Why Regular and Irregular Polygons Have Different Angle Formulas?

Explanation

In this lesson, we explain why regular and irregular polygons use different methods when finding missing interior angles.

Both methods come from the same original formula:

Sum of Interior Angles \(= (n − 2) × 180^\circ\)

where:

\(n\) is the number of sides of the polygon.

The difference is how we use the total sum.

Regular Polygons

A regular polygon has all sides equal and all angles equal.

So, if the polygon is regular, all the angles have the same value.

That is why we divide the total sum by the number of angles:

\(x = \frac{\text{Sum}}{n}\)

Irregular Polygons

An irregular polygon does not have all angles equal.

Usually, some angles are given, and one angle is missing.

So, we subtract the known angles from the total sum:

\(x = \text{Sum}− \text{Known Angles}\)

So, regular and irregular polygons do not really have completely different formulas. They both start with the same total sum formula, but we use the sum in different ways.

Formula / Rule

For any polygon:

Sum of Interior Angles \(= (n − 2) × 180^\circ\)

For a regular polygon:

\(x = \frac{\text{Sum}}{n}\)

or:

\(x = \frac{(n − 2) × 180^\circ}{n}\)

For an irregular polygon with one missing angle:

\(x = \text{Sum}− \text{Sum of Known Angles}\)

Examples

Example 1: Regular Triangle

Find each interior angle of a regular triangle.

A regular triangle has \(3\) equal angles.

So:

Sum \(= x + x + x\)

Sum \(= 3x\)

Divide both sides by \(3\):

\(x = \frac{\text{Sum}}{3}\)

A triangle has an interior angle sum of:

\(180^\circ\)

So:

\(x = \frac{180^\circ}{3}\)

\(x = 60^\circ\)

So, each interior angle of a regular triangle is:

\(60^\circ\)

Example 2: Irregular Triangle

Find the missing angle in a triangle with angles:

\(90^\circ, 40^\circ,\) and \(x\)

The sum of the interior angles of a triangle is:

\(180^\circ\)

So:

Sum \(= 90^\circ + 40^\circ + x\)

To find \(x\), subtract the known angles from the sum:

\(x = 180^\circ − (90^\circ\ + 40^\circ)\)

\(x = 180° − 130^\circ\)

\(x = 50^\circ\)

So, the missing angle is:

\(50^\circ\)

Video Explanation

Mulkek