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A polygon is a closed shape that is made up of at least three or more straight lines and angles. There exists a relationship between the number of sides, the number of angles, and the values of angles in a polygon. Learn a general formula to find the sum of all interior angles of a polygon....Read MoreRead Less

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 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 \(\frac{(n~-~2)\times 180}{n}\)

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.

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.

**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.

Frequently Asked Questions on Polygon Angles

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.