Regular Polygon Angles

\( \text{Exterior}=\tfrac{360^{\circ}}{n},\qquad \text{Interior}=180^{\circ}-\tfrac{360^{\circ}}{n} \)
Geometry GCSE
Question 1 of 20
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A regular polygon has each interior angle 135°. How many sides does it have?

Hint (H)
\( Interior → Exterior = 45° → n=360/45 \)

Explanation

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Statement

In a regular polygon with \(n\) sides:

\[ \text{Exterior angle} = \frac{360^\circ}{n}, \qquad \text{Interior angle} = 180^\circ - \frac{360^\circ}{n}. \]

Exterior and interior angles are supplementary since they form a straight line at each vertex.

Why it’s true

  • The exterior angles of any polygon always sum to 360°.
  • In a regular polygon, all exterior angles are equal, so each is \(360/n\).
  • The interior angle is the supplement: \(180 - \text{exterior}\).

Recipe (how to use it)

  1. Count the number of sides \(n\).
  2. Divide 360° by \(n\) to get the exterior angle.
  3. Subtract from 180° to get the interior angle.

Spotting it

These formulas apply to regular polygons (all sides and angles equal). They are often used in GCSE problems about tessellations, polygon classification, or angle sums.

Common pairings

  • Tessellation problems (e.g. which polygons fit around a point).
  • Finding number of sides from an angle.
  • Angle sum of polygons.

Mini examples

  1. Square (\(n=4\)): Exterior = 360/4 = 90°, Interior = 180-90=90°.
  2. Pentagon (\(n=5\)): Exterior = 360/5 = 72°, Interior = 180-72=108°.
  3. Hexagon (\(n=6\)): Exterior = 360/6 = 60°, Interior = 120°.

Pitfalls

  • Using the formula for irregular polygons (not valid).
  • Forgetting exterior + interior = 180°.
  • Mixing up with total interior angle sum \((n-2)\times 180\).

Exam strategy

  • If asked for “an angle of a regular polygon”, check if they mean exterior or interior.
  • If given interior, solve for \(n\) using \(180 - \tfrac{360}{n}\).
  • For tessellations, test whether the interior angles fit around 360°.

Summary

Regular polygon angles are easy to find: divide 360° by the number of sides for the exterior, subtract from 180° for the interior. These appear frequently in geometry and tessellation problems.

Worked examples

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  1. Find the interior angle of a regular pentagon
    1. \( n=5 \)
    2. \( Exterior=360/5=72° \)
    3. \( Interior=180-72=108° \)
    Answer: 108°
  2. Find the exterior angle of a regular hexagon
    1. \( n=6 \)
    2. \( Exterior=360/6=60° \)
    Answer: 60°
  3. Find the interior angle of a regular octagon
    1. \( n=8 \)
    2. \( Exterior=360/8=45° \)
    3. \( Interior=180-45=135° \)
    Answer: 135°
  4. How many sides does a regular polygon have if each exterior angle is 30°?
    1. \( Exterior=30° \)
    2. \( n=360/30=12 \)
    Answer: 12
  5. Find the interior angle of a regular 12-sided polygon
    1. \( n=12 \)
    2. \( Exterior=360/12=30° \)
    3. \( Interior=180-30=150° \)
    Answer: 150°
  6. Find the number of sides if each interior angle of a regular polygon is 140°
    1. \( Interior=140 → Exterior=40 \)
    2. \( n=360/40=9 \)
    Answer: 9
  7. Find the interior angle of a regular 20-gon
    1. \( n=20 \)
    2. \( Exterior=360/20=18° \)
    3. \( Interior=180-18=162° \)
    Answer: 162°
  8. How many sides does a regular polygon have if its interior angle is 120°?
    1. \( Interior=120 → Exterior=60 \)
    2. \( n=360/60=6 \)
    Answer: 6
  9. Find the exterior angle of a regular 15-sided polygon
    1. \( n=15 \)
    2. \( Exterior=360/15=24° \)
    Answer: 24°
  10. How many sides does a regular polygon have if each interior angle is 175.5°?
    1. \( Interior=175.5 → Exterior=4.5 \)
    2. \( n=360/4.5=80 \)
    Answer: 80