How to work out how many sides a polygon has

The size of each interior angle of a regular polygon is 11 times the size of each exterior angle. Work out how many sides the polygon has.

Polygons in Maths consist of straight lines (sides) and angles, measured in degrees. An interesting aspect of these shapes for students is the proportional relationship between their interior and exterior angles. Understanding this relationship is crucial in Maths and is an integral part of student revision. 

In this article, we will explore the relationship between the angles of a regular polygon and how it can be a helpful tool for students in their Maths revision.

How To Calculate Each Angle

The internal angle is inside the shape, while the external angle lies outside it. For every regular polygon, the sum of all internal angles equals (n-2 ) multiplied by 180 degrees, where n represents the number of sides. This formula is key in working out the measure of each internal angle.

When dealing with these shapes, it's essential to understand the relationship between inside and outside angles. To determine the size of each internal angle, we first need to know the number of sides or vertices of the shape in question. 

For any regular polygon, the size of each interior angle can be calculated using the formula: 180(n-2)/n degrees, where n represents the number of sides. This formula is derived by dividing the sum of all internal angles in a polygon by the number of sides. 

On the other hand, the size of each outside angle of a regular polygon is equal to 360/n degrees.

According to the given problem, the size of each interior angle of a regular polygon equals 11 times the size of each exterior angle, which can be expressed as 11(360/n) degrees. Using the formula for the size of each interior angle, which is 180(n-2)/n, we can equate the two angles to form an equation, i.e., 180(n-2)/n = 11(360/n).

Now we need to simplify the equation by eliminating the denominator. This can be done by multiplying both sides of the equation by n, which results in 180n - 360 = 3960. We can then rearrange the equation, giving us 180n = 360 + 3960. Finally, by solving for n, we get 24. Therefore, this shape has 24 sides.

Conclusion

Figuring out the number of sides of a regular polygon involves grasping the proportional connection between its interior and exterior angles. Every Maths student should be familiar with applying this relationship to study and find the number of sides, especially when the size of each interior angle is 11 times the size of each exterior angle.

In summary:

• If each interior angle equals 11 times each exterior angle, the equation is 180°(n - 2) / n = 11 × (360° / n).

• Multiply both sides by n to eliminate the denominator.

• Rearrange to form 180n = 360 + 3960.

• Solve for n to find the number of sides.

A profound understanding of angle relationships can help solve any challenging questions around polygons. Therefore, as a student, it's essential to master this relationship as a vital part of your Maths study and revision.