Geometry & Trigonometry – Page 6

When dealing with polygons in geometry, it sometimes helps to know the total measure of all interior angles. We can also find the exact measure of each angle in a regular polygon. So, how many degrees are in a polygon? If we sum the interior angles of a polygon with N sides (an N-gon), there are 180(N-2) degrees. If the polygon is regular, then every interior angle has the same measure: 180(N-2)/n. The exterior angles of an N-sided polygon always sum to 360 degrees, regardless of the value of N. Of course, for a polygon that is not regular, we would need to do a little more work to find the measure of an individual interior angle. In this article, we’ll talk about how many degrees are in a polygon. We’ll also look at some specific examples for polygons from N = 3 to N = 10 sides. Let’s get started. How Many Degrees In A Polygon? There are 180(N - 2) degrees in a polygon if we add up the measures of every interior angle: Sum of Interior Angles of an N-gon = 180(N – 2) degrees. For example, a polygon with N = 22 sides has 180(22 – 2) = 180(20) = 3600 degrees. That is, the sum of all interior angles in a 22-sided polygon is 3600 degrees. If we add up the measure of every exterior angle, we get a total of 360 degrees: Sum of Exterior Angles of an N-gon = 360 degrees. The image below illustrates interior and exterior angles (in a hexagon). This picture illustrates an interior and exterior angle of a hexagon. The two must sum to 180 degrees. Remember that for a regular polygon: every side has the same length every angle has the same measure Given this second fact, we can easily calculate the measures of each interior and exterior angle for a regular polygon with N sides. Since there are N angles of the same measure, we only need to divide the sum of the angles by N. Therefore: Measure of One Interior Angle of a Regular N-gon = 180(N – 2) / N degrees. Measure of One Exterior Angle of a Regular N-gon = 360 / N degrees. Note what happens when we take the sum of an interior angle and an exterior angle for a regular N-gon: [Interior Angle of Regular N-gon] + [Exterior Angle of Regular N-gon] =[180(N – 2) / N] + [360 / N] =[(180N – 360) / N] + [360 / N] =(180N – 360 + 360) / N =180N / N =180 As expected, we get a sum of 180 degrees for the sum of an interior angle in a polygon and its corresponding exterior angle. How Many Degrees In A Triangle? There are 180 degrees in a triangle. This figure illustrates the interior and exterior angles of a triangle. A triangle has three angles (or if you prefer, a trigon has 3 sides – this is where trigonometry comes from!). We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(3 – 2) =180(1) =180 degrees For a regular 3-gon (that is, an equilateral triangle), the measure of each interior angle is: 180(N – 2) / N =180(3 – 2) / 3 =60 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 3-gon (equilateral triangle) is: 360 / N =360 / 3 =120 degrees How Many Degrees In A Quadrilateral, Rectangle, Or Square? There are 360 degrees in a quadrilateral, rectangle, or square (that is, a 4-gon). A rectangle has 4 sides and 4 angles. The sum of the interior angles is 360. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(4 – 2) =180(2) =360 degrees For a regular 4-gon (that is, a square), the measure of each interior angle is: 180(N – 2) / N =180(4 – 2) / 4 =180(2) / 4 =90 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 4-gon (square) is: 360 / N =360 / 4 =90 degrees Note: this is the only case where the interior and exterior angles for a regular N-gon have the same measure. Here is the proof: Measure of Interior Angle of Regular N-gon = Measure of Exterior Angle of Regular N-gon 180(N – 2) / N = 360 / N 180(N – 2) = 360 180N – 360 = 360 180N = 720 N = 4 How Many Degrees In A Pentagon? There are 540 degrees in a pentagon (that is, a 5-gon). A pentagon has 5 sides and 5 angles. The interior angles sum to 540. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(5 – 2) =180(3) =540 degrees For a regular 5-gon (that is, a regular pentagon), the measure of each interior angle is: 180(N – 2) / N =180(5 – 2) / 5 =180(3) / 5 =108 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 5-gon (regular pentagon) is: 360 / N =360 / 5 =72 degrees How Many Degrees In A Hexagon? There are 720 degrees in a hexagon (that is, a 6-gon). A hexagon has 6 sides and 6 angles. The interior angles sum to 720. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(6 – 2) =180(4) =720 degrees For a regular 6-gon (that is, a regular hexagon), the measure of each interior angle is: 180(N – 2) / N =180(6 – 2) / 6 =180(4) / 6 =120 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 6-gon (regular hexagon) is: 360 / N =360 / 6 =60 degrees You can learn about uses of hexagons in real life (and where they appear in nature) here. How Many Degrees In A Heptagon? There are 900 degrees in a heptagon (that is, a 7-gon). A heptagon has 7 sides and 7 angles. The interior angles sum to 900. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(7 – 2) =180(5) =900 degrees For a regular 7-gon (that is, a regular heptagon), the measure of each interior angle is: 180(N – 2) / N =180(7 – 2) / 7 =180(5) / 7 =900 / 7 ~128.57 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 7-gon (regular heptagon) is: 360 / N =360 / 7 ~51.43 degrees How Many Degrees In An Octagon? There are 1080 degrees in an octagon (that is, an 8-gon). An octagon has 8 sides and 8 angles. The interior angles sum to 1080. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(8 – 2) =180(6) =1080 degrees For a regular 8-gon (that is, a regular octagon), the measure of each interior angle is: 180(N – 2) / N =180(8 – 2) / 8 =180(6) / 8 =1080 / 8 =135 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 8-gon (regular octagon) is: 360 / N =360 / 8 =45 degrees How Many Degrees In A Nonagon? There are 1260 degrees in a nonagon (that is, a 9-gon). A nonagon has 9 sides and 9 angles. The interior angles sum to 1260. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(9 – 2) =180(7) =1260 degrees For a regular 9-gon (that is, a regular nonagon), the measure of each interior angle is: 180(N – 2) / N =180(9 – 2) / 9 =180(7) / 9 =1260 / 9 =140 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 9-gon (regular nonagon) is: 360 / N =360 / 9 =40 degrees How Many Degrees In A Decagon? There are 1440 degrees in a decagon (that is, a 10-gon). A decagon has 10 sides and 10 angles. The interior angles sum to 1440. We can use the formula for the sum of interior angles to verify this: 180(N – 2) =180(10 – 2) =180(8) =1440 degrees For a regular 10-gon (that is, a regular decagon), the measure of each interior angle is: 180(N – 2) / N =180(10 – 2) / 10 =180(8) / 10 =1440 / 10 =144 degrees As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 10-gon (regular decagon) is: 360 / N =360 / 10 =36 degrees The table below summarizes the interior angles and exterior angles for polygons with N sides, for N = 3 to 10. NInterior Angle SumInterior Angle Measure(Regular N-gon)Exterior Angle Measure (Regular N-gon)318060120436090905540108726720120607900~128.57~51.438108013545912601404010144014436 The table below summarizes the interior angles and exterior angles for polygons with N sides, for N = 3 to 10. Conclusion Now you know how many degrees are in a polygon (that is, the sum of the interior angles), depending on the number of sides. You also know how to find the measures of interior and exterior angles for regular polygons. I hope you found this article helpful. If so, please share it with someone who can use the information. Don’t forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon