Angles within polygons are not solely determined by their sides. There can be multiple shapes with the same number of sides, making it important to consider the angles to differentiate them. In this article, we will dive into the concept of angles in polygons and explore how to calculate and understand them.
How to Find the Angles of a Polygon?
A polygon is a two-dimensional figure composed of straight-line segments. The sum of angles within a polygon is equal to the total measure of all interior angles. To find the angles of a regular polygon, we can use the formula:
*Sum of interior angles = 180° (n - 2)
Let's explore some examples to understand this concept better:
Angles of a Triangle
A triangle has 3 sides, so n = 3. By substituting this value into the formula, we can find the sum of interior angles:
Sum of interior angles = 180° (3 - 2) = 180° 1 = 180°
Angles of a Quadrilateral
A quadrilateral is a 4-sided polygon, hence n = 4. Substituting this value into the formula:
Sum of interior angles = 180° (4 - 2) = 180° 2 = 360°
Angles of a Pentagon
A pentagon is a 5-sided polygon, so n = 5. By substituting this value:
Sum of interior angles = 180° (5 - 2) = 180° 3 = 540°
Angles of an Octagon
An octagon has 8 sides, so n = 8. Substituting this value:
Sum of interior angles = 180° (8 - 2) = 180° 6 = 1080°
Angles of a Hectagon
A hectagon is a 100-sided polygon, meaning n = 100. By substituting this value:
Sum of interior angles = 180° (100 - 2) = 180° 98 = 17640°
Interior Angles of Polygons
The interior angle is the angle formed within a polygon, between two sides. The number of sides in a polygon equals the number of angles formed. The measure of each interior angle of a polygon is given by:
*Measure of each interior angle = 180° (n - 2)/n
Let's consider some examples:
Size of the Interior Angle of a Decagon
A decagon is a 10-sided polygon, so n = 10. By substituting this value:
*Measure of each interior angle = 180° (10 - 2)/10 = 180° 8/10 = 18° 8 = 144°
Interior Angle of a Hexagon
A hexagon has 6 sides, thus n = 6. Substituting this value:
*Measure of each interior angle = 180° (6 - 2)/6 = 180° 4/6 = 60° 2 = 120°
Interior Angle of a Rectangle
A rectangle is an example of a quadrilateral with 4 sides, hence n = 4. By substituting this value:
Measure of each interior angle = 180° (4 - 2)/4 = 180° 1/2 = 90°
Interior Angle of a Pentagon
A pentagon consists of 5 sides, thus n = 5. By substituting this value:
Measure of each interior angle = 180° (5 - 2)/5 = 180° 3/5 = 108°
Exterior Angles of Polygons
The exterior angle is the angle formed outside a polygon between one side and an extended side. The measure of each exterior angle of a regular polygon is given by:
Measure of each exterior angle = 360°/n
One important property of regular polygons is that the sum of the measures of the exterior angles always equals 360°.
Let's explore some examples:
Exterior Angle of a Triangle
For a triangle, n = 3. Substituting this value:
Measure of each exterior angle = 360°/3 = 120°
Exterior Angle of a Pentagon
With n = 5, we can substitute this value:
Measure of each exterior angle = 360°/5 = 72°
It's important to note that the interior angle and exterior angle formulas only apply to regular polygons. Irregular polygons have different measures of angles.
Now, let's solve some example problems related to interior and exterior angles of polygons.
Example 1: Finding an Angle in an Irregular Polygon
The interior angles of an irregular 6-sided polygon measure 80°, 130°, 102°, 36°, x°, and 146°. We need to calculate the size of angle x.
For a polygon with 6 sides, n = 6. We can find the sum of interior angles using the formula:
Sum of interior angles = 180° (6 - 2) = 180° 4 = 720°
To find x, we need to subtract the sum of the given angles from 720°:
494° + x = 720°
Simplifying:
x = 226°
Example 2: Finding the Exterior Angle of a Regular Polygon
We need to find the exterior angle of a regular polygon with 11 sides.
For this problem, n = 11. By substituting this value into the formula, we can find the measure of each exterior angle:
Measure of each exterior angle = 360°/11 ≈ 32.73°
Example 3: Determining the Value of x
Given that the exterior angles of a polygon are 7x°, 5x°, x°, 4x°, and x°, we need to determine the value of x.
The sum of exterior angles is always 360°:
7x° + 5x° + x° + 4x° + x° = 360°
Simplifying:
18x = 360°
Dividing both sides by 18:
x = 360°/18 = 20°
Therefore, the value of x is 20°.
Example 4: Identifying the Polygon with 140° Interior Angles
Given that the interior angles of a polygon each measure 140°, we need to determine the name of the polygon.
The formula to find the size of each interior angle is:
*Size of each interior angle = 180° (n - 2)/n
By substituting 140° into the formula:
*140° = 180° (n - 2)/n
Simplifying:
140°n = 180° (n - 2)
140°n = 180°n - 360°
Subtracting both sides by 180°n:
140°n - 180°n = 180°n - 180°n - 360°
-40°n = -360°
Dividing both sides by -40°:
n = -360°/-40° = 9
Therefore, the number of sides in the polygon is 9, known as a nonagon.
As we have explored, angles play a crucial role in differentiating polygons. Understanding their properties and formulas can help in solving various problems related to polygons.