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Polygons (Interior/Exterior Angles)

Introduction

Polygons are closed plane figures with three or more straight sides. In aptitude and geometry problems, questions often involve finding the measure of interior or exterior angles and the number of diagonals of regular polygons.

Understanding angle relationships helps solve geometry and pattern-based reasoning questions quickly.

Pattern: Polygons (Interior/Exterior Angles)

Pattern

The key idea is that all interior and exterior angles of a polygon are related to its number of sides (n).

Key Formulas:

  • Sum of interior angles: (n - 2) × 180°
  • Each interior angle (regular polygon): [(n - 2) × 180°] ÷ n
  • Each exterior angle (regular polygon): 360° ÷ n
  • Number of diagonals: n(n - 3) ÷ 2

Step-by-Step Example

Question

Find the measure of each interior angle of a regular hexagon.

Solution

  1. Step 1: Identify number of sides (n).

    For a hexagon, n = 6.

  2. Step 2: Use interior angle formula.

    Each interior angle = [(n - 2) × 180°] ÷ n

  3. Step 3: Substitute and compute.

    Each interior angle = [(6 - 2) × 180°] ÷ 6 = (4 × 180°) ÷ 6 = 720° ÷ 6 = 120°.

  4. Final Answer:

    Each interior angle = 120°

  5. Quick Check:

    Exterior angle = 360° ÷ 6 = 60°, and 180° - 60° = 120° ✅

Quick Variations

1. Find the number of sides when each interior/exterior angle is known.

2. Compute sum of all interior or exterior angles.

3. Find number of diagonals for any polygon.

4. Compare interior and exterior angles of two polygons.

Trick to Always Use

  • Step 1 → Identify whether the polygon is regular (equal sides/angles) or irregular.
  • Step 2 → Use the appropriate formula for interior or exterior angles.
  • Step 3 → Remember that interior + exterior = 180° for regular polygons.

Summary

Summary

In Polygons (Interior/Exterior Angles):

  • Sum of interior angles = (n - 2) × 180°.
  • Each exterior angle = 360° ÷ n.
  • Interior + exterior = 180° (for regular polygons).
  • Number of diagonals = n(n - 3) ÷ 2.

These relationships form the foundation for many geometry and aptitude questions.

Practice

(1/5)
1. Find the sum of all interior angles of a pentagon.
easy
A. 540°
B. 360°
C. 720°
D. 900°

Solution

  1. Step 1: Recall formula for sum of interior angles.

    Sum = (n - 2) × 180°.

  2. Step 2: Substitute n = 5.

    Sum = (5 - 2) × 180° = 3 × 180°.

  3. Step 3: Compute.

    Sum = 540°.

  4. Final Answer:

    Sum of all interior angles = 540° → Option A.

  5. Quick Check:

    Each interior ≈ 108°; 108 × 5 = 540° ✅
Hint: Subtract 2 from number of sides and multiply by 180°.
Common Mistakes: Using n × 180° instead of (n - 2) × 180°.
2. Each exterior angle of a regular polygon is 60°. Find the number of sides of the polygon.
easy
A. 4
B. 6
C. 5
D. 8

Solution

  1. Step 1: Recall relation for regular polygon.

    Each exterior angle = 360° ÷ n.

  2. Step 2: Substitute given value.

    60 = 360 ÷ n.

  3. Step 3: Rearrange.

    n = 360 ÷ 60 = 6.

  4. Final Answer:

    Number of sides = 6 → Option B.

  5. Quick Check:

    360 ÷ 6 = 60° ✅
Hint: n = 360 ÷ each exterior angle.
Common Mistakes: Using 180° instead of 360° for total exterior sum.
3. Find each interior angle of a regular octagon.
easy
A. 108°
B. 120°
C. 135°
D. 150°

Solution

  1. Step 1: Use formula for interior angle.

    Each interior = [(n - 2) × 180°] ÷ n.

  2. Step 2: Substitute n = 8.

    Each interior = [(8 - 2) × 180°] ÷ 8 = (6 × 180°) ÷ 8.

  3. Step 3: Simplify.

    1080° ÷ 8 = 135°.

  4. Final Answer:

    Each interior = 135° → Option C.

  5. Quick Check:

    360 ÷ 8 = 45° exterior; 180 - 45 = 135° ✅
Hint: Interior = 180° - exterior angle.
Common Mistakes: Multiplying by n instead of dividing.
4. Find the number of diagonals in a decagon.
medium
A. 35
B. 40
C. 45
D. 50

Solution

  1. Step 1: Recall formula for diagonals.

    Number of diagonals = n(n - 3)/2.

  2. Step 2: Substitute n = 10.

    Diagonals = 10(10 - 3)/2 = 10 × 7 / 2.

  3. Step 3: Simplify.

    70 ÷ 2 = 35.

  4. Final Answer:

    Number of diagonals = 35 → Option A.

  5. Quick Check:

    10 × 7 / 2 = 35 ✅
Hint: Use n(n - 3)/2 to find diagonals instantly.
Common Mistakes: Using (n - 2) × 180° instead of diagonal formula.
5. If the sum of all interior angles of a polygon is 1620°, find the number of sides.
medium
A. 9
B. 10
C. 12
D. 11

Solution

  1. Step 1: Recall formula.

    Sum = (n - 2) × 180°.

  2. Step 2: Substitute and rearrange.

    1620 = (n - 2) × 180 ⇒ n - 2 = 1620 ÷ 180 = 9.

  3. Step 3: Add 2 to find n.

    n = 9 + 2 = 11 sides.

  4. Final Answer:

    Polygon has 11 sides → Option D.

  5. Quick Check:

    (11 - 2) × 180 = 9 × 180 = 1620 ✅
Hint: n = (Sum ÷ 180) + 2.
Common Mistakes: Dividing by 90 instead of 180.

Mock Test

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