Formula to Find Interior Angles of Any Polygon 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 How to Calculate the Sum of Interior Angles in a Polygon
1. How to find the sum of angles of a polygon?
The sum of interior angles of a polygon can be calculated using the formula: Sum = (n − 2) × 180°, where n is the number of sides of the polygon. Simply count how many sides the polygon has, subtract 2, and multiply the result by 180 degrees.
2. Do all angles in a polygon add up to 180°?
No, not all polygons have angles that add up to 180°. Only a triangle (which has 3 sides) has its interior angles summing up to 180°. For polygons with more sides, the sum increases according to the formula: (n − 2) × 180°, where n is the number of sides.
3. Do all angles in a polygon add up to 360°?
No, not all polygons have a sum of interior angles equal to 360°. Only a quadrilateral (4 sides) has interior angles that add up to 360°. Other polygons will have a different sum calculated by (n − 2) × 180°.
4. How to calculate the number of angles in a polygon?
The number of interior angles in a polygon is always equal to its number of sides. For example, a pentagon (5 sides) has 5 interior angles.
5. What is the formula for the sum of interior angles in a polygon?
The formula for sum of interior angles in a polygon is (n − 2) × 180°, where n is the number of sides of the polygon.
6. What is the sum of exterior angles of a polygon?
The sum of exterior angles of any polygon (one at each vertex) is always 360°, regardless of the number of sides.
7. How to calculate each interior angle of a regular polygon?
For a regular polygon, all interior angles are equal. Each interior angle = [(n − 2) × 180°] ÷ n, where n is the number of sides.
8. What is the sum of angles in a convex polygon?
The sum of interior angles of a convex polygon is given by the same formula: (n − 2) × 180°, where n is the number of sides.
9. What is the sum of individual exterior angles of a regular polygon?
Each exterior angle of a regular polygon is calculated as 360° ÷ n, where n is the number of sides, and the sum of all such angles is 360°.
10. What is the sum of the interior and exterior angles at each vertex in a polygon?
The interior and exterior angles at each vertex of any polygon always add up to 180°.
11. Can a polygon have interior angles less than 180°?
Yes, in a convex polygon, all interior angles are less than 180°. If any angle is greater than 180°, the polygon is called a concave polygon.
12. What is the sum of angles in a triangle, quadrilateral, and pentagon?
The sum of interior angles is 180° for a triangle, 360° for a quadrilateral, and 540° for a pentagon.
