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SSC CGL Maths · Lines, Angles and Polygons

Regular Polygons

In a regular polygon every side and every angle is equal, so each exterior angle is 360° divided by the number of sides, each interior angle is 180° minus that, and one side subtends the same 360°/n at the centre.

All Lines, Angles and Polygons notes

Why this matters

Ten questions in the bank, nine EASY and one MODERATE, all from the 2025 shifts. Most give one angle and ask for the number of sides, or give the sides and ask for an angle. The fast route for all of them is the exterior angle: it is 360°/n, it is the easy number to divide, and the interior angle is only 180° minus it.

Concept 1 of 3: Each interior and exterior angle of a regular polygon

The exterior angles of any polygon add to 360°. In a regular polygon all n of them are equal, so each one is 360° shared n ways. The interior angle at the same corner makes a straight line with it, so it is what is left of 180°.

Definition

  • Each exterior angle of a regular n-sided polygon =360∘n= \frac{360^\circ}{n}.
  • Each interior angle =180∘−360∘n= 180^\circ - \frac{360^\circ}{n}, which is the same as (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}.
  • Fast route: find the exterior angle first (one division of 360), then subtract from 180. It is quicker than dividing the interior sum by n.
  • Values worth knowing: triangle 60∘60^\circ, square 90∘90^\circ, pentagon 108∘108^\circ, hexagon 120∘120^\circ, octagon 135∘135^\circ, decagon 144∘144^\circ, dodecagon (12 sides) 150∘150^\circ.
  • As n grows, each exterior angle shrinks towards 0∘0^\circ and each interior angle grows towards 180∘180^\circ, but never reaches it.

One angle of a regular polygon

exterior=360∘ninterior=180∘−360∘n=(n−2)×180∘n\text{exterior} = \frac{360^\circ}{n} \qquad \text{interior} = 180^\circ - \frac{360^\circ}{n} = \frac{(n - 2) \times 180^\circ}{n}
  • nthe number of equal sides

Worked example

Find each interior angle of a regular decagon (10 sides).
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 1 · Q57Easy

Example 1 · Lines, Angles and Polygons · Regular Polygons

What is the ratio of the interior angle of a regular hexagon to the interior angle of a regular 12-sided polygon (dodecagon)?

Taking 360°/n as the interior angle

360∘n\frac{360^\circ}{n} is the exterior angle. The interior angle is 180∘180^\circ minus it. For a regular octagon, 45∘45^\circ is the exterior angle and 135∘135^\circ the interior one.

Using these formulas on a polygon that is not regular

The angles of an irregular polygon are not equal, so 360∘n\frac{360^\circ}{n} is only their average. Without the word 'regular', use the angle sums instead.
Drill 1 more on each interior and exterior angle of a regular polygon

Concept 2 of 3: Number of sides of a regular polygon from one angle

Turn the angle formula round. If each exterior angle is known, the number of sides is how many of those angles fit into 360°. Every other clue (an interior angle, a difference, a ratio) is only a route to the exterior angle.

Definition

  • From the exterior angle: n=360∘exteriorn = \frac{360^\circ}{\text{exterior}}.
  • From the interior angle: exterior =180∘−= 180^\circ - interior, then divide 360 by it.
  • Interior minus exterior =d= d: the two add to 180 and differ by d, so exterior =180∘−d2= \frac{180^\circ - d}{2}.
  • Interior =k×= k \times exterior: the two make k+1k + 1 parts of 180∘180^\circ, so exterior =180∘k+1= \frac{180^\circ}{k + 1} and n=2(k+1)n = 2(k + 1).
  • The answer must be a whole number of at least 3. If the exterior angle does not divide 360 exactly, no regular polygon has that angle.

Sides from an angle

n=360∘exteriorexterior=180∘−d2n=2(k+1)n = \frac{360^\circ}{\text{exterior}} \qquad \text{exterior} = \frac{180^\circ - d}{2} \qquad n = 2(k + 1)
  • dinterior angle minus exterior angle
  • khow many times the exterior angle the interior angle is

Worked example

Each interior angle of a regular polygon is 156∘156^\circ. How many sides does it have?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 3 · Q57Moderate

Example 2 · Lines, Angles and Polygons · Regular Polygons

If the interior angle of a regular polygon is kk times its exterior angle, express the number of sides (n)(n) of the polygon in terms of kk.

Dividing 360 by the interior angle

n=360exteriorn = \frac{360}{\text{exterior}}. If the interior angle is 150∘150^\circ, first find the exterior angle, 30∘30^\circ, and only then divide: 12 sides.

Forgetting to halve when the difference is given

If the interior angle exceeds the exterior one by d, the exterior angle is 180−d2\frac{180 - d}{2}, not 180−d180 - d. With d=100d = 100 it is 40∘40^\circ, not 80∘80^\circ.

Accepting a polygon that cannot exist

The number of sides must be a whole number. An exterior angle that does not divide 360 exactly, such as 50∘50^\circ or 70∘70^\circ, belongs to no regular polygon.
Drill 4 more on number of sides of a regular polygon from one angle

Concept 3 of 3: Centre, side and area of a regular polygon

A regular polygon has a centre that is the same distance from every corner. Join the centre to all the corners and you cut the polygon into n equal triangles, which share the full turn of 360° at the centre.

Definition

  • Each side subtends 360∘n\frac{360^\circ}{n} at the centre, the same number as the exterior angle.
  • Each of the n triangles from the centre is isosceles, and its two base angles are each half an interior angle.
  • The centre is the centre of the circle through the corners and of the circle inside touching every side.
  • All sides are equal, so side =perimetern= \frac{\text{perimeter}}{n}.
  • Regular hexagon: the angle at the centre is 60∘60^\circ, so the six triangles are equilateral with side a. Area =6×34a2=332a2= 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2, and the distance from the centre to each corner equals the side.

Centre angle and hexagon area

angle at centre=360∘nhexagon area=6×34a2=332a2\text{angle at centre} = \frac{360^\circ}{n} \qquad \text{hexagon area} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2
  • athe side of the regular hexagon

Worked example

A regular hexagon has side 4 cm. Find its area.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 3 · Q74Easy

Example 3 · Lines, Angles and Polygons · Regular Polygons

A circle is inscribed in a regular pentagon. What is the angle subtended by one side of the pentagon at the center of the circle?

Giving the interior angle for the angle at the centre

A side of a regular pentagon subtends 72∘72^\circ at the centre. Its interior angle, 108∘108^\circ, is a different angle, at a corner, and is printed as a wrong option.

Treating the hexagon's centre distance as different from its side

The six triangles in a regular hexagon are equilateral, so the distance from the centre to any corner equals the side. A circle through the corners of a hexagon of side 5 cm has radius 5 cm.
Drill 2 more on centre, side and area of a regular polygon

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

  • Each interior and exterior angle of a regular polygon

    One angle of a regular polygon

    exterior=360∘ninterior=180∘−360∘n=(n−2)×180∘n\text{exterior} = \frac{360^\circ}{n} \qquad \text{interior} = 180^\circ - \frac{360^\circ}{n} = \frac{(n - 2) \times 180^\circ}{n}
  • Number of sides of a regular polygon from one angle

    Sides from an angle

    n=360∘exteriorexterior=180∘−d2n=2(k+1)n = \frac{360^\circ}{\text{exterior}} \qquad \text{exterior} = \frac{180^\circ - d}{2} \qquad n = 2(k + 1)
  • Centre, side and area of a regular polygon

    Centre angle and hexagon area

    angle at centre=360∘nhexagon area=6×34a2=332a2\text{angle at centre} = \frac{360^\circ}{n} \qquad \text{hexagon area} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

Watch out for (7)

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