Interior and Exterior Angles of Polygons

What you’ll learn

Calculate angle sums and use regularity only when justified.

គណនាផលបូកមុំពហុកោណ និងប្រើភាពនិយ័តតាមលក្ខខណ្ឌ។

Meaning and use

A simple convex polygon with n sides can be divided into n − 2 triangles. The interior-angle sum follows from triangle sums. A regular polygon has equal angles, so divide the sum by n only in that case. One consistently chosen exterior turn at each vertex totals 360 degrees.

ពហុកោណប៉ោងមាន n ជ្រុង អាចចែកជា n−2 ត្រីកោណ។ ពហុកោណនិយ័តមានមុំស្មើគ្នា ទើបអាចចែកផលបូកមុំតាមចំនួនជ្រុង។

Formulas

Interior sum

\[S=(n-2)180^\circ\]

Regular polygon angles

\[I=\frac{(n-2)180^\circ}{n},\qquad E=\frac{360^\circ}{n}\]

Symbols and units

Symbol	Meaning	Unit or restriction
n	Number of sides	Integer at least 3
S	Sum of interior angles	Degrees
I, E	One interior and adjacent exterior angle in a regular polygon	Degrees

Conditions and limits

• The individual-angle formulas require a regular polygon.
• Use simple convex polygons for the basic turning interpretation here.
• មុំក្នុង និងមុំក្រៅជាប់គ្នាបំពេញ 180 ដឺក្រេ។

Worked examples

Interior sum of a hexagon

A convex hexagon has n = 6.

1. Use \(S=(6-2)180^\circ\).
2. Calculate \(S=720^\circ\).

720°

ផលបូកមុំមិនទាមទារថាពហុកោណនិយ័តទេ។

A regular pentagon

Find one interior and one exterior angle.

1. Interior \(I=(5-2)180^\circ/5=108^\circ\).
2. Exterior \(E=360^\circ/5=72^\circ\).

Interior 108°; exterior 72°

ព្រោះនិយ័ត មុំទាំងអស់ស្មើគ្នា។

Common mistakes

Instead of	Use	Why
Every hexagon has each interior angle 120°.	A regular hexagon has each interior angle 120°.	Equal individual angles require regularity.
The exterior-angle total is (n − 2)180°.	The consistently chosen exterior turns total 360°.	Do not confuse interior sums with turning sums.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Find the interior sum of an octagon.
2. Find one exterior angle of a regular hexagon.
3. A regular polygon has exterior angle 45°. How many sides?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 1080°

   (8 − 2) × 180 = 1080.

2. 60°

   360/6 = 60.

3. 8

   n = 360/45.

Quick recap

• Distinguish a sum from one angle.
• Regularity is an extra condition.
• Interior and adjacent exterior angles sum to 180°.