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CDS Mathematics · Lines, Angles and Polygons

Angles of Polygons

The interior angles of an n-sided polygon add to (n − 2) × 180°, and its exterior angles always add to 360°.

All Lines, Angles and Polygons notesCDS Mathematics strategy

Why this matters

Ten PYQs, eight of them EASY. Work with the exterior angle: in a regular polygon it is 360°/n, and the interior angle is 180° minus it. Most items then take one line.

Concept 1 of 1: Interior and exterior angle sums

Walk round a polygon and you turn through one full circle, so the exterior angles add to 360∘360^\circ whatever the shape. Each interior angle is 180∘180^\circ minus its exterior angle.

Definition

  • Sum of interior angles =(n−2)×180∘=(2n−4)= (n - 2) \times 180^\circ = (2n - 4) right angles.
  • Sum of exterior angles =360∘= 360^\circ for every polygon.
  • Regular: exterior =360∘n= \dfrac{360^\circ}{n}; interior =180∘−= 180^\circ - exterior.
  • Interior : exterior =n−22= \dfrac{n - 2}{2}.
  • An angle ee is a possible exterior angle exactly when 360e\dfrac{360}{e} is a whole number of at least 33.

Interior-angle sum

(n−2)×180∘(n - 2) \times 180^\circ

Worked example

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

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q84Easy

Example 1 · Lines, Angles and Polygons · Interior and Exterior Angles of Polygons

A closed polygon has six sides and one of its angles is 30∘30^\circ greater than each of the other five equal angles. What is the value of one of the equal angles ?

Exterior angles sum to 360° always

For two polygons together, the exterior sums give 720∘720^\circ, not a figure that depends on the sides. The interior sum is the one that grows with nn.
Drill 9 more on interior and exterior angle sums

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 (1)

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Test yourself on Lines, Angles and Polygons

10 past CDS questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Lines, Angles and Polygons chapter testDrill all 10 questions