Calculating the interior and exterior angles of a polygon

Let us begin with the basics, shall we? "What are Polygons?" The question may seem tough, but it can be simplified to make more sense. Polygons are a fascinating sight, but a bit complex as you move onto the shapes with more than six sides.

Find Maths Tutors

It doesn't matter if you're up against a simple triangle or a more complex decagon, a maths tutor can guide you in calculating the interior and exterior angles of a polygon, making the process as smooth as water flowing from a clean pond.

Understanding Polygons

Polygons are two-dimensional (2D) shapes having straight sides. A triangle is the only polygon that has 3 sides. If you keep adding more sides to a polygon, you get shapes such as quadrilaterals (4 sides), pentagons (5 sides), and so on.

There are two crucial angles associated with polygons:

• Interior Angles: These are found inside the polygon, and are formed by two adjacent sides.
• Exterior Angles: As the name suggests, these angles are found outside the polygon when you extend one of its sides. 

Related articles Why do so many students need maths tuition?

Maths is often seen as the nemesis of many primary and secondary school students, who are forced to take private lessons and maths tuition. This complex, yet fascinating subject 'haunts' students unt...

How to Calculate Interior Angles

You can easily calculate the sum of these angles if you know the number of sides the polygon has. Here's a simple formula:

Sum of interior angles= (n - 2) x 180∘

For example, let's say you have:

• A quadrilateral (4 sides): 4 sides? That means n = 2 right?

Let's input the value in the above formula; 

Sum of Interior Angles = (4 - 2) x 180∘ = 360∘

• A pentagon (5 sides):  Sum of Interior Angles = (5 - 2) x 180∘ = 540∘

An octagon (8 sides): Sum of Interior Angles = (8 - 2) x 180∘ = 1080∘

Now, if you need to find the measure of each interior angle in a polygon (applies where all angles are equal), divide the sum of interior angles by the number of sides, like this:

Each interior angle = { (n - 2) x 180∘ } / n

For example,

You need to find each interior of an octagon (8 sides ---> n = 6 i.e.):

Each interior angle = { (n - 2) x 180∘ } / n

= { (8 - 2) x 180∘ } / 8

= 1080∘ / 8 

= 135∘

Now, let's discuss how we can calculate the exterior angles of any polygon.

Calculating Exterior Angles

These angles are formed when you extend one side of the polygon at each vertex. Let me tell you an interesting fact, the sum of the exterior angles of any polygon is always 360∘, irrespective of the sides it has. 

Here's the formula to find the measure of each exterior angle:

Each exterior angle = 360∘ / n

Let's calculate for a decagon (ooo a complex figure):

Remember, a decagon has 10 sides so n = 2.

so,

Each exterior angle = 360∘ / 10

= 36∘

Summing It All Up

• Sum of interior angles= (n - 2) x 180∘
• Each interior angle = { (n - 2) x 180∘ } / n
• Each exterior angle = 360∘ / n

Find Geometry Tutors

Why Do These Calculations Matter?

Understanding how to calculate the interior and exterior angles of a polygon is not only important for geometry but also for multiple real-world applications. For example, a professional architect or an engineer uses these calculations to design structures to ensure each angle and the sides meet precisely. For robotics and computer graphics (You may have a knack for these professions), knowing these angles allows for more accurate models.