What Is the Sum of Interior Angles in a Polygon Formula with Examples
Knowing the Sum Of Angles In A Polygon is crucial for maths exams and geometric problem-solving. This concept helps students calculate angles in various shapes, spot patterns, and avoid common errors. Mastering it supports quick thinking for board or competitive exams and builds confidence working with polygons and complex figures.
Formula Used in Sum Of Angles In A Polygon
The standard formula is: \( (n - 2) \times 180^\circ \), where n is the number of sides in the polygon.
Here’s a helpful table to understand Sum Of Angles In A Polygon more clearly:
Sum Of Angles In A Polygon Table
| Polygon | Number of Sides (n) | Sum of Interior Angles |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| Heptagon | 7 | 900° |
| Octagon | 8 | 1080° |
This table shows how the pattern of the sum of angles in a polygon grows as the number of sides increases. It’s very useful for quick revision and in competitive exams.
Worked Example – Solving a Problem
Let’s find the sum of the interior angles of a hexagon using the standard formula and then solve for a missing angle.
1. Identify the number of sides in a hexagon:2. Apply the formula for the sum of interior angles:
3. Multiply:
4. If five of the interior angles are each 120°, find the sixth angle.
Thus, each interior angle of a regular hexagon is 120°, and the total sum matches the table above. Always show each step clearly in exams to avoid mistakes!
Practice Problems
- What is the sum of the interior angles of a decagon?
- A quadrilateral has angles of 90°, 85°, and 95°. Find the fourth angle.
- How many sides does a polygon have if the sum of its interior angles is 1260°?
Common Mistakes to Avoid
- Using \( n + 2 \) instead of the correct formula \( (n - 2) \times 180^\circ \).
- Mixing up interior and exterior angles—remember, the sum of exterior angles in any polygon is always 360°.
- Forgetting to count all sides correctly; always check n before applying the formula!
- Not showing all working steps—this is important for getting full marks.
- Assuming all polygons are regular unless stated. For irregular polygons, angles can be different.
Real-World Applications
Knowing how to find the sum of angles in a polygon is essential in fields like architecture, design, and engineering. When creating floor plans, tiling, or making structures, correct angle calculation ensures accuracy and safety. Vedantu provides many practical examples and stepwise lessons so learners see how maths extends far beyond the classroom.
We explored the idea of Sum Of Angles In A Polygon, its formula, uses, and solving strategies. Understanding these concepts helps with geometric questions in exams and real-world tasks. For more detailed practice, visit Vedantu’s interactive resources and boost your maths confidence.
Want to learn more? Dive deeper into related topics such as Interior Angles of a Polygon, Exterior Angle Theorem, or review sum of angles in a triangle. If you're studying special polygon shapes, check out pentagons or angle sum property of quadrilaterals for focused examples.
FAQs on Sum of Angles in a Polygon Explained with Formula and Proof
1. What is the sum of angles in a polygon?
The sum of angles in a polygon is given by the formula (n − 2) × 180°, where n is the number of sides.
- This formula works for any polygon with 3 or more sides.
- It calculates the total of all interior angles.
- Example: For a pentagon (n = 5), sum = (5 − 2) × 180° = 540°.
2. What is the formula for the sum of interior angles of a polygon?
The formula for the sum of interior angles of a polygon is (n − 2) × 180°.
- n represents the number of sides (or vertices).
- The formula comes from dividing the polygon into triangles.
- Each triangle contributes 180° to the total.
3. How do you find the sum of angles in a regular polygon?
To find the sum of angles in a regular polygon, use the formula (n − 2) × 180°.
- Step 1: Count the number of sides (n).
- Step 2: Substitute into the formula.
- Example: A hexagon (n = 6) → (6 − 2) × 180° = 720°.
4. What is the sum of interior angles of a triangle, quadrilateral, and pentagon?
The sums are 180° for a triangle, 360° for a quadrilateral, and 540° for a pentagon.
- Triangle: (3 − 2) × 180° = 180°
- Quadrilateral: (4 − 2) × 180° = 360°
- Pentagon: (5 − 2) × 180° = 540°
5. Why is the sum of interior angles of a polygon (n − 2) × 180°?
The formula (n − 2) × 180° works because any polygon can be divided into (n − 2) triangles.
- Each triangle has an angle sum of 180°.
- Multiplying (n − 2) by 180° gives the total interior angle sum.
- This method works for all simple polygons.
6. How do you find each interior angle of a regular polygon?
Each interior angle of a regular polygon is found using [(n − 2) × 180°] ÷ n.
- First calculate the total interior angle sum.
- Then divide by the number of sides (n).
- Example: For a regular hexagon → 720° ÷ 6 = 120°.
7. What is the sum of exterior angles of a polygon?
The sum of exterior angles of any polygon is always 360°.
- This is true for all polygons, regular or irregular.
- One exterior angle is taken at each vertex.
- The angles are measured in the same direction around the shape.
8. What is the difference between interior and exterior angles of a polygon?
An interior angle is inside a polygon, while an exterior angle is formed outside between a side and its extension.
- Interior angle sum = (n − 2) × 180°.
- Exterior angle sum = 360°.
- Interior angle + exterior angle at a vertex = 180° (linear pair).
9. How many sides does a polygon have if the sum of interior angles is 900°?
A polygon with an interior angle sum of 900° has 7 sides.
- Use the formula: (n − 2) × 180° = 900°
- Divide both sides by 180° → n − 2 = 5
- Add 2 → n = 7
10. Can you give an example of finding the sum of angles in a polygon?
Yes, for an octagon (8 sides), the sum of interior angles is 1080°.
- Use the formula: (n − 2) × 180°
- Substitute n = 8 → (8 − 2) × 180°
- 6 × 180° = 1080°
