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°.
