Angle Sum Property of Polygons: Formulas, Videos and Solved Examples (2024)

Understanding Quadrilaterals

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.

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Angle Sum Property of Polygons

We have learned about the angle sum property in triangles! According to the angle sum property of atriangle, 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 theangle 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 analyzea 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 hexagonis 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 thenumber of triangles, we can find the sum of angles in a polygon.

You can downloadPolygon Cheat Sheet by clicking on the download button below

Solved Example for You

Question 1: Find the sum of angles for the following polygons

Answer :

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.

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Angle Sum Property of Polygons: Formulas, Videos and Solved Examples (2024)

FAQs

What is the formula to find angle sum property of a polygon? ›

To find the interior angle sum of a polygon, we can use a formula: interior angle sum = (n - 2) x 180°, where n is the number of sides. For example, a pentagon has 5 sides, so its interior angle sum is (5 - 2) x 180° = 3 x 180° = 540°. Created by Sal Khan.

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How to solve polygon formula? ›

A regular polygon is a flat shape whose sides are all equal and whose angles are all equal. The formula for finding the sum of the measure of the interior angles is (n - 2) * 180. To find the measure of one interior angle, we take that formula and divide by the number of sides n: (n - 2) * 180 / n.

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What is an example of the angle sum property? ›

For example, if two angles of a triangle are 70° and 60°, we will add these, 70 + 60 = 130°, and we will subtract it from 180°, which is the sum of the angles of a triangle. So, the third angle = 180° - 130° = 50°.

How to solve the angle of polygons? ›

The formula for calculating the sum of interior angles is ( n − 2 ) × 180 ∘ where is the number of sides. All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is: interior angle of a polygon = sum of interior angles ÷ number of sides.

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What is the formula used to find the angle sums of regular polygons? ›

The sum of all interior angles of a regular polygon is calculated by the formula S=(n-2) × 180°, where 'n' is the number of sides of a polygon. For example, to find the sum of interior angles of a pentagon, we will substitute the value of 'n' in the formula: S=(n-2) × 180°; in this case, n = 5.

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What is the formula for the sum of exterior angles in a polygon? ›

So, the sum of exterior angles is a + b + c + d + e = 5(180) – sum of interior angles. So, we do from (1) a + b + c + d + e = 5(180) – 540 = 900 – 540 = 360 degrees. Therefore sum of exterior angles in any of the polygon is 360 degrees.

What is an example of an angle property? ›

The angle properties of lines are: Vertically opposite angles are equal, for example a = d, b = c. Adjacent angles add to 180^o^, for example a + b = 180^o, a + c = 180. Corresponding angles are equal, for example a = e, b = f, c = g, d= h.