What is the sum of all the interior angles of an n-sided polygon
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(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°) Show
The Interior Angles of a Pentagon add up to 540° The General RuleEach time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total: In Mathematics, an angle is defined as the figure formed by joining the two rays at the common endpoint. An interior angle is an angle inside a shape. The polygons are the closed shape that has sides and vertices. A regular polygon has all its interior angles equal to each other. For example, a square has all its interior angles equal to the right angle or 90 degrees.The interior angles of a polygon are equal to a number of sides. Angles are generally measured using degrees or radians. So, if a polygon has 4 sides, then it has four angles as well. Also, the sum of interior angles of different polygons is different. Table of Contents: What is Meant by Interior Angles of a Polygon?An interior angle of a polygon is an angle formed inside the two adjacent sides of a polygon. Or, we can say that the angle measures at the interior part of a polygon are called the interior angle of a polygon. We know that the polygon can be classified into two different types, namely:
For a regular polygon, all the interior angles are of the same measure. But for irregular polygon, each interior angle may have different measurements. Also, read:
Sum of Interior Angles of a PolygonThe Sum of interior angles of a polygon is always a constant value. If the polygon is regular or irregular, the sum of its interior angles remains the same. Therefore, the sum of the interior angles of the polygon is given by the formula: Sum of the Interior Angles of a Polygon = 180 (n-2) degrees As we know, there are different types of polygons. Therefore, the number of interior angles and the respective sum of angles is given below in the table. Polygon NameNumber of Interior AnglesSum of Interior Angles = (n-2) x 180° Triangle3180°Quadrilateral4360°Pentagon5540°Hexagon6720°Septagon7900°Octagon81080°Nonagon91260°Decagon101440° Interior angles of TrianglesA triangle is a polygon that has three sides and three angles. Since, we know, there is a total of three types of triangles based on sides and angles. But the angle of the sum of all the types of interior angles is always equal to 180 degrees. For a regular triangle, each interior angle will be equal to: 180/3 = 60 degrees 60°+60°+60° = 180° Therefore, no matter if the triangle is an acute triangle or obtuse triangle or a right triangle, the sum of all its interior angles will always be 180 degrees. Interior Angles of QuadrilateralsIn geometry, we have come across different types of quadrilaterals, such as:
All the shapes listed above have four sides and four angles. The common property for all the above four-sided shapes is the sum of interior angles is always equal to 360 degrees. For a regular quadrilateral such as square, each interior angle will be equal to: 360/4 = 90 degrees. 90° + 90° + 90° + 90° = 360° Since each quadrilateral is made up of two triangles, therefore the sum of interior angles of two triangles is equal to 360 degrees and hence for the quadrilateral. Interior angles of PentagonIn case of the pentagon, it has five sides and also it can be formed by joining three triangles side by side. Thus, if one triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of three triangles will be: 3 x 180 = 540 degrees Thus, the angle sum of the pentagon is 540 degrees. For a regular pentagon, each angle will be equal to: 540°/5 = 108° 108°+108°+108°+108°+108° = 540° Sum of Interior angles of a Polygon = (Number of triangles formed in the polygon) x 180° Interior angles of Regular PolygonsA regular polygon has all its angles equal in measure. Regular Polygon NameEach interior angleTriangle60°Quadrilateral90°Pentagon108°Hexagon120°Septagon128.57°Octagon135°Nonagon140°Decagon144° Interior Angle FormulasThe interior angles of a polygon always lie inside the polygon. The formula can be obtained in three ways. Let us discuss the three different formulas in detail. Method 1: If “n” is the number of sides of a polygon, then the formula is given below: Interior angles of a Regular Polygon = [180°(n) – 360°] / n Method 2: If the exterior angle of a polygon is given, then the formula to find the interior angle is Interior Angle of a polygon = 180° – Exterior angle of a polygon Method 3: If we know the sum of all the interior angles of a regular polygon, we can obtain the interior angle by dividing the sum by the number of sides. Interior Angle = Sum of the interior angles of a polygon / n Where “n” is the number of polygon sides. Interior Angles TheoremBelow is the proof for the polygon interior angle sum theorem Statement: In a polygon of ‘n’ sides, the sum of the interior angles is equal to (2n – 4) × 90°. To prove: The sum of the interior angles = (2n – 4) right angles Proof:  ABCDE is a “n” sided polygon. Take any point O inside the polygon. Join OA, OB, OC. For “n” sided polygon, the polygon forms “n” triangles. We know that the sum of the angles of a triangle is equal to 180 degrees Therefore, the sum of the angles of n triangles = n × 180° From the above statement, we can say that Sum of interior angles + Sum of the angles at O = 2n × 90° ——(1) But, the sum of the angles at O = 360° Substitute the above value in (1), we get Sum of interior angles + 360°= 2n × 90° So, the sum of the interior angles = (2n × 90°) – 360° Take 90 as common, then it becomes The sum of the interior angles = (2n – 4) × 90° Therefore, the sum of “n” interior angles is (2n – 4) × 90° So, each interior angle of a regular polygon is [(2n – 4) × 90°] / n Note: In a regular polygon, all the interior angles are of the same measure. Exterior AnglesExterior angles of a polygon are the angles at the vertices of the polygon, that lie outside the shape. The angles are formed by one side of the polygon and extension of the other side. The sum of an adjacent interior angle and exterior angle for any polygon is equal to 180 degrees since they form a linear pair. Also, the sum of exterior angles of a polygon is always equal to 360 degrees. What is the sum of interior angles of an n sided polygon?Interior Angles Theorem
In a polygon of 'n' sides, the sum of the interior angles is equal to (2n – 4) × 90°.
What is an N sided polygon?The N-Sided primitive is a regular polygon with a given number of sides. The polygon can be made from a Bezier curve or a Polyline. The Bezier type is equivalent to a curve made by the Bezier tool. The Polyline is a connected sequence of line segments.
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