What is a polygon?
Polygons are one of the most well-known geometric shapes. A polygon is a closed shape that does not intersect itself and has straight sides connected end-to-end. Polygons consist of only straight lines and have no curves.
The properties of polygons
The properties of polygons are determined by their sides and angles.
- The sum of the interior angles of an n-sided polygon is (n – 2) × \(180^\circ\) .
- There are \(\frac{n~(n~- ~3)}{2}\) diagonals in a polygon with n sides.
- The interior angle of a n-sided regular polygon is measured as \(\frac{(n~-~2)\times 180}{n}\)
What is a regular polygon?
A regular polygon is one that has all of its sides and interior angles equal. Regular polygons include squares, equilateral triangles, regular hexagons, and others.
Measurement of the interior angles of a polygon
The sum of a polygon’s interior angles is always a constant value. The sum of the interior angles of a polygon is the same, whether it is regular or irregular. As a result, the formula for the sum of the polygon’s interior angles is:
S = (n – 2) X \(180^\circ\) .
Where S is the sum of the interior angles and n is the number of sides of the polygon.
Solved Polygon Angles Examples
Example 1:
Calculate the interior angles of the convex polygon shown below.
Solution:
The polygon has eight sides. Calculate the total measurement of the interior angles.
S = (n – 2)\(\times\) \(180^\circ\) Formula for the sum of the interior angles of a polygon.
= (8 -2)\(\times\)\(180^\circ\) Substitute 8 for n.
= 6\(\times\)\(180^\circ\) Subtract.
= \(1080^\circ\) Multiply.
The sum of the interior angles is 1080 degrees.
Example 2:
Calculate the value of z.
Solution:
Step 1: A polygon has seven sides. Calculate the sum of its interior angles.
S = (n – 2) \(\times\)\(180^\circ\) Write the formula.
= (7 – 2)\(\times\) \(180^\circ\) Substitute 7 for n.
= \(900^\circ\) Simplify.
The sum of the interior angles is \(900^\circ\).
Step 2: Write an equation and solve it.
125 + 110 + 95 + 170 + 155 + 115 + z = 900
770 + z = 900
z = 900 – 770 = 130
Hence, the value of z is \(130^\circ\).
Example 3:
A regular octagonal swimming pool is built by a company. Calculate the measure of each of the octagon’s interior angles.
Solution:
An octagon has eight sides. Calculate the sum of the interior angles .
S = (n – 2)\(\times\) \(180^\circ\) Write the formula.
= (8 – 2)\(\times\) \(180^\circ\) Substitute 8 for n.
= 6\(\times\)\(180^\circ\) Subtract.
= \(1080^\circ\) Multiply.
The sum of the interior angles is 1080 degrees.
Each interior angle of a regular polygon is congruent. Therefore, divide the sum of the interior angles by the number of interior angles.
\(1080^\circ\) ÷ 8 = \(135^\circ\)
Each interior angle is 135 degrees.
n-gon is the name for an n-sided polygon. For example, a 16-gon is a polygon with 16 sides.
Yes. A regular polygon is a shape that has all of its sides and all of its angles are equal. A square has four equal side lengths, and all its angles are equal to 90 degrees.
A circle is not a polygon. Since a circle is curved, it can’t be made out of line segments. Thus, circles don’t meet the criteria for being a polygon.
Convex polygons are those that have all their interior angles less than 180 degrees.