How to Find the Sum of Interior Angles in Any Polygon?
The concept of interior angles of a polygon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding this concept helps students quickly determine angles in polygons and solve geometry problems efficiently.
What Is Interior Angle of a Polygon?
An interior angle of a polygon is the angle formed between two adjacent sides of a polygon that meets inside the shape. Polygons such as triangles, quadrilaterals, pentagons, and hexagons all have interior angles. You’ll find this concept applied in areas such as types of angles, geometry problem-solving, and calculating properties of regular and irregular polygons.
Key Formula for Interior Angles of a Polygon
Here’s the standard formula: \( \text{Sum of Interior Angles} = (n - 2) \times 180° \), where ‘n’ is the number of sides of the polygon. For a regular polygon (all sides and angles equal), each interior angle is \( \dfrac{(n - 2) \times 180°}{n} \).
Polygon Types and Angle Table
| Polygon | Number of Sides (n) | Sum of Interior Angles | Each Interior Angle (Regular) |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Step-by-Step Illustration: Calculating Interior Angles
- Identify the number of sides (n) of the polygon.
Example: Pentagon, n = 5. - Apply the sum formula:
Sum = (n - 2) × 180° = (5 - 2) × 180° = 540°. - If regular, divide the sum by n for each angle:
Each interior angle = 540° ÷ 5 = 108°.
Speed Trick or Vedic Shortcut
To save time in exams, remember:
Shortcut: For any polygon with ‘n’ sides, interior angle sum = (n – 2) × 180°. For each angle in a regular polygon, just divide the sum by n.
For polygons with large n, practice multiplying quickly, e.g. for n = 12, (12 – 2) × 180 = 1800°.
Real-Life and Exam Application
Whether you’re designing tiles (tessellation), solving geometry questions, or checking building corners, knowing how to calculate the interior angles of a polygon is essential. This topic is tested in boards, JEE Main, and olympiads. Practicing problem-sets with stepwise solutions greatly increases speed and accuracy.
Exterior Angles vs. Interior Angles
| Aspect | Interior Angles | Exterior Angles |
|---|---|---|
| Definition | Angle inside the polygon | Angle formed by extending one side at a vertex |
| Sum for n Sides | (n-2) × 180° | 360° (always) |
| Each Angle (Regular) | \((n-2) × 180°/n\) | \(360°/n\) |
Stepwise Example Solution
Question: What is the measure of each interior angle in a regular octagon?
1. Number of sides \(n = 8\)2. Find the sum: \((8 - 2) × 180° = 6 × 180° = 1080°\)
3. Each angle: \(1080° ÷ 8 = 135°\)
Answer: Each angle is 135°.
Try These Yourself
- Find the sum of interior angles of a hexagon.
- If each interior angle of a regular polygon is 150°, how many sides does it have?
- What is the measure of each interior angle in a regular dodecagon (12 sides)?
- Show stepwise calculation for the sum of angles in a pentagon.
Frequent Errors and Misunderstandings
- Confusing sum of interior angles (depends on n) and sum of exterior angles (always 360°).
- Forgetting to use brackets: (n-2) × 180°, not n-2 × 180°.
- Mixing regular and irregular polygons when asked for each interior angle.
Relation to Other Concepts
The idea of interior angles of a polygon connects closely with triangle properties (since polygons are divided into triangles for the formula), and with quadrilateral angle sum property. Mastering this helps with more advanced geometry and coordinate geometry topics in higher classes.
Classroom Tip
A quick way to remember: "For any polygon, subtract 2 from the number of sides, and multiply by 180° to get the angle sum." Vedantu’s teachers often show polygons visually—drawing a triangle or quadrilateral, dividing them into triangles—for instant recall of the formula.
We explored interior angles of a polygon—definition, formula, tables, tricks, examples, and common errors. Continue practicing with Vedantu and use the stepwise approach to boost confidence in all geometry topics.
Helpful Study Links
FAQs on Understanding Interior Angles of a Polygon: Formula, Calculation & Examples
1. What is the formula for the sum of interior angles of a polygon?
The sum of the interior angles of a polygon with n sides is calculated using the formula: (n - 2) × 180°. This formula is derived by dividing the polygon into triangles. A triangle has 180°, and you can always divide an n-sided polygon into (n-2) triangles.
2. How do you find each interior angle in a regular polygon?
For a regular polygon (where all sides and angles are equal), find the sum of the interior angles using the formula (n - 2) × 180°, then divide by the number of sides (n) to find the measure of each interior angle: [(n - 2) × 180°] / n.
3. Do all polygons' interior angles add up to 360°?
No. Only quadrilaterals (four-sided polygons) have interior angles that add up to 360°. The sum of interior angles for other polygons depends on the number of sides; use the formula (n - 2) × 180° to calculate the sum.
4. How many degrees are in the interior angles of a pentagon?
A pentagon has five sides (n=5). Using the formula (n - 2) × 180°, the sum of its interior angles is (5 - 2) × 180° = 540°.
5. What is the difference between interior and exterior angles?
Interior angles are the angles formed inside a polygon by its sides. Exterior angles are formed by extending one side of the polygon and measuring the angle between the extension and the adjacent side. The sum of exterior angles of any polygon is always 360°.
6. How is the polygon angle sum formula derived from triangles?
Any polygon can be divided into triangles by drawing diagonals from one vertex to all other non-adjacent vertices. The number of triangles formed is always two less than the number of sides (n-2). Since each triangle has 180°, the total sum is (n - 2) × 180°.
7. Can the sum of interior angles be a non-integer value?
No, the sum of interior angles will always be an integer value because it's calculated by multiplying integers (number of sides minus two and 180).
8. For an irregular polygon, does this formula still hold?
Yes, the formula (n - 2) × 180° applies to both regular and irregular polygons. It gives the total sum of interior angles, although the individual angles will differ in an irregular polygon.
9. Why does the exterior angle sum always remain 360°?
Imagine walking around the perimeter of a polygon. At each vertex, you turn through an exterior angle. By the time you return to your starting point, you've made a complete 360° turn.
10. How do interior angles relate to tessellation and tiling problems?
In tessellations, shapes fit together without gaps or overlaps. The interior angles of the shapes used in tessellations must add up to 360° at each vertex point to create a complete tiling.
11. What is the measure of each interior angle of a regular hexagon?
A hexagon has six sides (n=6). Using the formula for each angle in a regular polygon, [(n - 2) × 180°] / n, we get [(6 - 2) × 180°] / 6 = 120°.
12. How can I find the number of sides of a polygon if I know the sum of its interior angles?
Let 'S' be the sum of the interior angles. Use the formula (n - 2) × 180° = S, then solve for 'n' (the number of sides): n = (S / 180°) + 2
