We already know that a simple closed curve that is made up of more than three line segments is called a polygon. Every polygon has a set of angles that are a result of the line segments involved in the closed figure. In the chapter below we shall learn about the angle sum property of polygons, which indirectly depends on the number of sides in that polygon.
Angle Sum Property of Polygons
We have learned about the angle sum property in triangles! According to the angle sum property of a triangle, the sum of all the angles in a triangle is 180º. Since a triangle has three sides, we find the measurements of the angles accordingly.
Let’s recap the method. For example, if there is a triangle with angles 45º and 60º. The third angle is unknown. For finding the third angle we follow the given system of calculation:
A + B + C = 180º
A = 45º; B = 60º; C =?
45 + 60 + ? = 180º
? = 180º – 105º
? = 75º
So the third angle is 75º. Using the above-shown system of calculations we can find out the unknown angle in a triangle, but what about a polygon. Similarly, according to the angle sum property of a polygon, the sum of angles depends on the number of triangles in the polygon.
According to the Angle sum property of polygons, the sum of all the angles in a polygon is the multiple the number of triangles constituting the polygon. We use the angle sum property of triangles while calculating the unknown angles of a polygon.
Browse more Topics under Understanding Quadrilaterals
- Polygon and Its Types
- Properties of Trapezium and Kite
- Properties of Parallelogram, Rhombus, Rectangle and Square
Relation of Angle Sum Property of Triangles and Polygons
When we analyze a polygon we come to know that it is a compilation of many triangles. Let’s see how? Take a polygon and draw diagonals that divide the structure into triangles. The number of triangles formed from this division gives us the idea of the total sum of angles in a polygon. See the figure below,
In the figures above, a is a hexagon while b is a pentagon. Hexagon when divided into diagonals, constitutes four triangles. The sum of angles in a triangle is 180 °. This means that the sum of angles in a hexagon is equal to 4 × 180° that is 720°.
Similarly, in figure b which is a pentagon, the number of triangles constituting the shape is three, so the sum of angles in a polygon shall be 3× 180 which equals 540°. Likewise, for a heptagon, the number of triangles formed after dividing into diagonals is five hence the sum of angles in a heptagon shall be 5 × 180° which equals 900°.
In the above discussion, one thing worth noting is that the number of angles = number of sides – 2. So for every polygon with x number of sides, the number of triangles is 2 less than the number of sides.
Polygons can have any number of sides greater than three, and when we find the sum of angles in a polygon we study the number of triangles constituting the closed shape. It is only after the study of the number of triangles, we can find the sum of angles in a polygon.
Solved Example for You
Question 1: Find the sum of angles for the following polygons
- for a polygon with 9 sides, the number of angles is 7. Therefore the sum of angles in a triangle shall be 7 × 180 = 1260°
- for a polygon with 8 sides, the number of angles is 6. Therefore the sum of angles in a triangle shall be 6 × 180 = 1080°
Question 3: What is the formula of angle sum property?
Answer: The sum of interior angles in a triangle refers to 180°. In order to find the sum of interior angles of a polygon we need to multiply the number of triangles in the polygon by 180°. Further, the sum of exterior angles of a polygon will be 360°. In other words, the formula to calculate the size of an exterior angle will be exterior angle of a polygon = 360 ÷ number of sides
Question 4: What is angle sum property of quadrilateral?
Answer: As per the angle sum property of a quadrilateral, the sum of all the four interior angles will be 360 degrees.
Question 5: What is the sum of parallelogram?
Answer: Firstly, please note that sum of the internal angles of any four-sided figure whether regular or irregular will be 360 degrees. However, regular figures like square, rectangle, parallelogram, or rhombus consist of an additional characteristic that the sum of any two adjacent angles is 180 degrees.
Question 6: What is the sum of all angles in a triangle?
Answer: When we look at a Euclidean space we see that the sum of measures of these three angles of any triangle is consistently equal to the straight angle which we also express as 180 °, π radians, two right angles, or a half-turn. However, it was not known for a long period whether other geometries exist having different sums.