About Regular & Irregular Polygons
Definition of Polygons
A two-dimensional closed shape made up of line segments is called a polygon. We can find the sum of the interior angles of a polygon with ‘n’ sides by S = (n − 2).180°. Based on the sides and angles, polygons are classified as regular and irregular polygons.
Regular Polygons
A regular polygon is a polygon that has equal sides and angles. Let’s look at some of the regular polygons that we have in geometry.
Square:
A square is a quadrilateral with four equal sides and four equal angles.
From the above figure, we can see that all the angles ∠PQS, ∠QSR, ∠SRP and ∠RPQ are equal to 90 degrees.
Equilateral Triangle:
An equilateral triangle is a triangle in which all the three sides and angles are equal.
As we can see, AB = BC = CA and the angles ∠ABC, ∠BCA and ∠CAB are equal to 60 degrees.
Regular Pentagon:
A regular pentagon is a polygon that has five sides with equal lengths and equal angles.
From the above image, it is clear that AB = BC = CD = DE = AE and the angles ∠ABC, ∠BCD, ∠CDE, ∠DEA and ∠EAB are equal to 108 degrees.
Now that we have knowledge of regular polygons, let’s move on to irregular polygons.
Irregular Polygons
A polygon in which all of its sides have unequal lengths and have unequal angles is called an irregular polygon. Irregular polygons are also known as non-regular polygons. Let’s take a look at a few irregular polygons.
Scalene Triangle:
A scalene triangle is a triangle in which all three sides have unequal lengths and angles of such a triangle are also unequal.
The above picture depicts that all sides of this triangle are unequal and it also has unequal angles, that is AB \(\neq\) BC \(\neq\) CA and
∠ABC \(\neq\) ∠BCA \(\neq\) ∠CAB.
Rectangle:
The rectangle is a four-sided polygon in which all the angles are right angles. The difference between a rectangle and a square is that the adjacent sides of a rectangle are not equal, hence they are irregular.
From the above rectangle, we can clearly see that every corner of the rectangle has a right angle with equal opposite sides. Therefore, AB = CD and AC = BD
Irregular Hexagon:
An irregular hexagon is a polygon having six sides of unequal lengths and all the angles are unequal.
The irregular hexagon shown in the image is made up of unequal lengths and angles.
Solved Examples
Example 1:
If two angles of a triangle are 60° and 80°, what would be the third angle of the triangle? Also, determine whether it is a regular or irregular polygon.
Solution:
The sum of interior angles of a polygon with ‘n’ sides is S = (n − 2).180°
A triangle has three sides, that is, n = 3
S = (3 – 2) x 180°
= 180°
Let the measure if the third angle be \(x\)
60 + 80 + \(x\) = 180
140 + \(x\) = 180
\(x\) = 180 – 140
\(x\) = 40°
So, the third side has an angle of 40°.
Since all the angles of the triangle are unequal, it is a scalene triangle and thus an irregular polygon.
Example 2:
Sort the polygons below into regular and irregular polygons.
Solution:
Regular Polygons:
Reason: Fig (a), All the sides in the given polygon are equal. Hence, it is a regular polygon.
Reason: Fig (b), The given octagon is a regular polygon as all the angles in it are equal.
Irregular Polygons:
Reason: Fig (d), Here, the polygon is irregular as it has unequal angles.
Reason: Fig (e), The sides of the given polygon are of different lengths. Therefore, it is an irregular polygon.
Reason: Fig (c), All the angles in the above polygon are different. Thus, we can say it as an irregular polygon.
Example 3:
Jack owns a pool that is shaped like a regular polygon. The sum of all the angles of the pool equals 720°. Can you determine the pool’s total number of sides and interior angle measure of the pool?
Solution:
The sum of all the angles of the pool = 720°
We already know, the formula for sum of interior angles of a polygon
S = (n − 2) x 180°
720 = (n − 2) x 180°
\(\frac{720}{180}\) = (n − 2)
4 = (n − 2)
4 + 2 = n
n = 6
Hence, the pool is in the shape of a hexagon as it has 6 sides.
As stated, the swimming pool is a regular polygon. That is, each angle will have an equal measure.
Then, the angle of each side = \(\frac{\text{Sum of all the angles}}{\text{number of sides}}\)
= \(\frac{720}{6}\)
= 120°
Each angle of the hexagon is 120°.
A regular polygon has sides of equal lengths and angles of equal measure, but an irregular polygon has sides of unequal lengths and unequal angles.
A quadrilateral is a four-sided polygon whose inner angles add up to 360°.
Both a square and a rhombus have equal-length sides. All the angles in a square measure 90 degrees whereas all the angles of a rhombus are not equal to 90 degrees.
No, as a trapezoid has sides of unequal lengths and angles, it is said to be an irregular polygon.