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Irregular Polygons – Definition, Examples, Angles and Area

Camille Ira B. Mendoza

reviewed by Jessica Kaminski

Updated on October 14, 2026

An irregular polygon is a polygon whose sides or angles are not all equal. A rectangle (equal angles, different side lengths), a rhombus that is not a square (equal sides, different angles), and a scalene triangle are all irregular polygons.

Below, you’ll find the main types of irregular polygons, their angle and diagonal formulas, how to find their perimeter and area, and practice problems with answers. These shapes come up again and again in math class, from naming shapes in elementary school to finding areas in middle school, and an online math tutor can help if your child gets stuck along the way.

What are irregular polygons?

Irregular polygons are a type of polygon where not all sides or angles are equal. Every polygon is a closed 2D shape with at least 3 straight sides, so triangles, rectangles, pentagons, and hexagons can all be irregular when their sides or angles differ.

A polygon only needs one side or one angle that is different from the others to count as irregular. That is why most shapes you see in everyday life, like a door, a kite, or the side of a house with a pointed roof, are irregular polygons.

Irregular shapes examples

Definition of irregular polygons

An irregular polygon (also called a non-regular polygon) is a polygon in which the sides, the angles, or both are not all equal. A regular polygon is the opposite: all of its sides and all of its angles are equal, like a square or an equilateral triangle.

A shape needs only one of the two to be uneven. A rectangle has four equal 90° angles, but its sides are not all the same length, so it is an irregular polygon. A rhombus that is not a square has four equal sides, but its angles are different, so it is irregular too.

To tell the two apart, check both things: are all the sides the same length, and are all the angles the same size? If the answer to either question is “no,” the polygon is irregular.

Types of irregular polygons

• Irregular triangles: Any triangle that is not equilateral is irregular. Scalene triangles (no equal sides) and isosceles triangles (two equal sides) both belong here.
• Irregular quadrilaterals: These have four sides and four angles that are not all equal. Rectangles, kites, trapezoids, and scalene quadrilaterals (all four sides different) are all irregular quadrilaterals.
• Irregular pentagon: The irregular pentagon is a shape with 5 sides and angles that will not all measure the same. This irregular 5-sided shape may not look like a traditional pentagon, because its five sides can form many different shapes.
• Irregular hexagon: This type of irregular polygon differs from a regular hexagon because its six sides or angles are not all the same size.

Formula of irregular polygons

Regular polygons have ready-made formulas for their area because every side and angle is the same. Irregular polygons don’t have one standard area formula, because their sides and angles can change from shape to shape.

So, how do you find the area of an irregular polygon? The most common way is to split it into simpler shapes that do have formulas, such as triangles, rectangles, and squares. For example, a house-shaped irregular pentagon can be split into a square and a triangle. The perimeter is even easier: just add up the lengths of all the sides.

Properties of irregular polygons

Irregular polygons don’t follow the “all equal” rule of regular polygons, but they still share a few properties that always hold true.

Some of these properties are as follows:

• Irregular polygons can have sides of any length. Some sides can be the same length, or every side can be a different length.
• Their angles can be different or partly the same, but the sum of all interior angles always equals (n − 2) × 180°, where n is the number of sides. This is the same formula used for every polygon.
• You can calculate the number of diagonals in an irregular polygon using the formula n(n − 3) ÷ 2.

Differences between types of irregular polygons

All types of irregular polygons follow the properties above, but each one is unique. The differences between them lie in their shapes, number of sides, areas, and perimeters.

For example, when you think of a pentagon, you probably picture a shape with a flat base and a point at the top, like a house. But an irregular pentagon can also look like an arrow or a crown with two points. That’s the beauty of irregular polygons: they go far beyond the shapes we usually think of.

Equations for calculating properties of irregular polygons

There are equations for calculating some of the properties of irregular polygons. In each formula, n stands for the number of sides:

• The sum of interior angles: (n − 2) × 180°
• The number of diagonals: n(n − 3) ÷ 2
• The perimeter: add the lengths of all the sides

The area is the only one of these properties without a single formula. To find it, split the polygon into simpler shapes, find the area of each one, and then add the results together.

Let’s use the example of a house-shaped irregular pentagon that can be split into a square and a triangle. First, find the area of the square using the formula s², where s is the length of one side. Then, find the area of the triangle using the formula (base × height) ÷ 2. Finally, add the two areas together.

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Writing equations for irregular polygons

Writing equations for irregular polygons is easy when you know the formulas. Let’s take the example of finding the sum of the interior angles of an irregular 6-sided shape (hexagon). The equation is (6 − 2) × 180° = 720°.

It’s just as easy to write the equation for the number of diagonals. For the same irregular hexagon, the equation is 6(6 − 3) ÷ 2 = 9 diagonals.

For the area, you combine equations after splitting the shape. Take the house-shaped pentagon from above: a square with sides of 5 units and a triangle on top with a base of 5 units and a height of 3 units. The equation is 5² + (5 × 3) ÷ 2 = 25 + 7.5 = 32.5 square units.

Practice problems on irregular polygons

Now, it’s your turn! Try the problems below to see how well you understand irregular polygons. Write down your answers, and use a sheet of paper to draw the shapes or show your work if you need to.

1. You have an irregular quadrilateral with side lengths of 3 cm, 5 cm, 7 cm, and 9 cm. What is the perimeter of this irregular quadrilateral?
2. You have an irregular pentagon with the interior angles of 100°, 120°, 110°, 130°, and 80°. Do these angles follow the formula that determines the sum of interior angles in irregular polygons?

Answers:

1. 3 + 5 + 7 + 9 = 24, so the perimeter is 24 cm.
2. Yes. 100° + 120° + 110° + 130° + 80° = 540°, and (5 − 2) × 180° = 540°, so the angles follow the formula.

Conclusion

Here, we’ve taken you through the definition of an irregular polygon, the properties of these shapes, and the equations associated with them. We also shared some irregular polygon examples so you can see for yourself how different irregular polygons can be. Now, you should be confident in identifying and working with them in math class!

Finding the area of an irregular polygon might seem tricky at first, but once you split the shape into simpler parts, it becomes a step-by-step task. Keep practicing, and you’ll be ready for any irregular polygon problem in your geometry lessons.

Frequently asked questions on irregular polygons

What are irregular shapes?

Irregular shapes are shapes whose sides or angles are not all equal. Among polygons, this includes irregular triangles, irregular quadrilaterals, irregular pentagons, and irregular hexagons.

What is an irregular polygon?

An irregular polygon is a 2D shape with 3 or more straight sides whose sides or angles are not all equal. It can have some equal sides and some different ones, or every side can be a different length. A rectangle is an irregular polygon because its sides are not all the same length, even though all its angles are 90°.

How to find the area of irregular polygons?

There is no single formula for the area of an irregular polygon because these shapes vary so much. To find the area, divide the shape into simpler shapes, such as triangles, squares, and rectangles, find the area of each part, and add them together.

What is the difference between regular and irregular polygons?

In a regular polygon, all sides and all angles are equal. In an irregular polygon, the sides, the angles, or both are not all equal. For example, a square is regular, while a rectangle (equal angles, different sides) and a rhombus that is not a square (equal sides, different angles) are irregular.

Camille Ira B. Mendoza

56 articles

I am a seasoned math tutor with over seven years of experience in the field. Holding a Master’s Degree in Education, I take great joy in nurturing young math enthusiasts, regardless of their age, grade, and skill level. Beyond teaching, I am passionate about spending time with my family, reading, and watching movies. My background also includes knowledge in child psychology, which aids in delivering personalized and effective teaching strategies.