non-equal sides and non-equal angles. www.mathssupport.org
Polygons In a convex polygon all of the interior angles are less than 180 . All regular polygons are convex. www.mathssupport.org
Polygons In a concave polygon some of the interior angles are more than 180 . www.mathssupport.org
Naming polygons Polygons are named according to their number of sides. Number of sides 3 4 5 6 7 8 9 10 Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon www.mathssupport.org
Interior angles in polygons The angles inside a polygon are called interior angles. b c a The sum of the interior angles of a triangle is 180 . www.mathssupport.org
Sum of the interior angles in a quadrilateral What is the sum of the interior angles in a quadrilateral? d c f a e b We can work this out by dividing the quadrilateral into two triangles. a + b + c = 180 and So, (a + b + c)+ (d + e + f )= 360 d + e + f = 180 The sum of the interior angles in a quadrilateral is 360 . www.mathssupport.org
Sum of interior angles in a polygon We already know that the sum of the interior angles in any triangle is 180 . c a + b + c = 180 a b We have just shown that the sum of the interior angles in any quadrilateral is 360 . d a c b a + b + c + d = 360 Do you know the sum of the interior angles for any other polygons? www.mathssupport.org
Sum of the interior angles in a polygon A quadrilateral can be divided into twotriangles a pentagon can be divided into threetriangles How many triangles can a hexagon be divided into? divided into four triangles. and a hexagon can be www.mathssupport.org
Sum of the interior angles in a polygon We can work out the sum of the interior angles in a polygon as follows: Name of regular polygon of sides diagonals from a point Number Number of Number of triangles Angle sum of the polygon 2 180 2= 1 4 360 Quadrilateral 3 5 2 180 3= 540 Pentagon 3 6 4 180 4= 720 Hexagon 5 7 180 5= 4 900 Heptagon www.mathssupport.org
Sum of the interior angles in a polygon The number of triangles that a polygon can be divided into is always two less than the number of sides. We can say that: A polygon with n sides can be divided into (n 2) triangles. The sum of the interior angles in a triangle is 180 . So, The sum of the interior angles in an n-sided polygon is (n 2) 180 . www.mathssupport.org
Interior angles in regular polygons A regular polygon has equal sides and equal angles. We can work out the size of the interior angles in a regular polygon as follows: Name of regular polygon Sum of the interior angles Size of each interior angle Equilateral triangle 180 180 3 = 60 Square 2 180 = 360 360 4 = 90 Regular pentagon 3 180 = 540 540 5 = 108 Regular hexagon 4 180 = 720 720 6 = 120 www.mathssupport.org
Sum of the interior angles in a polygon x Find x x 132 x The figure has 5 sides. the sum of its interior angles is 3 180 = 540 x + x + x + 132 + 90 = 540 3x + 222 = 540 3x = 318 x = 106 www.mathssupport.org
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